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Combine like terms

Learning goals

  • Identifies like terms correctly.
  • Combines coefficients while preserving the variable part.
  • Explains why unlike terms cannot be combined by addition.
  • Checks equivalence of the simplified expression.

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Combine like terms 1
Combine like terms. a) \(8x+5x\) b) \(14y-8y\) c) \(11a-11a\) d) \(3c+7c\)

Hints

- Add or subtract the coefficients of like terms. - Keep the common variable part unchanged. - Equal opposite terms cancel to zero.

Solution

1. \(8x+5x=(8+5)x=13x\). 2. \(14y-8y=(14-8)y=6y\). 3. \(11a-11a=0\). 4. \(3c+7c=(3+7)c=10c\).

Answer

a) \(13x\) b) \(6y\) c) \(0\) d) \(10c\)
Combine like terms 2
Three bags each hold \(m\) marbles. Write the total as a repeated sum and then as a single term.

Hints

- Write one \(m\) for each bag. - How can repeated addition of the same quantity be written more briefly?

Solution

1. One bag contributes \(m\) marbles, so three bags contribute \(m+m+m\). 2. Three copies of the same quantity make \(3m\).

Answer

\(m+m+m=3m\)
Combine like terms 3
Simplify \(7p-7p\). Explain why the answer does not depend on \(p\).

Hints

- Compare the two complete terms. - What is any quantity minus itself?

Solution

1. Both terms contain the same quantity \(7p\). 2. Subtracting a quantity from itself gives \(0\), for every value of \(p\).

Answer

\(0\). The two terms are equal and cancel each other.
Combine like terms 4
Simplify \(5x-2x\).

Hints

- Count the copies of the same variable in each term. - Only the number of copies changes.

Solution

1. Both terms contain the same variable \(x\). 2. Subtract the coefficients: \((5-2)x=3x\).

Answer

\(3x\)
Combine like terms 5
Fill in the missing term so the two expressions always have the same value: \(2x+\square=7x\).

Hints

- What quantity is counted on both sides? - How many more copies must be added to the first expression?

Solution

1. Both sides must contain the same total number of copies of \(x\). 2. The difference between seven copies and two copies is five copies, so the missing term is \(5x\).

Answer

\(5x\)
Combine like terms 6
Simplify \(2p+0p\). Explain whether the zero term changes the value.

Hints

- Find the value of the second term for any \(p\). - Does adding zero change a quantity?

Solution

1. Multiplying any number by \(0\) gives \(0\), so \(0p=0\). 2. The expression remains \(2p\).

Answer

\(2p\). The term \(0p\) adds nothing.
Combine like terms 7
A display border is assembled from four strips that are each \(x\) centimeters long and two strips that are each \(8\) centimeters long. a) Write and simplify an expression for the total length \(L\) of the strips. b) Find the total length when \(x=5\) and when \(x=2.4\).

Hints

- Group the strips that have the same length. - Use multiplication to represent repeated equal lengths. - Substitute each value only after simplifying the expression.

Solution

1. The four variable-length strips have total length \(x+x+x+x=4x\). 2. The two fixed-length strips have total length \(8+8=16\). 3. Therefore, \(L=4x+16\). 4. When \(x=5\), \(L=4\times5+16=36\,\text{cm}\). 5. When \(x=2.4\), \(L=4\times2.4+16=25.6\,\text{cm}\).

Answer

a) \(L=4x+16\) b) \(36\,\text{cm}\) and \(25.6\,\text{cm}\)
Combine like terms 8
Combine like terms. \(12m-5n+3m-8n+13n\)

Hints

- Group terms that contain the same variable. - Keep each coefficient’s sign. - Terms cancel when their coefficients add to zero.

Solution

1. Combine the \(m\)-terms: \(12m+3m=15m\). 2. Combine the \(n\)-terms: \(-5n-8n+13n=(-5-8+13)n=0n\). 3. Therefore, the expression simplifies to \(15m\).

Answer

\(15m\)
Combine like terms 9
Combine like terms. 1) \(a+a+a+a\) 2) \(10x-4x+5y+y\) 3) \(\frac{z}{6}+\frac{z}{6}+\frac{z}{6}+\frac{z}{6}+\frac{z}{6}\)

Hints

- Count how many copies of each variable are present. - Combine only terms with the same variable part. - Fractions with the same denominator are added by adding their numerators.

Solution

1. Four equal \(a\)-terms combine to \(4a\). 2. Combine each variable separately: \(10x-4x=6x\) and \(5y+y=6y\). The result is \(6x+6y\). 3. Add the fractions with the same denominator: \(\frac{z+z+z+z+z}{6}=\frac{5z}{6}\).

Answer

1) \(4a\) 2) \(6x+6y\) 3) \(\frac{5z}{6}\)
Combine like terms 10
Write each sum more concisely by using coefficients and combining like terms. 1) \(uv+uv+uv+uv\) 2) \(xyz+xyz+xyz\) 3) \(a+a-b-b-b\)

Hints

- Count the number of identical terms. - Keep each term’s sign. - Only terms with identical variable parts can be combined.

