Linear Expression Operations

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Answer Key & Explanations

Linear Expression Operations · Grade 7 · Worksheet 1

1. (47x - 29) + (38x + 16) = ? Answer: 85x - 13 Solution: Write the original expression: (47x - 29) + (38x + 16) Since we are adding, we can remove the parentheses without changing any signs: 47x - 29 + 38x + 16 Combine the x terms: 47x + 38x = 85x Combine the constant terms: -29 + 16 = -13 Write the final simplified expression: 85x - 13 The answer is…
   Full step-by-step solution

   Step 1: Write the original expression: (47x - 29) + (38x + 16) Step 2: Since we are adding, we can remove the parentheses without changing any signs: 47x - 29 + 38x + 16 Step 3: Combine the x terms: 47x + 38x = 85x Step 4: Combine the constant terms: -29 + 16 = -13 Step 5: Write the final simplified expression: 85x - 13 The answer is 85x - 13.

2. (12x - 7) - (8x + 5) + (3x - 2) = ? Answer: 7x - 14 Solution: Write the original expression: (12x - 7) - (8x + 5) + (3x - 2) Distribute the negative sign to the second expression: 12x - 7 - 8x - 5 + (3x - 2) Remove the remaining parentheses: 12x - 7 - 8x - 5 + 3x - 2 Combine x terms: 12x - 8x + 3x = 7x Combine constant terms: -7 - 5 - 2 = -14 Write the…
   Full step-by-step solution

   Step 1: Write the original expression: (12x - 7) - (8x + 5) + (3x - 2) Step 2: Distribute the negative sign to the second expression: 12x - 7 - 8x - 5 + (3x - 2) Step 3: Remove the remaining parentheses: 12x - 7 - 8x - 5 + 3x - 2 Step 4: Combine x terms: 12x - 8x + 3x = 7x Step 5: Combine constant terms: -7 - 5 - 2 = -14 Step 6: Write the final simplified expression: 7x - 14

3. (3x + 7) + (2x - 5) = ? Answer: 5x + 2 Solution: (3x + 7) + (2x - 5) Since we are adding the two expressions, the parentheses do not change the signs: 3x + 7 + 2x - 5 Like terms are terms with the same variable part and constant terms.
   Full step-by-step solution

   Let's solve the problem step by step. We are given: (3x + 7) + (2x - 5) --- **Step 1: Remove the parentheses** Since we are adding the two expressions, the parentheses do not change the signs: 3x + 7 + 2x - 5 --- **Step 2: Group like terms** Like terms are terms with the same variable part and constant terms. - Terms with x: 3x and 2x - Constant terms: 7 and -5 So we rewrite: (3x + 2x) + (7 - 5) --- **Step 3: Combine like terms** 3x + 2x = 5x 7 - 5 = 2 So we have: 5x + 2 --- **Step 4: Write the final answer** The simplified expression is: 5x + 2

4. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). The gardener wants to create a border by planting flowers along the perimeter. If each unit on the coordinate plane represents 3 feet, what is the total length of the border in feet? Answer: 60 Solution: Identify the coordinates of the garden's vertices. The vertices are: (2, 1), (8, 1), (8, 5), (2, 5). - (2, 1) and (8, 1) have the same y-coordinate → bottom side.
   Full step-by-step solution

   Step 1: Identify the coordinates of the garden's vertices. The vertices are: (2, 1), (8, 1), (8, 5), (2, 5). Step 2: Understand the shape. Plotting these points: - (2, 1) and (8, 1) have the same y-coordinate → bottom side. - (8, 1) and (8, 5) have the same x-coordinate → right side. - (8, 5) and (2, 5) have the same y-coordinate → top side. - (2, 5) and (2, 1) have the same x-coordinate → left side. This is a rectangle. Step 3: Find the side lengths in coordinate units. Bottom side length: from (2, 1) to (8, 1) → difference in x: 8 - 2 = 6 units. Right side length: from (8, 1) to (8, 5) → difference in y: 5 - 1 = 4 units. Top side length: same as bottom = 6 units. Left side length: same as right = 4 units. Step 4: Calculate the perimeter in coordinate units. Perimeter = 6 + 4 + 6 + 4 = 20 units. Step 5: Convert to feet. Each unit represents 3 feet, so total length in feet = 20 × 3 = 60 feet. Final answer: 60

