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Linear Expression Operations

Grade 7 · Algebra · Worksheet 1

  1. Mere is organizing the bookshelves in her classroom. The fiction section has (8x + 24) books, and the non-fiction section has (6x - 12) books. During a book drive, the class adds (4x + 18) more fiction books, but then removes (2x - 6) non-fiction books to make space for a new display. What simplified expression represents the total number of books on the shelves after these changes? Answer: ______________
  2. Maya is designing a community mural project that will cover two adjacent walls. The first wall has a length of (5x + 12) feet, and the second wall has a length of (3x - 7) feet. She needs to calculate the total length of trim needed to frame both walls. What is the simplified expression for the total length of trim Maya needs? Answer: ______________
  3. (12x - 7) - (8x + 5) = ? Answer: ______________
  4. Sophia is helping her school’s art club create a large mosaic for the gym wall. The mosaic will have two sections. The left section’s area is represented by the expression (9x + 15) square feet, and the right section’s area is represented by (7x - 8) square feet. After planning, the club decides to add a border around the entire mosaic that adds (4x + 12) square feet to the total area. What is the simplified expression for the total area of the mosaic with the border added? Answer: ______________
  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: ______________
  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: ______________
  7. (12x - 8) - (7x + 15) = ? Answer: ______________
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Answer Key & Explanations

Linear Expression Operations · Grade 7 · Worksheet 1

  1. Mere is organizing the bookshelves in her classroom. The fiction section has (8x + 24) books, and the non-fiction section has (6x - 12) books. During a book drive, the class adds (4x + 18) more fiction books, but then removes (2x - 6) non-fiction books to make space for a new display. What simplified expression represents the total number of books on the shelves after these changes? Answer: 16x + 36 Solution: Find the new number of fiction books. Original fiction: (8x + 24) Added fiction: (4x + 18) New fiction = (8x + 24) + (4x + 18) = 8x + 4x + 24 + 18 = 12x + 42 Find the new number of non-fiction books.
    Full step-by-step solution

    Step 1: Find the new number of fiction books. Original fiction: (8x + 24) Added fiction: (4x + 18) New fiction = (8x + 24) + (4x + 18) = 8x + 4x + 24 + 18 = 12x + 42 Step 2: Find the new number of non-fiction books. Original non-fiction: (6x - 12) Removed non-fiction: (2x - 6) New non-fiction = (6x - 12) - (2x - 6) = 6x - 12 - 2x + 6 = 4x - 6 Step 3: Add the new totals together. Total books = (12x + 42) + (4x - 6) = 12x + 4x + 42 - 6 = 16x + 36 The simplified expression for the total number of books is 16x + 36.

  2. Maya is designing a community mural project that will cover two adjacent walls. The first wall has a length of (5x + 12) feet, and the second wall has a length of (3x - 7) feet. She needs to calculate the total length of trim needed to frame both walls. What is the simplified expression for the total length of trim Maya needs? Answer: 8x + 5 Solution: Identify the two lengths that need to be combined: (5x + 12) and (3x - 7) Add the expressions together: (5x + 12) + (3x - 7) Combine like terms: 5x + 3x = 8x Combine the constant terms: 12 + (-7) = 12 - 7 = 5 Write the simplified expression: 8x + 5 The total length of trim needed is 8x + 5 feet.
    Full step-by-step solution

    Step 1: Identify the two lengths that need to be combined: (5x + 12) and (3x - 7) Step 2: Add the expressions together: (5x + 12) + (3x - 7) Step 3: Combine like terms: 5x + 3x = 8x Step 4: Combine the constant terms: 12 + (-7) = 12 - 7 = 5 Step 5: Write the simplified expression: 8x + 5 The total length of trim needed is 8x + 5 feet.

  3. (12x - 7) - (8x + 5) = ? Answer: 4x - 12 Solution: Write the original expression: (12x - 7) - (8x + 5) Distribute the subtraction sign: 12x - 7 - 8x - 5 Combine like terms for x: 12x - 8x = 4x Combine constant terms: -7 - 5 = -12 Write the final expression: 4x - 12 The answer is 4x - 12.
    Full step-by-step solution

    Step 1: Write the original expression: (12x - 7) - (8x + 5) Step 2: Distribute the subtraction sign: 12x - 7 - 8x - 5 Step 3: Combine like terms for x: 12x - 8x = 4x Step 4: Combine constant terms: -7 - 5 = -12 Step 5: Write the final expression: 4x - 12 The answer is 4x - 12.

  4. Sophia is helping her school’s art club create a large mosaic for the gym wall. The mosaic will have two sections. The left section’s area is represented by the expression (9x + 15) square feet, and the right section’s area is represented by (7x - 8) square feet. After planning, the club decides to add a border around the entire mosaic that adds (4x + 12) square feet to the total area. What is the simplified expression for the total area of the mosaic with the border added? Answer: 20x + 19 Solution: Write the expression for the total area of the two sections: (9x + 15) + (7x - 8) Combine like terms for the x terms: 9x + 7x = 16x Combine the constant terms: 15 - 8 = 7 So, the total area of the two sections is 16x + 7 square feet.
    Full step-by-step solution

    Step 1: Write the expression for the total area of the two sections: (9x + 15) + (7x - 8) Step 2: Combine like terms for the x terms: 9x + 7x = 16x Step 3: Combine the constant terms: 15 - 8 = 7 So, the total area of the two sections is 16x + 7 square feet. Step 4: Add the border area (4x + 12) to the total: (16x + 7) + (4x + 12) Step 5: Combine like terms again: 16x + 4x = 20x and 7 + 12 = 19 The simplified expression for the total area is 20x + 19 square feet.

  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. (12x - 8) - (7x + 15) = ? Answer: 5x - 23 Solution: Write the original expression: (12x - 8) - (7x + 15) Distribute the negative sign to the second expression: 12x - 8 - 7x - 15 Combine like terms for x: 12x - 7x = 5x Combine constant terms: -8 - 15 = -23 Write the simplified expression: 5x - 23 The answer is 5x - 23.
    Full step-by-step solution

    Step 1: Write the original expression: (12x - 8) - (7x + 15) Step 2: Distribute the negative sign to the second expression: 12x - 8 - 7x - 15 Step 3: Combine like terms for x: 12x - 7x = 5x Step 4: Combine constant terms: -8 - 15 = -23 Step 5: Write the simplified expression: 5x - 23 The answer is 5x - 23.