Solution

1. Four identical \(uv\)-terms combine to \(4uv\). 2. Three identical \(xyz\)-terms combine to \(3xyz\). 3. Two positive \(a\)-terms combine to \(2a\), and three negative \(b\)-terms combine to \(-3b\). The result is \(2a-3b\).

Answer

1) \(4uv\) 2) \(3xyz\) 3) \(2a-3b\)
Combine like terms 11
Combine like terms. 1) \(1.5x+3.2x-0.7x\) 2) \(7y-4y\) 3) \(12a-15a+3a\)

Hints

- Combine the numerical coefficients of like terms. - Keep the common variable part unchanged. - A term with coefficient zero has value zero.

Solution

1. Combine the coefficients: \(1.5+3.2-0.7=4\). The result is \(4x\). 2. Combine the coefficients: \(7-4=3\). The result is \(3y\). 3. Combine the coefficients: \(12-15+3=0\). Therefore, the expression equals \(0\).

Answer

1) \(4x\) 2) \(3y\) 3) \(0\)
Combine like terms 12
Find each sum and simplify. a) \(12a\) and \(5a\) b) \(4x\) and \(-9x\) c) \(-3m\) and \(3m\) d) \(7y\) and \(2z\)

Hints

- Like terms have the same variable part. - Add the coefficients of like terms. - Unlike terms cannot be combined.

Solution

1. \(12a+5a=17a\). 2. \(4x+(-9x)=-5x\). 3. \(-3m+3m=0\). 4. The terms \(7y\) and \(2z\) are not like terms, so their sum remains \(7y+2z\).

Answer

a) \(17a\) b) \(-5x\) c) \(0\) d) \(7y+2z\)
Combine like terms 13
Find the missing expression in each equation. a) \(5b+\square=12b\) b) \(12k-\square=3k\) c) \(\square+8m=10m\)

Hints

- Treat each blank as a missing like term. - Compare the coefficient of the variable on both sides of each equation. - Substitute each result to check the equation.

Solution

1. For part a, \(12b-5b=7b\). 2. For part b, the amount subtracted is \(12k-3k=9k\). 3. For part c, \(10m-8m=2m\).

Answer

a) \(7b\) b) \(9k\) c) \(2m\)
Combine like terms 14
Combine like terms: \(4a+3b+2a-b\).

Hints

- Group terms that have exactly the same variable part. - Combine coefficients within each group and keep unlike groups separate.

Solution

1. Combine the \(a\)-terms: \(4a+2a=6a\). 2. Combine the \(b\)-terms: \(3b-b=2b\). 3. The simplified expression is \(6a+2b\).

Answer

\(6a+2b\)
Combine like terms 15
Which expression is equivalent to \(9x-2y-4x+7y\)? a) \(5x+5y\) b) \(13x+9y\) c) \(5x-9y\) d) \(13x+5y\)

Hints

- Keep each sign attached to its coefficient. - Simplify the two variable groups independently before choosing.

Solution

1. Combine \(x\)-terms: \(9x-4x=5x\). 2. Combine \(y\)-terms: \(-2y+7y=5y\). 3. The equivalent expression is \(5x+5y\), choice a).

Answer

a) \(5x+5y\)
Combine like terms 16
Simplify \(5c-8+3c+11-2c\).

Hints

- Separate variable terms from constant terms. - The constants form their own like-term group.

Solution

1. Combine variable terms: \(5c+3c-2c=6c\). 2. Combine constants: \(-8+11=3\). 3. The simplified expression is \(6c+3\).

Answer

\(6c+3\)
Combine like terms 17
Combine like terms in \(6uv-2u+5uv+9u-uv\).

Hints

- Compare the complete variable parts, not just one letter. - Terms with \(uv\) are not like terms with terms containing only \(u\).

Solution

1. Combine \(uv\)-terms: \(6uv+5uv-uv=10uv\). 2. Combine \(u\)-terms: \(-2u+9u=7u\). 3. The simplified expression is \(10uv+7u\).

Answer

\(10uv+7u\)
Combine like terms 18
Simplify \(12r-12r+5s-3s+4\). What feature of the \(r\)-terms affects the result?

Hints

- Look for a pair of like terms whose coefficients sum to zero. - After cancellation, continue simplifying the remaining groups.

Solution

1. The \(r\)-terms cancel: \(12r-12r=0\). 2. The \(s\)-terms combine: \(5s-3s=2s\). 3. The constant remains \(4\). 4. The simplified expression is \(2s+4\).

Answer

\(2s+4\). The \(r\)-terms have opposite coefficients, so they cancel.
Combine like terms 19
The table lists three expressions. Fill the simplified expression in each row. <table> <thead> <tr> <th>Expression</th> <th>Simplified expression</th> </tr> </thead> <tbody> <tr> <td>\(2a+a+5a\)</td> <td>?</td> </tr> <tr> <td>\(9b-4b-b\)</td> <td>?</td> </tr> <tr> <td>\(3c+2-7c+6\)</td> <td>?</td> </tr> </tbody> </table>

Hints

- Use the same variable part as the signal for a like-term group. - In the last row, remember that constants combine with constants.

