Expand and factor linear expressions

Learning goals

Expands linear expressions using the distributive property.
Factors common numerical or variable factors from linear expressions.
Recognizes expansion and factoring as inverse processes.
Checks equivalence of the resulting forms.

Difficulty: 1 – 5

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Expand and factor linear expressions 1

Expand $3(x+4)$. Show what the $3$ multiplies.

Hints

- Could an area model help you check the expression you write? - Test a value of $x$ in both forms. Do they agree?

Solution

1. Multiply $3$ by $x$ and by $4$. 2. $3(x+4)=3x+3\cdot4=3x+12$.

Answer

$3$ multiplies both $x$ and $4$: $3x+3\cdot4=3x+12$.

Expand and factor linear expressions 2

Complete the factored form: $8x+12=4(\square x+\square)$.

Hints

- Work backward from multiplication by $4$. - Which term inside parentheses would produce each term on the left?

Solution

1. Divide each term by the outside factor $4$. 2. $8x\div4=2x$ and $12\div4=3$. 3. Therefore $8x+12=4(2x+3)$.

Answer

$2$ and $3$, giving $4(2x+3)$.

Expand and factor linear expressions 3

The drawings are not to scale. Write the sum of their areas. Which factor do the two terms share? Factor the sum.

Hints

- Write a product for each rectangle's area. - Look for a factor that appears in both products.

Solution

1. The areas are $3x$ and $5x$, so their sum is $3x+5x$. 2. Each term contains $x$, giving $x(3+5)$, or $8x$.

Answer

Sum: $3x+5x$. Shared factor: $x$. Factored form: $x(3+5)$.

Expand and factor linear expressions 4

Expand $-2(y-5)$.

Hints

- Try a small value of $y$ to check your expanded expression. - Think about which signs appear in the separate products.

Solution

1. Multiply $-2$ by $y$ to get $-2y$. 2. Multiply $-2$ by $-5$ to get $+10$. 3. The expanded form is $-2y+10$.

Answer

$-2y+10$

Expand and factor linear expressions 5

Nina writes $5(y+1)=5y+1$. What did she forget? Write the correct expansion.

Hints

- Try $y=0$ in Nina's two expressions. Are the values equal? - What could you change on the right to make them agree for every $y$?

Solution

1. Nina multiplied $y$ by $5$ but did not multiply the $1$ by $5$. 2. Both terms must be multiplied, giving $5y+5$.

Answer

Nina forgot to multiply $1$ by $5$. The correct expression is $5y+5$.

Expand and factor linear expressions 6

Expand and simplify $7(3x - 4) - 5(2x + 8)$.

Hints

- Pay close attention to the negative factor before the second set of parentheses. - Group the variable terms and the constant terms. - Remember that a negative number times a positive number is negative.

Solution

1. Distribute through the first parentheses: $7(3x - 4) = 21x - 28$. 2. Distribute the negative factor through the second parentheses: $-5(2x + 8) = -10x - 40$. 3. Combine like terms: $21x - 28 - 10x - 40 = 11x - 68$.

Answer

$11x - 68$

Expand and factor linear expressions 7

A square mosaic is made from an $n \times n$ array of small tiles, where $n\ge2$ is a whole number. The four corner tiles are gold. All other tiles along the outside edge are silver. The remaining interior tiles are white. Write and simplify an expression for the number of silver tiles in terms of $n$.

Hints

- How many tiles remain on one side after the two corner tiles are removed? - The same count appears on each of the four sides. - Write a product before you simplify. - Check your expression with a small value such as $n=4$.

Solution

1. Each side has $n-2$ silver tiles after the two gold corner tiles are excluded. 2. There are four sides, so the number of silver tiles is $4(n-2)$. 3. Distribute: $4(n-2)=4n-8$.

Answer

$4n-8$ silver tiles

Expand and factor linear expressions 8

At a school store, a granola bar costs $\$p$, and a bottle of juice costs $\$s$. A recycling deposit of $\$x$ is added to each bottle. a) Write an expression for the total cost of $12$ granola bars and $10$ bottles of juice. b) Simplify the expression. c) Find the total cost when $p=1.30$, $s=1.15$, and $x=0.25$.

Hints

- Add the bottle deposit to the juice price first. - Multiply each item price by the number purchased. - Use the distributive property to remove parentheses. - Line up decimal points when calculating.

Solution

1. Each bottle costs $s+x$, so the total cost is $12p+10(s+x)$. 2. Distribute: $12p+10(s+x)=12p+10s+10x$. 3. Substitute the values: $12\cdot1.30+10\cdot(1.15+0.25)=15.60+14.00=29.60$.