5. Hana is organizing the school's recycling drive. On Monday, a certain number of kilograms of paper were collected, represented by (6x + 15) kg. On Tuesday, the amount collected was represented by (4x - 9) kg. On Wednesday, the collection was represented by (9x + 22) kg. Hana needs to report the total amount of paper collected over these three days. What simplified expression represents the total kilograms of paper collected on Monday, Tuesday, and Wednesday? Answer: 19x + 28 Solution: Write the expressions for each day. Monday: (6x + 15) kg Tuesday: (4x - 9) kg Wednesday: (9x + 22) kg Add all three expressions together to find the total. Total = (6x + 15) + (4x - 9) + (9x + 22) Group like terms.
   Full step-by-step solution

   Step 1: Write the expressions for each day. Monday: (6x + 15) kg Tuesday: (4x - 9) kg Wednesday: (9x + 22) kg Step 2: Add all three expressions together to find the total. Total = (6x + 15) + (4x - 9) + (9x + 22) Step 3: Group like terms. First, group the terms with x: 6x, 4x, and 9x. Then, group the constant terms: 15, -9, and 22. Total = (6x + 4x + 9x) + (15 - 9 + 22) Step 4: Combine the x terms. 6x + 4x + 9x = 19x Step 5: Combine the constant terms. 15 - 9 = 6 6 + 22 = 28 Step 6: Write the simplified expression. Total = 19x + 28 The simplified expression for the total kilograms of paper collected is 19x + 28.

6. Liam is designing a rectangular garden with a length of (3x + 7) meters and a width of (2x - 4) meters. He wants to install a decorative border along the entire perimeter. Write a simplified expression for the total length of border Liam needs to purchase. Answer: 10x + 6 Solution: To find the total length of the border, we need to calculate the perimeter of the rectangular garden. Recall the formula for the perimeter of a rectangle. P = 2 * (length + width) Substitute the given expressions for length and width into the formula.
   Full step-by-step solution

   To find the total length of the border, we need to calculate the perimeter of the rectangular garden. Step 1: Recall the formula for the perimeter of a rectangle. The perimeter P is given by: P = 2 * (length + width) Step 2: Substitute the given expressions for length and width into the formula. Length = (3x + 7) meters Width = (2x - 4) meters So, P = 2 * ( (3x + 7) + (2x - 4) ) Step 3: Simplify the expression inside the parentheses. Combine like terms: 3x + 2x = 5x, and 7 + (-4) = 3. So, (3x + 7) + (2x - 4) = 5x + 3 Step 4: Multiply the simplified expression by 2. P = 2 * (5x + 3) Apply the distributive property: 2 * 5x = 10x, and 2 * 3 = 6. So, P = 10x + 6 Step 5: State the final simplified expression. The total length of border Liam needs is (10x + 6) meters.

7. Liam is designing a rectangular garden with a length of (3x + 7) meters and a width of (2x - 4) meters. He wants to install a decorative border around the entire perimeter. Write a simplified expression that represents the total length of border Liam needs for his garden. Answer: 10x + 6 Solution: To find the total length of border needed, we need to calculate the perimeter of the rectangular garden. Recall the formula for the perimeter of a rectangle. P = 2 * (length + width) Substitute the given expressions for length and width into the formula.
   Full step-by-step solution

   To find the total length of border needed, we need to calculate the perimeter of the rectangular garden. Step 1: Recall the formula for the perimeter of a rectangle. The perimeter P of a rectangle is given by: P = 2 * (length + width) Step 2: Substitute the given expressions for length and width into the formula. Length = (3x + 7) meters Width = (2x - 4) meters So, P = 2 * ( (3x + 7) + (2x - 4) ) Step 3: Simplify the expression inside the parentheses. First, add the two expressions: (3x + 7) + (2x - 4) Combine the like terms: 3x + 2x = 5x 7 + (-4) = 3 So, (3x + 7) + (2x - 4) simplifies to 5x + 3 Step 4: Multiply the simplified expression by 2. P = 2 * (5x + 3) Apply the distributive property: 2 * 5x = 10x 2 * 3 = 6 So, P = 10x + 6 Step 5: State the final answer. The simplified expression for the total length of border needed is 10x + 6 meters.

8. (12x - 7) - (5x + 9) = ? Answer: 7x - 16 Solution: Write the original expression: (12x - 7) - (5x + 9) Distribute the negative sign to the second expression: 12x - 7 - 5x - 9 Combine like terms for x: 12x - 5x = 7x Combine constant terms: -7 - 9 = -16 Write the final simplified expression: 7x - 16 The answer is 7x - 16.
   Full step-by-step solution

   Step 1: Write the original expression: (12x - 7) - (5x + 9) Step 2: Distribute the negative sign to the second expression: 12x - 7 - 5x - 9 Step 3: Combine like terms for x: 12x - 5x = 7x Step 4: Combine constant terms: -7 - 9 = -16 Step 5: Write the final simplified expression: 7x - 16 The answer is 7x - 16.