Solution

1. \(2a+a+5a=(2+1+5)a=8a\). 2. \(9b-4b-b=(9-4-1)b=4b\). 3. \(3c+2-7c+6=(3-7)c+(2+6)=-4c+8\).

Answer

First row: \(8a\) Second row: \(4b\) Third row: \(-4c+8\)
Combine like terms 20
Emre combines \(3x+4+2x-9\) as \(5x+13\). Find the correct simplified expression and identify the sign error.

Hints

- Track each term together with its sign before combining like terms. - Check the variable terms and constant terms independently.

Solution

1. The variable terms combine to \(3x+2x=5x\). 2. The constants combine to \(4-9=-5\). 3. The correct result is \(5x-5\). 4. Emre treated \(-9\) as \(+9\).

Answer

\(5x-5\). Emre changed the sign of the \(-9\) term.
Combine like terms 21
Simplify \(a+2b+3a-5b+6a+b\), then state the coefficient of each variable in the result.

Hints

- Remember the implied coefficient on a variable written alone. - After simplifying, read the signed numerical factors from the result.

Solution

1. Combine \(a\)-terms: \(a+3a+6a=10a\). 2. Combine \(b\)-terms: \(2b-5b+b=-2b\). 3. The simplified expression is \(10a-2b\). 4. The coefficients are \(10\) for \(a\) and \(-2\) for \(b\).

Answer

Simplified expression: \(10a-2b\) Coefficients: \(10\) and \(-2\)
Combine like terms 22
Three expressions are listed: A: \(7k-2k+k\) B: \(4k+3k-k\) C: \(9k-4k+2k\) Which two simplify to the same expression?

Hints

- Reduce each expression to one coefficient times the variable. - Compare the resulting coefficients rather than the original term counts.

Solution

1. A simplifies to \((7-2+1)k=6k\). 2. B simplifies to \((4+3-1)k=6k\). 3. C simplifies to \((9-4+2)k=7k\). 4. A and B simplify to the same expression.

Answer

A and B; both simplify to \(6k\).
Combine like terms 23
Simplify \(0.25x+\frac{1}{2}x+1.25x\). Give the coefficient in decimal form.

Hints

- Express all coefficients in a common number form. - Only the numerical factors need to be combined because the variable parts match.

Solution

1. Convert \(\frac{1}{2}\) to \(0.5\). 2. Add coefficients: \(0.25+0.5+1.25=2\). 3. The simplified expression is \(2x\).

Answer

\(2x\)
Combine like terms 24
The expression \(8p-3q-5p+q+2p\) is simplified in two stages. After combining only the \(p\)-terms, what expression remains? Then finish the simplification.

Hints

- Work with one variable group while leaving the other terms unchanged. - Then repeat the same process for the remaining like terms.

Solution

1. The \(p\)-terms combine as \(8p-5p+2p=5p\). 2. After that stage, the expression is \(5p-3q+q\). 3. The \(q\)-terms combine as \(-3q+q=-2q\). 4. The final simplified expression is \(5p-2q\).

Answer

After the first stage: \(5p-3q+q\) Final result: \(5p-2q\)
Combine like terms 25
A student combines \(3x+2y\) into \(5xy\). Explain why these are not like terms and cannot be combined this way. Give values of \(x\) and \(y\) that show the expressions are different.

Hints

- Which variable belongs to each term? - Try choosing unequal values for the two variables to check the claimed equality.

Solution

1. The terms \(3x\) and \(2y\) count different quantities, so they are not like terms. 2. Their sum cannot generally be replaced by one term \(5xy\). For example, when \(x=1\) and \(y=2\), the sum is \(7\) but \(5xy=10\).

Answer

No. The terms have different variable parts, so they cannot be combined. For \(x=1\) and \(y=2\), the values are \(7\) and \(10\).
Combine like terms 26
Simplify \(2n+3n+4+1\). Explain which numbers can be combined with the variable terms.

Hints

- Group terms that contain exactly the same variable part. - Find the terms without any variable.

Solution

1. The variable terms combine to \((2+3)n=5n\). 2. The constants combine to \(4+1=5\). 3. Constants cannot be added to the coefficient of \(n\), so the result is \(5n+5\).

Answer

\(5n+5\). Combine the coefficients of \(n\) with each other, and combine the constants with each other.
Combine like terms 27
Find the combined area of the two rectangles as one term, in square centimeters. Use the marked side lengths.
Figure for problem 559230

Hints

- Multiply length by width for each rectangle. - Do both resulting terms have the same variable part?

Solution

1. The areas are \(2x\) and \(3x\) square centimeters. 2. Because both areas contain the same \(x\)-wide strip, the total is \(2x+3x=5x\) square centimeters.

