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Build your own math worksheets from 21,000 problems for grades 3 to 12, from fractions to calculus. Every problem includes step-by-step solutions.

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5124607
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

\(11x - 68\)
5125027
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
5225497
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) \(\$29.60\)
5230517
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) \(40 - 15y\)
5231177
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) \(13x - 4\)
5234717
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) \(6(3m - 2n)\)
5279897
Factor out the greatest common factor. 1) \(4a + 4b\) 2) \(15x - 5y\) 3) \(12m + 18n\) 4) \(20r - 30s\)

Hints

- Find the greatest number dividing both coefficients. - Apply the distributive property in reverse. - Distribute to check your result. - Divide each original term by the common factor.

Solution

1. The greatest common factor is \(4\): \(4(a + b)\). 2. The greatest common factor is \(5\): \(5(3x - y)\). 3. The greatest common factor is \(6\): \(6(2m + 3n)\). 4. The greatest common factor is \(10\): \(10(2r - 3s)\).

Answer

1) \(4(a + b)\) 2) \(5(3x - y)\) 3) \(6(2m + 3n)\) 4) \(10(2r - 3s)\)
5279907
Factor out the greatest common numerical and variable factor. 1) \(8ax + 12ay\) 2) \(14m - 21mn\) 3) \(-9p - 27q\) 4) \(15uv + 10uw - 5u\)

Hints

- Check for variables shared by every term, not only numerical factors. - If an entire term is factored out, a \(1\) or \(-1\) remains. - Use the greatest positive common factor of the coefficients. - Distribute to check your result.

Solution

1. The greatest common factor is \(4a\): \(4a(2x + 3y)\). 2. The greatest common factor is \(7m\): \(7m(2 - 3n)\). 3. The greatest common factor is \(9\): \(9(-p - 3q)\). 4. The greatest common factor is \(5u\): \(5u(3v + 2w - 1)\).

Answer

1) \(4a(2x + 3y)\) 2) \(7m(2 - 3n)\) 3) \(9(-p - 3q)\) 4) \(5u(3v + 2w - 1)\)
5113697
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.
5119877
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) \(3x + 4y - 6\)
5119887
Factor out the greatest common factor. 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 factor shared by every term. - Write the common factor outside parentheses and determine what remains inside. - Evaluate the expression inside the parentheses before multiplying.

Solution

1. a) Factor: \(17x + 17y = 17(x + y)\). Evaluate: \(17(0.4 + 0.6) = 17\). 2. 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. 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: \(90\)
5120817
Let \(T(x) = 3(x + 4) - 2x\). a) Evaluate the expression for \(x = 5\) and \(x = 10\). b) Expand and simplify the expression. c) Use your result from part b) to explain why the value of the expression increases by exactly \(5\) when \(x\) increases by \(5\).

Hints

- Substitute each given number everywhere \(x\) appears, and follow the order of operations. - Use the distributive property, then combine like terms. - In the simplified expression, compare the value at \(x\) with the value at \(x + 5\).

Solution

1. For \(x = 5\), \(T(5) = 3(5 + 4) - 2 \cdot 5 = 27 - 10 = 17\). 2. For \(x = 10\), \(T(10) = 3(10 + 4) - 2 \cdot 10 = 42 - 20 = 22\). 3. Expand and combine like terms: \(3(x + 4) - 2x = 3x + 12 - 2x = x + 12\). 4. Replacing \(x\) with \(x + 5\) gives \((x + 5) + 12 = x + 17\), which is \(5\) greater than \(x + 12\).

Answer

a) \(T(5) = 17\) and \(T(10) = 22\) b) \(T(x) = x + 12\) c) Increasing \(x\) by \(5\) changes \(x + 12\) to \(x + 17\), so the expression increases by \(5\).
5122527
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 \(x = 2\).
5124617
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

\(2 - a\)
5124737
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 \(-12\).
5124837
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 \(a = 5\): \(-5\)
5125037
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
5126167
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 \(-15\).
5126177
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 \(4x + 12\).
5126187
Consider the expression \(6ax + 9ay\). a) Factor out the greatest common factor. b) A student claims that the expression can be written as \(3a(2x + 3y)\). Is the student correct? Explain. c) Evaluate the original expression and your factored expression for \(a = 5\), \(x = 2\), and \(y = 4\).

Hints

- Identify the number and variable shared by both terms. - Find the greatest common factor of \(6\) and \(9\). - Use the order of operations when evaluating. - Distribute to check the factored form.