Answer

a) $12p+10(s+x)$ dollars b) $12p+10s+10x$ dollars c) $\$29.60$

Expand and factor linear expressions 9

Use the distributive property to expand each expression. 1) $7(a + 9)$ 2) $3(5x - 4)$ 3) $2(6m + 11n)$ 4) $5(8 - 3y)$

Hints

- Recall the distributive property. - A factor beside parentheses multiplies every term inside. - Break each expression into smaller products if helpful.

Solution

1. $7(a + 9) = 7a + 63$. 2. $3(5x - 4) = 15x - 12$. 3. $2(6m + 11n) = 12m + 22n$. 4. $5(8 - 3y) = 40 - 15y$.

Answer

1) $7a + 63$ 2) $15x - 12$ 3) $12m + 22n$ 4) $40 - 15y$

Expand and factor linear expressions 10

Expand and simplify. 1) $5(2x - 4) + 3x$ 2) $5(2x) - (4 + 3x)$ 3) $(5(2x) - 4) + 3x$

Hints

- Pay close attention to which terms are inside each set of parentheses. - Follow the order of operations. - A minus sign before parentheses changes every sign inside. - An outside factor applies only to the terms in the parentheses immediately beside it.

Solution

1. $5(2x - 4) + 3x = 10x - 20 + 3x = 13x - 20$. 2. $5(2x) - (4 + 3x) = 10x - 4 - 3x = 7x - 4$. 3. $(5(2x) - 4) + 3x = 10x - 4 + 3x = 13x - 4$.

Answer

1) $13x - 20$ 2) $7x - 4$ 3) $13x - 4$

Expand and factor linear expressions 11

Factor out the greatest common factor. 1) $12x - 30$ 2) $8a + 20$ 3) $15 - 10b$ 4) $7y + 7$ 5) $18m - 12n$

Hints

- Find the greatest number that divides both coefficients. - Apply the distributive property in reverse. - Distribute mentally to check each result. - Preserve the operation sign between the terms. - A number divided by itself equals $1$.

Solution

1. The greatest common factor of $12$ and $30$ is $6$: $12x - 30 = 6(2x - 5)$. 2. The greatest common factor is $4$: $8a + 20 = 4(2a + 5)$. 3. The greatest common factor is $5$: $15 - 10b = 5(3 - 2b)$. 4. The greatest common factor is $7$: $7y + 7 = 7(y + 1)$. 5. The greatest common factor is $6$: $18m - 12n = 6(3m - 2n)$.

Answer

1) $6(2x - 5)$ 2) $4(2a + 5)$ 3) $5(3 - 2b)$ 4) $7(y + 1)$ 5) $6(3m - 2n)$

Expand and factor linear expressions 12

Factor $9x-6$ by taking out the greatest whole-number factor shared by both terms. Expand your answer to check it.

Hints

- Look for a number that divides both coefficients. - After taking that number outside, what remains inside parentheses? - Check by multiplying back.

Solution

1. The greatest whole-number factor of $9$ and $6$ is $3$. 2. Dividing each term by $3$ gives $3(3x-2)$. 3. Expanding $3(3x-2)=9x-6$ verifies the result.

Answer

$3(3x-2)$. Check: $3(3x-2)=9x-6$.

Expand and factor linear expressions 13

Find the number in the box so the expressions are equivalent for every $x$: $6(2x-\square)-3x=9x-24$. Show that your value makes the expressions equal for every $x$, not just one value.

Hints

- Rewrite the left side with no parentheses. - Which terms contain $x$, and which depend on the missing number? - Compare with the constant on the right.

Solution

1. Let the box be $c$. Expanding the left side gives $12x-6c-3x=9x-6c$. 2. Match constant terms: $-6c=-24$, so $c=4$. 3. Checking gives $6(2x-4)-3x=12x-24-3x=9x-24$.

Answer

$4$. Checking: $6(2x-4)-3x=12x-24-3x=9x-24$.

Expand and factor linear expressions 14

A mural takes $2.5$ liters of paint per section. There are $x$ regular sections and $4$ extra sections. Write the total paint as a sum, then factor the sum to show the total number of sections.

Hints

- How much paint do the extra sections need? - What number is a factor of both terms in your sum? - Which form makes the number of sections easiest to see?

Solution

1. Regular sections use $2.5x$ liters; the extra sections use $2.5\cdot4=10$ liters. 2. The sum is $2.5x+10$. 3. Factor out $2.5$ to get $2.5(x+4)$, showing $x+4$ sections.

Answer

Sum: $2.5x+10$. Factored form: $2.5(x+4)$ liters.

Expand and factor linear expressions 15

Factor $-8u+12v$ by taking out $-4$. Explain why the sign before $v$ changes inside the parentheses.