Answer

\(5x\) square centimeters
Combine like terms 28
The drawings are not to scale. A student writes the combined area of the two rectangles as \(5xy\) square centimeters. Explain why that is not generally correct. Give the correct area expression using the marked side lengths.
Figure for problem 559231

Hints

- Find the area of each rectangle separately. - Can you add the coefficients as if the two widths represented the same unknown length?

Solution

1. The first area is \(2x\) and the second is \(3y\) square centimeters. 2. The widths use different variables, so their areas are unlike terms and cannot generally combine as \(5xy\). 3. The total is \(2x+3y\) square centimeters.

Answer

\(2x+3y\) square centimeters. The two terms have different variable parts and cannot be combined as \(5xy\).
Combine like terms 29
First combine like terms. Then evaluate the expression for \(x=1.2\). \(15x-7+5x+17\)

Hints

- Group the variable terms and constant terms separately. - Simplify the expression before substituting. - Multiply before adding.

Solution

1. Combine the variable terms: \(15x+5x=20x\). 2. Combine the constant terms: \(-7+17=10\). 3. The simplified expression is \(20x+10\). 4. Substitute \(x=1.2\): \(20\times1.2+10=34\).

Answer

Simplified: \(20x+10\) Value: \(34\)
Combine like terms 30
Let \(k\) be a whole number. Two expressions are A: \(3k+1\) B: \(2k+2\) a) Write and simplify an expression for A plus B. b) Write and simplify an expression for A minus B. c) Explain why \(k+k+k\) and \(3k\) have the same value for any whole number \(k\).

Hints

- Treat each given expression as a complete amount before adding or subtracting it. - When subtracting expression B, which terms change sign? - What does multiplying \(k\) by \(3\) mean?

Solution

1. A plus B is \((3k+1)+(2k+2)=5k+3\), after combining the \(k\)-terms and constants. 2. A minus B is \((3k+1)-(2k+2)=3k+1-2k-2=k-1\). 3. \(k+k+k\) contains three copies of the same amount; that is exactly what \(3k\) means.

Answer

a) \(5k+3\) b) \(k-1\) c) Both represent adding \(k\) three times.
Combine like terms 31
Combine like terms. a) \(1.5a+2.7a-0.8a\) b) \(-3k-4k+12k-k\) c) \(\frac{1}{2}x+\frac{3}{4}x-x\) d) \(12y-15y+3y\)

Hints

- Combine only terms with the same variable part. - A variable with no visible coefficient has coefficient \(1\). - Work with the numerical coefficients while keeping each sign.

Solution

1. For part a, combine the coefficients: \(1.5+2.7-0.8=3.4\), so the result is \(3.4a\). 2. For part b, \(-3-4+12-1=4\), so the result is \(4k\). 3. For part c, \(\frac{1}{2}+\frac{3}{4}-1=\frac{2}{4}+\frac{3}{4}-\frac{4}{4}=\frac{1}{4}\), so the result is \(\frac{1}{4}x\). 4. For part d, \(12-15+3=0\), so all variable terms cancel and the result is \(0\).

Answer

a) \(3.4a\) b) \(4k\) c) \(\frac{1}{4}x\) d) \(0\)
Combine like terms 32
Anna and Ben combine like terms in \(7x-4x+x\). Anna says, “The result is \(3x\).” Ben says, “The result is \(4x\).” Who is correct? Show the calculation and explain the likely error in the other student’s reasoning.

Hints

- Write the implied coefficient of the final \(x\). - Combine all three coefficients, not only the first two. - Check how adding one more \(x\) changes \(3x\).

Solution

1. The coefficients are \(7\), \(-4\), and \(1\) because \(x=1x\). 2. Combine them: \(7-4+1=4\). 3. Therefore, the expression simplifies to \(4x\), so Ben is correct. 4. Anna likely ignored the implied coefficient \(1\) on the final \(x\) and stopped after calculating \(7-4=3\).

Answer

Ben is correct: \(7x-4x+x=(7-4+1)x=4x\). Anna likely overlooked the coefficient \(1\) on the final \(x\).
Combine like terms 33
Find the expression that belongs in each blank. a) \(5m+\square=12m\) b) \(8.4t-\square=2.1t\) c) \(-3b+\square=0\) d) \(\square-6p=-10p\)

Hints

- Treat each blank as a missing term in an addition or subtraction statement. - Use an inverse operation to find the missing term. - Keep careful track of negative signs.

Solution

1. For part a, subtract the known term from the result: \(12m-5m=7m\). 2. For part b, the missing term is \(8.4t-2.1t=6.3t\). 3. For part c, add the opposite of \(-3b\), which is \(3b\). 4. For part d, add \(6p\) to the result: \(-10p+6p=-4p\).

Answer

a) \(7m\) b) \(6.3t\) c) \(3b\) d) \(-4p\)
Combine like terms 34
Simplify each expression, then evaluate it for the given variable value. a) \(6x\times2-4x+9\) for \(x=4\) b) \(3.5-8y+1.2-y\) for \(y=0.5\)

Hints

- Perform multiplication before addition or subtraction. - Combine variable terms and constants separately. - Substitute only after simplifying.