Solution

1. The greatest common factor of \(6ax\) and \(9ay\) is \(3a\). 2. Therefore, \(6ax + 9ay = 3a(2x + 3y)\). 3. The student is correct because distributing gives \(3a(2x) = 6ax\) and \(3a(3y) = 9ay\). 4. Original expression: \(6 \cdot 5 \cdot 2 + 9 \cdot 5 \cdot 4 = 60 + 180 = 240\). 5. Factored expression: \(3 \cdot 5\left(2 \cdot 2 + 3 \cdot 4\right) = 15 \cdot 16 = 240\).

Answer

a) \(3a(2x + 3y)\) b) Yes. Distributing \(3a\) reproduces the original expression. c) Both forms have the value \(240\).
5126327
A shipping company uses three package sizes: S, M, and L. - An S package weighs \(x\) pounds. - An M package weighs \(3\) pounds more than an S package. - An L package weighs twice as much as an M 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 an S package weighs \(4\) pounds.

Hints

- Write the expression for M before writing the expression for L. - Multiply each package weight by the number of packages of that size. - Use the distributive property before combining like terms.

Solution

1. An M package weighs \(x+3\) pounds. 2. An L package weighs \(2(x+3)=2x+6\) pounds. 3. The total shipment weight is \(G=2x+4(x+3)+(2x+6)\). 4. Distribute and combine like terms: \(G=2x+4x+12+2x+6=8x+18\). 5. For \(x=4\), \(G=8\cdot 4+18=50\) pounds.

Answer

a) M: \(x+3\); L: \(2x+6\), or \(2(x+3)\) b) \(G=8x+18\) c) \(50\) pounds
5229687
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

\(2.25a - 3.4b - 2\)
5230527
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) \(6x - 9\)
5230627
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) \(-3m + n - 2\)
5230657
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) \(2a - 6b\)
5230667
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) \(2u - 8v + 21\)
5230677
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

\(u + 4v\)
5230737
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

\(-x - 5y\)
5230747
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

\(2y - 4\)
5230797
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

\(-13x + 32y\)
5234137
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) \(8y + 18\)
5234147
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) \(8 - 4z\)
5234357
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

\(3x + 5y\)
5234727
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) \(3, 2c\)
5113707
The expression \(T(x) = 5(2x + 1.4) - 2\) represents a number process, where \(x\) is the starting number. a) Evaluate \(T(x)\) when \(x = 4.3\). b) Simplify \(T(x)\). c) Someone claims, “If I know the final result \(R\), I can subtract \(5\) and then divide by \(10\) to recover \(x\).” Determine whether this rule is correct.

Hints

- Substitute the given value for \(x\) and follow the order of operations. - Use the distributive property to remove the parentheses. - Set the final result equal to the simplified expression and undo the operations in reverse order.

Solution

1. For a), substitute \(4.3\): \(5(2 \cdot 4.3 + 1.4) - 2 = 5(8.6 + 1.4) - 2 = 5 \cdot 10 - 2 = 48\). 2. For b), distribute and combine constants: \(5(2x + 1.4) - 2 = 10x + 7 - 2 = 10x + 5\). 3. For c), write \(R = 10x + 5\). Then \(R - 5 = 10x\), so \(x = \frac{R - 5}{10}\). 4. This matches the stated rule, so the rule is correct.

Answer

a) \(48\) b) \(10x + 5\) c) The rule is correct because \(x = \frac{R - 5}{10}\).
5122617
Find the value of \(\square\) that makes each equation true. Use the distributive property. a) \(15 \times 7 + 15 \times \square = 150\) b) \(\square \times 1.2 - \square \times 0.2 = 8\) c) \((-4) \times 2.5 + \square \times 2.5 = -25\) d) \(0.8 \times \square - 0.8 \times 5 = 4\)

Hints

- Factor the expression on the left side first. - Then isolate the unknown value. - Determine what must be inside the parentheses to produce the value on the right.

Solution

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

Answer

a) \(\square = 3\) b) \(\square = 8\) c) \(\square = -6\) d) \(\square = 10\)
5230567
Use the distributive property. 1) Find the missing term: \(7(x + \Box) = 7x + 21y\). 2) Find the missing factor: \(\Box(4a - 5) = -8a + 10\). 3) Expand and simplify: \(5(2z - 1) - 4(z + 3)\).

Hints

- For a missing term, divide the corresponding product by the known factor. - Check that the factor in part 2 produces both terms with the correct signs. - In part 3, distribute \(-4\) to every term in the second set of parentheses. - Group like terms before combining.

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) \(\Box = 3y\) 2) \(\Box = -2\) 3) \(6z - 17\)

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