Hints

- Divide both terms by the outside factor, including its sign. - Which inside sign makes the $v$-term positive after multiplying by $-4$?

Solution

1. Divide each term by $-4$: $(-8u)\div(-4)=2u$ and $12v\div(-4)=-3v$. 2. The factored form is $-4(2u-3v)$. 3. Multiplying back gives $-8u+12v$, confirming the signs.

Answer

$-4(2u-3v)$. The $v$-term is negative inside because $(-4)(-3v)=12v$.

Expand and factor linear expressions 16

The two expressions $12p+18q$ and $2(6p+9q)$ are equivalent. Can you factor $12p+18q$ by taking out a larger whole number? Use the greatest whole-number factor common to both terms. Expand your answer to check.

Hints

- The given factor of $2$ works, but is it the largest factor of both coefficients? - Compare the factors of the two coefficients before choosing what goes outside. - Multiply your proposed factors to check the original terms return.

Solution

1. The coefficients $12$ and $18$ have greatest common factor $6$, which is larger than $2$. 2. Divide both terms by $6$: $12p+18q=6(2p+3q)$. 3. Check: $6(2p+3q)=12p+18q$. The given form with a factor of $2$ is also equivalent, but it does not use the greatest common numerical factor.

Answer

$6(2p+3q)$. Expanding gives $12p+18q$.

Expand and factor linear expressions 17

A number trick says: “Think of a number $x$. Multiply it by $4$. Add $1.2$. Multiply the result by $2.5$. Subtract $3$. Tell me the result, and I can immediately name your original number.” Write and simplify an expression for the steps. Then explain how the original number can be recovered from the final result.

Hints

- Translate the instructions in order and use parentheses for the intermediate result. - Distribute the factor outside the parentheses. - Compare the simplified expression with the original number $x$.

Solution

1. Translate the steps into an expression: $2.5(4x + 1.2) - 3$. 2. Distribute $2.5$: $10x + 3 - 3$. 3. Combine constants: the expression simplifies to $10x$. 4. The final result is ten times the original number, so divide the result by $10$ to recover $x$.

Answer

The expression simplifies to $10x$. Divide the final result by $10$ to find the original number.

Expand and factor linear expressions 18

Jada starts with a number $x$ and calculates $5(2x+1.4)-2$. a) Find the result when $x=4.3$. b) Expand and simplify the expression. c) If the result is $R$, write a rule to find the starting number $x$. Explain your rule.

Hints

- Follow the given operations in order for part a). - Distribute the outside factor before combining constants. - Work backward from the simplified expression to test the rule.

Solution

1. Substitute $x=4.3$: $5(2\cdot4.3+1.4)-2=5(10)-2=48$. 2. Expand: $5(2x+1.4)-2=10x+7-2=10x+5$. 3. If $R=10x+5$, subtract $5$: $R-5=10x$. Divide by $10$: $x=\frac{R-5}{10}$. 4. The reverse rule is to subtract $5$ from the result and then divide by $10$.

Answer

a) $48$ b) $10x+5$ c) $x=\frac{R-5}{10}$: subtract $5$ from the result, then divide by $10$.

Expand and factor linear expressions 19

Expand and simplify each expression. a) $6\left(2x - \frac{2}{3}\right)$ b) $\frac{3}{4}(8a + 4b - 12)$ c) $12\left(\frac{x}{4} + \frac{y}{3} - \frac{1}{2}\right)$

Hints

- Use the distributive property: multiply the factor outside the parentheses by every term inside. - Pay close attention to the signs inside the parentheses. - Before multiplying a whole number by a fraction, look for factors that cancel.

Solution

1. For a), distribute $6$: $6(2x) - 6\left(\frac{2}{3}\right) = 12x - 4$. 2. For b), distribute $\frac{3}{4}$: $\frac{3}{4}(8a) + \frac{3}{4}(4b) - \frac{3}{4}(12) = 6a + 3b - 9$. 3. For c), distribute $12$: $12\left(\frac{x}{4}\right) + 12\left(\frac{y}{3}\right) - 12\left(\frac{1}{2}\right) = 3x + 4y - 6$.

Answer

a) $12x - 4$ b) $6a + 3b - 9$ c) $3x + 4y - 6$

Expand and factor linear expressions 20

Factor out the common coefficient in every term. Then evaluate each expression for the given values. a) $17x+17y$, where $x=0.4$ and $y=0.6$ b) $\frac{2}{9}z-\frac{2}{9}\left(\frac{1}{2}\right)$, where $z=3.5$ c) $4.5a+4.5b+4.5c$, where $a=12$, $b=5$, and $c=3$

Hints

- Identify the coefficient shared by every term. - Write that common coefficient outside parentheses and determine what remains inside. - Evaluate the expression inside the parentheses before multiplying.