Solution

1. For part a, multiply first and combine like terms: \(6x\times2-4x+9=12x-4x+9=8x+9\). 2. Substitute \(x=4\): \(8\times4+9=32+9=41\). 3. For part b, combine the constants and variable terms: \(3.5-8y+1.2-y=4.7-9y\). 4. Substitute \(y=0.5\): \(4.7-9\times0.5=4.7-4.5=0.2\).

Answer

a) Simplified expression: \(8x+9\); value: \(41\) b) Simplified expression: \(4.7-9y\); value: \(0.2\)
Combine like terms 35
Combine like terms. a) \(A=15x-4\times2x-7x\) b) \(B=1.2z+0.8z-3\times0.5z\) Then evaluate \(B\) for \(z=4\).

Hints

- Perform each multiplication before combining like terms. - Variable terms can cancel completely. - Substitute the given value only after simplifying \(B\).

Solution

1. Simplify \(A\): \(15x-4\times2x-7x=15x-8x-7x=0\). 2. Simplify \(B\): \(3\times0.5z=1.5z\), so \(B=1.2z+0.8z-1.5z=0.5z\). 3. Evaluate \(B\) at \(z=4\): \(0.5\times4=2\).

Answer

a) \(A=0\) b) \(B=0.5z\) For \(z=4\), \(B=2\).
Combine like terms 36
A quadrilateral has side lengths \(x\), \(x+2\), \(2x\), and \(5\) centimeters. a) Write and simplify an expression for its perimeter \(P\). b) Find the perimeter when \(x=3.5\).

Hints

- Perimeter is the sum of all side lengths. - Write the sum before combining any terms. - Combine like terms before substituting the value of \(x\).

Solution

1. Add the four side lengths: \(P=x+(x+2)+2x+5\). 2. Combine like terms: \(P=4x+7\). 3. Substitute \(x=3.5\): \(P=4\times3.5+7=21\,\text{cm}\).

Answer

a) \(P=4x+7\) b) \(21\,\text{cm}\)
Combine like terms 37
A rectangular lot has length \(L\) meters. Its width is \(10\,\text{m}\) less than its length. a) Write and simplify an expression for the perimeter \(P\) using only \(L\). b) Write an expression for the area \(A\). c) Find the perimeter and area when \(L=25\,\text{m}\).

Hints

- Express the width in terms of \(L\). - Use the formulas for rectangle perimeter and area. - Keep the expression \(L-10\) grouped when multiplying.

Solution

1. The width is \(L-10\). 2. The perimeter is \(P=2L+2(L-10)=4L-20\). 3. The area is \(A=L(L-10)\). 4. When \(L=25\), \(P=4\times25-20=80\,\text{m}\), and \(A=25\times(25-10)=375\,\text{m}^2\).

Answer

a) \(P=4L-20\) b) \(A=L(L-10)\) c) \(P=80\,\text{m}\); \(A=375\,\text{m}^2\)
Combine like terms 38
Combine like terms. 1) \(6.4m-2.1m+3.5n-n\) 2) \(\frac{c}{3}+\frac{c}{3}+\frac{c}{3}-\frac{d}{2}-\frac{d}{2}\) 3) \(\frac{2}{9}k+\frac{7}{9}k-k\)

Hints

- Combine only terms with the same variable part. - Work with the numerical coefficients. - A variable with no visible coefficient has coefficient \(1\).

Solution

1. Combine each variable separately: \(6.4m-2.1m=4.3m\) and \(3.5n-n=2.5n\). The result is \(4.3m+2.5n\). 2. Three copies of \(\frac{c}{3}\) equal \(c\), and two copies of \(\frac{d}{2}\) equal \(d\). The result is \(c-d\). 3. \(\frac{2}{9}k+\frac{7}{9}k=k\), so \(k-k=0\).

Answer

1) \(4.3m+2.5n\) 2) \(c-d\) 3) \(0\)
Combine like terms 39
Simplify each expression. 1) \((c-d)+(c-d)+(c-d)\) 2) \(\frac{pq+pq+pq+pq+pq+pq}{9}\) 3) Subtract \(2ab\) from the sum \(ab+ab+ab+ab+ab\).

Hints

- Count how many times the entire grouped expression appears. - Combine the terms in the numerator before simplifying the fraction. - Rewrite repeated equal terms using a coefficient.

Solution

1. Three identical grouped expressions combine to \(3(c-d)\), which can also be written as \(3c-3d\). 2. The numerator contains six \(pq\)-terms, so \(\frac{pq+pq+pq+pq+pq+pq}{9}=\frac{6pq}{9}=\frac{2}{3}pq\). 3. The sum of five \(ab\)-terms is \(5ab\). Then \(5ab-2ab=3ab\).