Solution

1. For a), factor: $17x+17y=17(x+y)$. Evaluate: $17(0.4+0.6)=17$. 2. For b), factor: $\frac{2}{9}z-\frac{2}{9}\left(\frac{1}{2}\right)=\frac{2}{9}\left(z-\frac{1}{2}\right)$. Evaluate: $\frac{2}{9}(3.5-0.5)=\frac{2}{3}$. 3. For c), factor: $4.5a+4.5b+4.5c=4.5(a+b+c)$. Evaluate: $4.5(12+5+3)=90$.

Answer

a) Factored form: $17(x+y)$; value: $17$ b) Factored form: $\frac{2}{9}\left(z-\frac{1}{2}\right)$; value: $\frac{2}{3}$ c) Factored form: $4.5(a+b+c)$; value: $90$

Expand and factor linear expressions 21

Use the distributive property to simplify the expression $T = 4(2x - 5) + 3(10 - x)$. a) Expand and combine like terms. b) Check your result by substituting $x = 2$ into both the original expression and the simplified expression.

Hints

- Pay close attention to signs when distributing. - Combine only like terms. - If the two forms give different values in the check, revisit your simplification.

Solution

1. Expand: $4 \cdot 2x - 4 \cdot 5 + 3 \cdot 10 - 3x = 8x - 20 + 30 - 3x$. 2. Combine like terms: $8x - 3x - 20 + 30 = 5x + 10$. 3. In the original expression, substituting $x = 2$ gives $4(2 \cdot 2 - 5) + 3(10 - 2) = 4(-1) + 24 = 20$. 4. In the simplified expression, substituting $x = 2$ gives $5 \cdot 2 + 10 = 20$. 5. Both forms give the same value for the check.

Answer

a) $5x + 10$ b) Both expressions have value $20$ when $x = 2$.

Expand and factor linear expressions 22

Simplify the expression step by step. Pay close attention to the signs in the nested parentheses. $24 - \left(3(a + 5) - (2a - 7)\right)$

Hints

- With nested parentheses, work from the inside out. - What happens to every sign inside parentheses when a minus sign is directly in front? - Simplify the expression inside the outer parentheses before carrying out the outer subtraction.

Solution

1. Expand inside the outer parentheses: $3(a + 5) - (2a - 7) = 3a + 15 - 2a + 7$. 2. Combine like terms inside the outer parentheses: $3a - 2a + 15 + 7 = a + 22$. 3. Subtract the entire expression: $24 - (a + 22) = 24 - a - 22$. 4. Combine the constants: $24 - 22 = 2$, so the result is $2 - a$.

Answer

$2 - a$

Expand and factor linear expressions 23

Simplify each expression. Label your steps: distributive property, commutative property of addition, or combining like terms. a) $4(2x-5)+3x$ b) $15y-3(5y-2)$ c) $8z+4-2(4z+2)$

Hints

- Which property lets you change the form of an expression without changing its value? - When you change the order of added terms, how is that different from combining terms? - Could you test your simplified result with a particular variable value?

Solution

1. For a), distribute: $4(2x-5)+3x=8x-20+3x$. Reorder the added terms using the commutative property: $8x+3x-20$. Combine like terms: $11x-20$. 2. For b), distribute: $15y-3(5y-2)=15y-15y+6$. Combine like terms: $6$. 3. For c), distribute: $8z+4-2(4z+2)=8z+4-8z-4$. Reorder by the commutative property: $8z-8z+4-4$. Combine like terms: $0$.

Answer

a) Distributive property: $8x-20+3x$; commutative property of addition: $8x+3x-20$; combine like terms: $11x-20$. b) Distribute: $15y-15y+6$; combine: $6$. c) Distributive property: $8z+4-8z-4$; commutative property of addition: $8z-8z+4-4$; combine like terms: $0$.

Expand and factor linear expressions 24

Lucas and Mia simplify $6(a + 2) - 3(2a - 4)$. Lucas writes: $6a + 12 - 6a - 12 = 0$. Mia writes: $6a + 12 - 6a + 12 = 24$. Determine who is correct. Show how to simplify the expression, and identify the other student's error.

Hints

- Expand each product separately. - Pay close attention when multiplying two negative numbers. - Compare your expanded expression with each student's work one step at a time.

Solution

1. Expand the first product: $6(a + 2) = 6a + 12$. 2. Distribute the negative factor through the second set of parentheses: $-3(2a - 4) = -6a + 12$. 3. Combine like terms: $6a + 12 - 6a + 12 = 24$. 4. Mia is correct. Lucas treated $-3(-4)$ as $-12$, but the product of two negative numbers is positive.

Answer

Mia is correct. The expression simplifies to $24$. Lucas made a sign error because $-3(-4) = 12$, not $-12$.

Expand and factor linear expressions 25

Simplify the expression, then evaluate it for $a = -1$ and $a = 5$. $18 - 4(2a - 3) + a$

Hints

- Pay close attention to the negative factor $-4$ when distributing. - Where does the rule “a negative times a negative is positive” apply? - Remember that $a$ means $1a$.

Solution

1. Distribute the negative factor: $-4(2a - 3) = -8a + 12$. 2. Combine like terms: $18 - 8a + 12 + a = 30 - 7a$. 3. For $a = -1$, $30 - 7 \cdot (-1) = 37$. 4. For $a = 5$, $30 - 7 \cdot 5 = -5$.

Answer

Simplified expression: $30 - 7a$ For $a = -1$: $37$ For $a = 5$: $-5$

Expand and factor linear expressions 26

Crates are arranged in a rectangular array that is $x$ crates long and $y$ crates wide, where $x$ and $y$ are whole numbers and $x,y\ge2$. Only the crates along the outside edge receive labels. a) Write and simplify an expression for the number of labeled crates. b) Find the number of labels needed when $x=12$ and $y=8$.

Hints

- Count the crates in the top and bottom rows first. - When counting the two side columns, do not count the corner crates again. - Use the distributive property to simplify your expression. - Substitute $x=12$ and $y=8$ only after simplifying.

Solution

1. The top and bottom rows contain $2x$ labeled crates. 2. After the four corner crates have already been counted, the two side columns contribute $2(y-2)$ more crates. 3. The total is $2x+2(y-2)=2x+2y-4$. 4. For $x=12$ and $y=8$, $2\cdot12+2\cdot8-4=24+16-4=36$.

Answer

a) $2x+2y-4$ b) $36$ labels

Expand and factor linear expressions 27

Tim claims that $-3(x - 5)$ is equivalent to $-3x - 15$. a) Check Tim's claim by evaluating both expressions for $x = 10$. b) Without doing any additional calculations, explain the common distribution error Tim made.

Hints

- What happens to each sign when parentheses are multiplied by a negative number? - Evaluate the two expressions separately and compare the results. - Pay special attention to the product of two negative numbers.

Solution

1. For $x = 10$, the first expression is $-3(10 - 5) = -15$. 2. The second expression is $-3 \cdot 10 - 15 = -45$. 3. Since $-15 \ne -45$, Tim's claim is false. 4. Tim did not distribute the negative factor correctly. The product $-3(-5)$ is $+15$, so the correct expansion is $-3x + 15$.

Answer

a) The values are $-15$ and $-45$, so the expressions are not equivalent. b) Tim made a sign error: $-3(-5) = 15$, not $-15$.

Expand and factor linear expressions 28

Lina tries to solve $4(x + 3) = 20$ as follows: Step 1: $4x + 3 = 20$ Step 2: $4x = 17$ Step 3: $x = 4.25$ a) Check her answer by substituting $x = 4.25$ into the original equation. What do you find? b) In which step did Lina make an error? Describe the error precisely.

Hints

- A correct solution must make both sides of the original equation equal. - How is a factor distributed over a sum inside parentheses? - Examine Lina's first step carefully.

Solution

1. Substitute $x = 4.25$ into the left side: $4(4.25 + 3) = 4 \cdot 7.25 = 29$. 2. Since $29 \ne 20$, $x = 4.25$ is not a solution. 3. The error occurred in Step 1. Lina multiplied $x$ by $4$ but did not multiply $3$ by $4$. The correct expansion is $4x + 12 = 20$.

Answer

a) The check gives $29 \ne 20$, so the answer is incorrect. b) Lina's error is in Step 1. She failed to distribute $4$ to the $3$; $4(x + 3)$ should become $4x + 12$.

Expand and factor linear expressions 29

A shipping company uses small (S), medium (M), and large (L) packages. A medium package weighs $3$ pounds more than a small package. A large package weighs twice as much as a medium package. a) Complete the table with an expression for each package weight. <table><thead><tr><th>Package size</th><th>Weight in pounds</th></tr></thead><tbody><tr><td>S</td><td>$x$</td></tr><tr><td>M</td><td></td></tr><tr><td>L</td><td></td></tr></tbody></table> b) A shipment contains $2$ S packages, $4$ M packages, and $1$ L package. Write and simplify an expression for the total weight $G$. c) Find the total weight when a small package weighs $4$ pounds.