Answer

1) \(3(c-d)\), or \(3c-3d\) 2) \(\frac{2}{3}pq\) 3) \(3ab\)
Combine like terms 40
In algebra, a number can serve as a coefficient or an exponent. a) Explain the difference between \(4x\) and \(x^4\) by writing \(4x\) as repeated addition and \(x^4\) as repeated multiplication. b) Use repeated addition, without coefficients, to show that \(2a+3a=5a\). c) Use repeated addition to explain why \(3x+2y\) cannot be combined into one term such as \(5xy\).

Hints

- A coefficient tells how many copies are added; an exponent tells how many equal factors are multiplied. - Expand each coefficient into repeated addition. - Can terms with different variable parts be counted as copies of the same quantity? - Count the total number of \(a\)-terms in part b).

Solution

1. The coefficient \(4\) in \(4x\) counts four addends: \(x+x+x+x\). The exponent \(4\) in \(x^4\) counts four factors: \(x\times x\times x\times x\). 2. Write \(2a+3a\) as \((a+a)+(a+a+a)\). This is \(a+a+a+a+a=5a\). 3. Write \(3x+2y\) as \(x+x+x+y+y\). The \(x\)-terms and \(y\)-terms are unlike terms, so they cannot be combined into a single term.

Answer

a) \(4x=x+x+x+x\), while \(x^4=x\times x\times x\times x\). b) \(2a+3a=(a+a)+(a+a+a)=a+a+a+a+a=5a\). c) \(3x+2y=x+x+x+y+y\). Since \(x\) and \(y\) are different variables, the terms are not like terms and cannot be combined.
Combine like terms 41
Find the expression that belongs in each box. 1) \(7b-\square=-3b\) 2) \(\square+(-5k)=2k\) 3) \(4m-9m+\square=0\)

Hints

- Treat each box as an unknown term. - Use inverse operations to isolate the missing term. - To make a sum equal zero, add the opposite term.

Solution

1. The missing subtracted term is \(7b-(-3b)=10b\). Check: \(7b-10b=-3b\). 2. Subtract \(-5k\) from \(2k\): \(2k-(-5k)=7k\). Check: \(7k+(-5k)=2k\). 3. First combine \(4m-9m=-5m\). Its opposite, \(5m\), must be added to obtain \(0\).

Answer

1) \(10b\) 2) \(7k\) 3) \(5m\)
Combine like terms 42
Combine like terms. \(4.7m-2.15n+1.3m-3.85n-5.2m\)

Hints

- Group terms containing the same variable. - Align decimal points when combining coefficients. - Keep the negative signs on the \(n\)-terms.

Solution

1. Combine the \(m\)-terms: \(4.7+1.3-5.2=0.8\). 2. Combine the \(n\)-terms: \(-2.15-3.85=-6\). 3. The simplified expression is \(0.8m-6n\).

Answer

\(0.8m-6n\)
Combine like terms 43
Find the missing expression in each equation. a) \(8x+\square=15x\) b) \(5a+\square=-2a\) c) \(\square+(-4m)=6m\) d) \(3b+\square=3b-2c\)

Hints

- Treat each blank as the term needed to make both sides equivalent. - Keep careful track of signed variable terms. - Compare matching terms on both sides before deciding what is missing.

Solution

1. For part a, \(15x-8x=7x\). 2. For part b, \(-2a-5a=-7a\). 3. For part c, \(6m-(-4m)=10m\). 4. For part d, \(3b\) already appears on both sides, so the missing expression is \(-2c\).

Answer

a) \(7x\) b) \(-7a\) c) \(10m\) d) \(-2c\)
Combine like terms 44
Combine like terms. \(14x-8y+3-5x+2y-11\)

Hints

- Group terms with the same variable. - Combine constant terms separately. - Keep each term’s sign.

Solution

1. Combine the \(x\)-terms: \(14x-5x=9x\). 2. Combine the \(y\)-terms: \(-8y+2y=-6y\). 3. Combine the constants: \(3-11=-8\). 4. The simplified expression is \(9x-6y-8\).

Answer

\(9x-6y-8\)
Combine like terms 45
Combine like terms. \(\frac{3}{8}x-\frac{1}{3}y+\frac{1}{4}x-\frac{5}{6}y+x\)

Hints

- Group terms containing the same variable. - An \(x\) with no visible coefficient has coefficient \(1\). - Use common denominators when combining fractional coefficients.

Solution

1. Combine the \(x\)-coefficients: \(\frac{3}{8}+\frac{1}{4}+1=\frac{3}{8}+\frac{2}{8}+\frac{8}{8}=\frac{13}{8}\). 2. Combine the \(y\)-coefficients: \(-\frac{1}{3}-\frac{5}{6}=-\frac{2}{6}-\frac{5}{6}=-\frac{7}{6}\). 3. The simplified expression is \(\frac{13}{8}x-\frac{7}{6}y\).

Answer

\(\frac{13}{8}x-\frac{7}{6}y\)
Combine like terms 46
Wanjiku claims, “When I add \(3a-5\), \(2-4a\), and \(a+3\), the variable \(a\) disappears completely.” Check the claim by adding and simplifying the three expressions. What is the result?

Hints

- Combine all \(a\)-terms separately from the constants. - A variable term disappears when its coefficient is zero. - Check whether the constants also cancel.

Solution

1. Write the sum: \((3a-5)+(2-4a)+(a+3)\). 2. Combine the variable terms: \(3a-4a+a=0\). 3. Combine the constants: \(-5+2+3=0\). 4. The total is \(0\), so Wanjiku's claim is correct.

Answer

Wanjiku's claim is correct. The sum is \(0\).
Combine like terms 47
Find the expression that belongs in each box. 1) \(7x+\square=15x\) 2) \(4.5y+\square=10y\) 3) \(\square+\frac{1}{3}z=z\) 4) \(\square-8w=2w\)

Hints

- Treat each blank as a missing like term. - Compare the known coefficient with the coefficient on the other side of the equation. - Rewrite an implied coefficient of \(1\) in a convenient form when needed.

Solution

1. The missing term is \(15x-7x=8x\). 2. The missing term is \(10y-4.5y=5.5y\). 3. Rewrite \(z\) as \(\frac{3}{3}z\). The missing term is \(\frac{3}{3}z-\frac{1}{3}z=\frac{2}{3}z\). 4. Add \(8w\) to the result: \(2w+8w=10w\).

Answer

1) \(8x\) 2) \(5.5y\) 3) \(\frac{2}{3}z\) 4) \(10w\)
Combine like terms 48
Combine like terms. 1) \(\frac{3}{5}a+a\) 2) \(\frac{3}{4}b-\frac{1}{2}b\) 3) \(1\frac{2}{3}c+\frac{5}{6}c\)

Hints

- Write every coefficient as a fraction when helpful. - Use common denominators before adding or subtracting coefficients. - Convert a mixed-number coefficient to an improper fraction.

Solution

1. Rewrite \(a\) as \(\frac{5}{5}a\): \(\frac{3}{5}a+\frac{5}{5}a=\frac{8}{5}a\). 2. Rewrite \(\frac{1}{2}\) as \(\frac{2}{4}\): \(\frac{3}{4}b-\frac{2}{4}b=\frac{1}{4}b\). 3. Convert \(1\frac{2}{3}\) to \(\frac{5}{3}\): \(\frac{5}{3}c+\frac{5}{6}c=\frac{10}{6}c+\frac{5}{6}c=\frac{5}{2}c\).

Answer

1) \(\frac{8}{5}a\) 2) \(\frac{1}{4}b\) 3) \(\frac{5}{2}c\)
Combine like terms 49
Combine like terms and give each result as a fraction in simplest form. 1) \(\frac{4}{9}x+\frac{1}{3}x-\frac{5}{6}x\) 2) Find the expression that belongs in the blank: \(\frac{7}{10}y-\square=\frac{1}{5}y\)

Hints

- Use a common denominator for all fractional coefficients. - Treat the blank as a missing like term. - Check your result by substituting it into the original equation in part 2.

Solution

1. Use denominator \(18\): \(\frac{8}{18}x+\frac{6}{18}x-\frac{15}{18}x=-\frac{1}{18}x\). 2. The missing expression is the difference between \(\frac{7}{10}y\) and \(\frac{1}{5}y\): \(\frac{7}{10}y-\frac{2}{10}y=\frac{1}{2}y\).

Answer

1) \(-\frac{1}{18}x\) 2) \(\frac{1}{2}y\)
Combine like terms 50
Luis simplifies \(8m-3m+2m\) as \(8m-5m=3m\). Explain his error and give the correct result.

Hints

- Rewrite the coefficients in their original order with signs attached. - Check whether every operation sign was preserved in the student’s rewrite.

Solution

1. The final term is \(+2m\), so it should be added, not combined with \(-3m\) as another negative term. 2. Add the signed coefficients: \(8-3+2=7\). 3. The correct result is \(7m\).

Answer

Luis changed the sign of \(+2m\). The correct result is \(7m\).
Combine like terms 51
Find the term that makes the statement true: \(7p+\square-2p=12p\).

Hints

- First combine only the terms that are already known. - Compare the resulting coefficient with the target coefficient.

Solution

1. The known terms combine to \(7p-2p=5p\). 2. The missing term must raise \(5p\) to \(12p\). 3. The needed coefficient is \(12-5=7\), so the missing term is \(7p\).

Answer

\(7p\)
Combine like terms 52
Kai says \(2.4t+1.7-0.9t+3.3=6.5t\) because all four numbers can be combined. Explain Kai’s error and simplify the expression correctly.

Hints

- Sort the four terms by whether they contain the variable. - Check whether a numerical total may be attached to a variable after the groups are combined.

Solution

1. The terms \(2.4t\) and \(-0.9t\) are like terms, so their combined coefficient is \(2.4-0.9=1.5\). 2. The constants \(1.7\) and \(3.3\) combine to \(5\). 3. Variable terms and constants are not like terms, so the simplified expression is \(1.5t+5\).