Hints

- Write the expression for M before working out L. - Multiply each package weight by the number in the shipment. - Distribute the factors before combining like terms.

Solution

1. A medium package weighs $x+3$ pounds. 2. A large package weighs $2(x+3)=2x+6$ pounds. 3. The shipment weight is $G=2x+4(x+3)+(2x+6)$. 4. Expanding gives $G=2x+4x+12+2x+6=8x+18$. 5. At $x=4$, $G=8\cdot4+18=50$ pounds.

Answer

a) M: $x+3$; L: $2(x+3)=2x+6$. b) $G=8x+18$. c) $50$ pounds.

Expand and factor linear expressions 30

Simplify the expression and write the result using decimals. $1\frac{1}{4}(2.4a - 4b) - \left(0.75a - 1\frac{3}{5}b + 2\right)$

Hints

- Convert the mixed numbers to decimals first. - Distribute through the first parentheses, then remove the parentheses being subtracted. - Change every sign inside parentheses that are preceded by a minus sign.

Solution

1. Convert the mixed numbers to decimals: $1\frac{1}{4} = 1.25$ and $1\frac{3}{5} = 1.6$. 2. Distribute in the first product: $1.25(2.4a - 4b) = 3a - 5b$. 3. Subtract the second expression: $-(0.75a - 1.6b + 2) = -0.75a + 1.6b - 2$. 4. Combine like terms: $(3 - 0.75)a + (-5 + 1.6)b - 2 = 2.25a - 3.4b - 2$.

Answer

$2.25a - 3.4b - 2$

Expand and factor linear expressions 31

Expand each expression. 1) $-3(4k - 7)$ 2) $1.2(5p + 10q)$ 3) $(2r - 5s + 6)(-4)$ 4) $\frac{3}{4}(8x - 12)$

Hints

- Apply the sign rules when multiplying by a negative factor. - Multiply a decimal or fraction by each coefficient inside the parentheses. - The outside factor must multiply every term, whether there are two terms or three.

Solution

1. $-3(4k - 7) = -12k + 21$. 2. $1.2(5p + 10q) = 6p + 12q$. 3. $(2r - 5s + 6)(-4) = -8r + 20s - 24$. 4. $\frac{3}{4}(8x - 12) = 6x - 9$.

Answer

1) $-12k + 21$ 2) $6p + 12q$ 3) $-8r + 20s - 24$ 4) $6x - 9$

Expand and factor linear expressions 32

Each equation contains an error in the expansion. Correct the right side. 1) $5(x - 4) = 5x - 4$ 2) $-2(a + 3) = -2a + 6$ 3) $-(3m - n + 2) = -3m - n - 2$

Hints

- Check whether the outside factor was applied to every term. - Verify the sign of each product. - A minus sign directly before parentheses is the same as multiplying every term by $-1$.

Solution

1. The factor $5$ must multiply both terms: $5(x - 4) = 5x - 20$. 2. Since $-2 \cdot 3 = -6$, $-2(a + 3) = -2a - 6$. 3. A minus sign before parentheses changes every sign: $-(3m - n + 2) = -3m + n - 2$.

Answer

1) $5x - 20$ 2) $-2a - 6$ 3) $-3m + n - 2$

Expand and factor linear expressions 33

Expand and combine like terms. 1) $5(x + 2y) + 3(2x - y)$ 2) $8(a - 2b) - 2(3a - 5b)$

Hints

- Expand each set of parentheses first. - Pay close attention to the factor before each set of parentheses. - Combine only like terms. - Check that every term was multiplied by its outside factor.

Solution

1. $5(x + 2y) + 3(2x - y) = 5x + 10y + 6x - 3y = 11x + 7y$. 2. $8(a - 2b) - 2(3a - 5b) = 8a - 16b - 6a + 10b = 2a - 6b$.

Answer

1) $11x + 7y$ 2) $2a - 6b$

Expand and factor linear expressions 34

Expand and combine like terms. 1) $9(k - 2m) - 3(2k - 5m)$ 2) $3(4u - 2v + 5) - 2(5u + v - 3)$

Hints

- A negative outside factor changes the signs of its products. - Multiply the outside factor by each term inside the parentheses. - Combine only terms with exactly the same variable part. - Remember to combine the constants.

Solution

1. $9(k - 2m) - 3(2k - 5m) = 9k - 18m - 6k + 15m = 3k - 3m$. 2. $3(4u - 2v + 5) - 2(5u + v - 3) = 12u - 6v + 15 - 10u - 2v + 6 = 2u - 8v + 21$.

Answer

1) $3k - 3m$ 2) $2u - 8v + 21$

Expand and factor linear expressions 35

Expand and simplify. $4(3u - 2v) - 5(u - 2v) + 2(v - 3u)$

Hints

- Pay close attention to factors with a negative sign. - Group terms by variable before combining. - A variable with no written coefficient has coefficient $1$. - Remember that a negative times a negative is positive.

Solution

1. Expand all three products: $12u - 8v - 5u + 10v + 2v - 6u$. 2. Combine the $u$-terms: $12u - 5u - 6u = u$. 3. Combine the $v$-terms: $-8v + 10v + 2v = 4v$. 4. The simplified expression is $u + 4v$.

Answer

$u + 4v$

Expand and factor linear expressions 36

Expand and simplify. $6(x - 2y) - 2(2x - 5y) - 3(x + y)$

Hints

- Pay close attention to negative factors before parentheses. - Combine only like terms. - Group the terms by variable before combining.

Solution

1. Expand: $6x - 12y - 4x + 10y - 3x - 3y$. 2. Combine the $x$-terms: $(6 - 4 - 3)x = -x$. 3. Combine the $y$-terms: $(-12 + 10 - 3)y = -5y$. 4. The simplified expression is $-x - 5y$.

Answer

$-x - 5y$

Expand and factor linear expressions 37

Expand and simplify. $-3(2x - y + 4) + 2(3x + 2y - 1) - (5y - 10)$

Hints

- Pay close attention when removing parentheses preceded by a minus sign. - A minus sign before parentheses changes every sign inside. - Combine constants only with other constants.

Solution

1. Expand: $-6x + 3y - 12 + 6x + 4y - 2 - 5y + 10$. 2. The $x$-terms cancel: $-6x + 6x = 0$. 3. Combine the $y$-terms and constants: $3y + 4y - 5y = 2y$ and $-12 - 2 + 10 = -4$. 4. The result is $2y - 4$.

Answer

$2y - 4$

Expand and factor linear expressions 38

Expand and simplify. $5\left[2x - 3(x - 4y)\right] - 4(2x + 7y)$

Hints

- With nested grouping symbols, work from the inside out. - Pay close attention to negative factors. - Simplify inside the brackets before multiplying by $5$. - Combine the $x$-terms and $y$-terms separately.

Solution

1. Simplify inside the brackets: $2x - 3(x - 4y) = 2x - 3x + 12y = -x + 12y$. 2. Distribute the outside factors: $5(-x + 12y) - 4(2x + 7y) = -5x + 60y - 8x - 28y$. 3. Combine like terms: $-13x + 32y$.

Answer

$-13x + 32y$

Expand and factor linear expressions 39

Simplify each expression. 1) $(18x - 12) \div 6 + 4(x - 2)$ 2) $5(2y + 3) - (12y - 18) \div 6$

Hints

- Pay attention to the operation before each set of parentheses. - A minus sign before parentheses changes both signs when the parentheses are removed. - Use the order of operations. - Combine only like terms.

Solution

1. Divide and distribute: $(18x - 12) \div 6 = 3x - 2$ and $4(x - 2) = 4x - 8$. Then $3x - 2 + 4x - 8 = 7x - 10$. 2. Distribute and divide: $5(2y + 3) = 10y + 15$ and $(12y - 18) \div 6 = 2y - 3$. Then $10y + 15 - (2y - 3) = 8y + 18$.

Answer

1) $7x - 10$ 2) $8y + 18$

Expand and factor linear expressions 40

Combine like terms and simplify. 1) $(24w - 16) \div 4 - (15w + 10) \div 5$ 2) $0.2(10x - 50) - (4x - 12) \div 4$ 3) $7 - (9z - 6) \div 3 - (z + 1)$

Hints

- When an expression in parentheses is divided by a number, divide each term. - A minus sign before parentheses changes every sign inside when the parentheses are removed. - Group variable terms and constants before combining.

Solution

1. Divide each expression: $(6w - 4) - (3w + 2) = 6w - 4 - 3w - 2 = 3w - 6$. 2. Multiply and divide: $(2x - 10) - (x - 3) = 2x - 10 - x + 3 = x - 7$. 3. Divide first: $7 - (3z - 2) - (z + 1) = 7 - 3z + 2 - z - 1 = 8 - 4z$.

Answer

1) $3w - 6$ 2) $x - 7$ 3) $8 - 4z$

Expand and factor linear expressions 41

Simplify the expression. $4(x + 2y) + (12x - 6y) \div 3 - (5x + y)$

Hints

- Simplify one part of the expression at a time. - A minus sign before parentheses changes every sign inside. - Divide every term inside the parentheses by the divisor. - Combine only terms with matching variable parts.