Answer

Kai combined unlike terms. The correct simplified expression is \(1.5t+5\).
Combine like terms 53
Mina claims that \(\frac{3}{4}q+\frac{1}{6}q-\frac{5}{12}q\) simplifies to more than \(\frac{1}{2}q\). Is she correct? Simplify the expression and compare the coefficients.

Hints

- Put the fractional coefficients into a form that makes their sizes directly comparable. - Decide whether the final coefficient is below, equal to, or above the stated benchmark.

Solution

1. Rewrite the coefficients with denominator \(12\): \(\frac{9}{12}+\frac{2}{12}-\frac{5}{12}\). 2. Combine them: \(\frac{9+2-5}{12}=\frac{6}{12}=\frac{1}{2}\). 3. The expression equals \(\frac{1}{2}q\), so it is not greater than \(\frac{1}{2}q\).

Answer

Mina is not correct. The expression simplifies to \(\frac{1}{2}q\), exactly equal to \(\frac{1}{2}q\).
Combine like terms 54
Without evaluating any variable, decide whether \(4x+7x-3\) and \(12x-x-3\) are equivalent. Justify by combining like terms.

Hints

- Simplify each expression independently. - Equivalent expressions must give the same simplified form for every value of the variable.

Solution

1. The first expression simplifies to \(11x-3\). 2. The second expression simplifies to \(11x-3\). 3. Since both simplify to the same expression, they are equivalent.

Answer

Yes. Both expressions simplify to \(11x-3\).
Combine like terms 55
A rectangle has side lengths \(3x+2\), \(x+5\), \(3x+2\), and \(x+5\) centimeters. Combine like terms to simplify an expression for its perimeter.

Hints

- Write the sum of all four side expressions before simplifying. - Combine variable terms and constants in separate groups.

Solution

1. Add the four sides: \((3x+2)+(x+5)+(3x+2)+(x+5)\). 2. Combine variable terms: \(3x+x+3x+x=8x\). 3. Combine constants: \(2+5+2+5=14\). 4. The perimeter is \(8x+14\) centimeters.

Answer

\(8x+14\,\text{cm}\)
Combine like terms 56
Which term must be removed from \(5x+2y-3x+4y\) so that the remaining expression simplifies to \(2x+2y\)?

Hints

- Simplify the complete expression first to see each combined coefficient. - Compare the current simplified form with the target one variable group at a time.

Solution

1. The full expression simplifies to \(2x+6y\). 2. To leave \(2x+2y\), the removed term must account for \(4y\). 3. Removing the term \(4y\) leaves \(5x+2y-3x=2x+2y\).

Answer

Remove \(4y\).
Combine like terms 57
Decide whether the statement is always true: \(2m+5n+3m=5(m+n)\). Show the like-term combination on the left.

Hints

- Begin with the terms that share the same variable part. - Compare the simplified sum with what the grouped product represents.

Solution

1. Combine the \(m\)-terms on the left: \(2m+3m=5m\). 2. The left side becomes \(5m+5n\). 3. The right side \(5(m+n)\) represents \(5m+5n\). 4. The statement is always true.

Answer

Yes. The left side simplifies to \(5m+5n\), which is equivalent to \(5(m+n)\).
Combine like terms 58
Which expression does not simplify to \(4x+3\)? a) \(x+3x+3\) b) \(6x-2x+8-5\) c) \(9x-5x+3\) d) \(2x+2x+6-2\)

Hints

- Simplify both the variable terms and constants in every choice. - One choice differs only in its constant group, so check both parts carefully.

Solution

1. a) simplifies to \(4x+3\). 2. b) simplifies to \(4x+3\). 3. c) simplifies to \(4x+3\). 4. d) simplifies to \(4x+4\). 5. Choice d) does not match.

Answer

d) \(2x+2x+6-2\)
Combine like terms 59
Caio claims that \(3(a+5)+2a\) is equivalent to \(5a+15\). Decide whether the claim is correct and justify your decision using properties of operations.

Hints

- First remove the parentheses using the property that applies to multiplication over addition. - After distributing, identify which terms are like terms. - Compare the simplified result with Caio's expression.

Solution

1. Distribute the \(3\): \(3(a+5)=3a+15\). 2. The full expression becomes \(3a+15+2a\). 3. Combine the like terms \(3a\) and \(2a\) to get \(5a+15\). 4. Caio's claim is correct.

Answer

Caio's claim is correct: \(3(a+5)+2a=3a+15+2a=5a+15\).
Combine like terms 60
Simplify \(4(x+3)+2x\) to an equivalent expression with no parentheses and no like terms left to combine.

Hints

- Remove the parentheses before trying to combine terms. - After distributing, identify the terms that have the same variable part. - Leave the constant term separate from the variable terms.

Solution

1. Distribute \(4\) across \(x+3\): \(4(x+3)=4x+12\). 2. The expression becomes \(4x+12+2x\). 3. Combine the like terms \(4x\) and \(2x\) to get \(6x+12\).

Answer

\(6x+12\)

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