Solution

1. Distribute: $4(x + 2y) = 4x + 8y$. 2. Divide each term: $(12x - 6y) \div 3 = 4x - 2y$. 3. Remove the last parentheses: $-(5x + y) = -5x - y$. 4. Combine like terms: $4x + 8y + 4x - 2y - 5x - y = 3x + 5y$.

Answer

$3x + 5y$

Expand and factor linear expressions 42

Fill in each blank to make the equation true. a) $12z + 18 = \square(2z + 3)$ b) $5x - 15y + 10 = 5(\square - \square + \square)$ c) $\square(4a - 1) = 8a - 2$ d) $24 - 16c = 8(\square - \square)$

Hints

- Relate the outside factor and the terms inside the parentheses to the original expression. - Distribute to check each completed equation. - Use division to determine a missing outside factor. - Keep each variable in its corresponding term.

Solution

1. a) Since $12z \div 2z = 6$, the outside factor is $6$. Check: $6 \cdot 3 = 18$. 2. b) Divide each term by $5$: $x$, $3y$, and $2$. 3. c) Since $8a \div 4a = 2$, the outside factor is $2$. Check: $2 \cdot (-1) = -2$. 4. d) Divide each term by $8$: $3$ and $2c$.

Answer

a) $6$ b) $x, 3y, 2$ c) $2$ d) $3, 2c$

Expand and factor linear expressions 43

The diagram partitions one rectangular garden into two adjacent rectangles of the same height. a) Use the labeled dimensions to write the total area as a sum of the two smaller rectangle areas. b) Write the total area as one product using the full width. c) Explain why your two expressions are equivalent and identify the distributive-property structure shown by the partition.

Hints

- Read the width of each colored rectangle separately. - Multiply each smaller rectangle's width by the common height. - Add the two widths to find the full width. - Compare “area by pieces” with “area as one rectangle.”

Solution

1. The left rectangle has width $x$ and height $4$, so its area is $4x$. 2. The right rectangle has width $3$ and height $4$, so its area is $12$. 3. Adding the pieces gives $4x+12$. 4. The full width is $x+3$, so the whole rectangle's area is $4(x+3)$. 5. Therefore, $4(x+3)=4x+12$. The partition shows the distributive property geometrically.

Answer

a) $4x+12$ b) $4(x+3)$ c) The partition shows $4(x+3)=4x+12$, an application of the distributive property.

Expand and factor linear expressions 44

For each equation: 1. Use the distributive property in reverse to rewrite the left side as a single product. 2. Then solve for $\square$. a) $15\cdot7+15\cdot\square=150$ b) $\square\cdot1.2-\square\cdot0.2=8$ c) $(-4)\cdot2.5+\square\cdot2.5=-25$ d) $0.8\cdot\square-0.8\cdot5=4$

Hints

- Do not solve the original equation first; begin by identifying the common factor on the left. - Write the factored equation explicitly before isolating the blank. - Distribute your factored form mentally to check that it reproduces the original left side. - Then solve the simpler product equation.

Solution

1. a) Factor $15$: $15(7+\square)=150$. Divide by $15$: $7+\square=10$, so $\square=3$. 2. b) Factor $\square$: $\square(1.2-0.2)=8$. Thus $\square(1)=8$, so $\square=8$. 3. c) Factor $2.5$: $2.5(-4+\square)=-25$. Divide by $2.5$: $-4+\square=-10$, so $\square=-6$. 4. d) Factor $0.8$: $0.8(\square-5)=4$. Divide by $0.8$: $\square-5=5$, so $\square=10$.

Answer

a) $15(7+\square)=150$, so $\square=3$ b) $\square(1.2-0.2)=8$, so $\square=8$ c) $2.5(-4+\square)=-25$, so $\square=-6$ d) $0.8(\square-5)=4$, so $\square=10$

Expand and factor linear expressions 45

Use the distributive property. 1) Find the missing term: $7(x + \square) = 7x + 21y$. 2) Find the missing factor: $\square(4a - 5) = -8a + 10$. 3) Expand and simplify: $5(2z - 1) - 4(z + 3)$.

Hints

- Compare each distributed product with the matching term on the right. - A missing factor must make every resulting term agree, including the signs. - In the final expression, expand both grouped terms before combining like terms.

Solution

1. Since $7 \cdot 3y = 21y$, the missing term is $3y$. 2. Since $-2(4a) = -8a$ and $-2 \cdot (-5) = 10$, the missing factor is $-2$. 3. Expand and combine like terms: $5(2z - 1) - 4(z + 3) = 10z - 5 - 4z - 12 = 6z - 17$.

Answer

1) $\square = 3y$ 2) $\square = -2$ 3) $6z - 17$

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Grade 7 math worksheets

Expand and factor linear expressions

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