Equivalent Expressions

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Answer Key & Explanations

Equivalent Expressions · Grade 7 · Worksheet 2

1. 8(2x - 9) + 5x = ? Answer: 21x - 72 Solution: Distribute the 8 to each term inside the parentheses: 8(2x) = 16x and 8(-9) = -72. Combine like terms (the x terms): 16x + 5x = 21x. The simplified expression is 21x - 72.
   Full step-by-step solution

   Step 1: Distribute the 8 to each term inside the parentheses: 8(2x) = 16x and 8(-9) = -72. So the expression becomes 16x - 72 + 5x. Step 2: Combine like terms (the x terms): 16x + 5x = 21x. Step 3: The simplified expression is 21x - 72. Final answer: 21x - 72.

2. A school is planning a field trip and needs to transport 245 students. Each school bus can carry 42 students, and each minivan can carry 8 students. If the school wants to use exactly 9 vehicles in total, how many buses and how many minivans should they use? Answer: 5 buses and 4 minivans Solution: Let b represent the number of buses and m represent the number of minivans. Write the equation for the total vehicles: b + m = 9 Write the equation for the total students: 42b + 8m = 245 Solve the first equation for m: m = 9 - b Substitute into the second equation: 42b + 8(9 - b) = 245 Simplify:…
   Full step-by-step solution

   Step 1: Let b represent the number of buses and m represent the number of minivans. Step 2: Write the equation for the total vehicles: b + m = 9 Step 3: Write the equation for the total students: 42b + 8m = 245 Step 4: Solve the first equation for m: m = 9 - b Step 5: Substitute into the second equation: 42b + 8(9 - b) = 245 Step 6: Simplify: 42b + 72 - 8b = 245 Step 7: Combine like terms: 34b + 72 = 245 Step 8: Subtract 72 from both sides: 34b = 173 Step 9: Divide both sides by 34: b = 173 ÷ 34 = 5.088 Step 10: Since we need whole vehicles, try b = 5: 42(5) + 8m = 245 → 210 + 8m = 245 → 8m = 35 → m = 4.375 (not whole) Step 11: Try b = 5 and check total students: 42(5) + 8(4) = 210 + 32 = 242 (too few) Step 12: Try b = 5 and m = 4: Total vehicles = 5 + 4 = 9, Total students = 42(5) + 8(4) = 210 + 32 = 242 (3 students short) Step 13: Try b = 4 and m = 5: Total vehicles = 4 + 5 = 9, Total students = 42(4) + 8(5) = 168 + 40 = 208 (too few) Step 14: Try b = 6 and m = 3: Total vehicles = 6 + 3 = 9, Total students = 42(6) + 8(3) = 252 + 24 = 276 (too many) Step 15: The only combination that uses exactly 9 vehicles and comes closest to 245 students is 5 buses and 4 minivans, which transports 242 students. The school will need to make arrangements for the 3 additional students.

3. Mason draws a rectangle on a coordinate plane with vertices at (2, 7), (17, 7), (17, 12), and (2, 12). He then draws a smaller rectangle inside it with vertices at (7, 9), (12, 9), (12, 12), and (7, 12). Write an expression in expanded form for the area of the shaded region (the larger rectangle minus the smaller rectangle). Then, factor that expression completely. Answer: (15)(5) - (5)(3) = 75 - 15 = 60; factored: 5(15 - 3) = 5(12) = 60 Solution: Find the dimensions of the large rectangle. Length = difference in x-coordinates: 17 - 2 = 15. Width = difference in y-coordinates: 12 - 7 = 5.
   Full step-by-step solution

   Step 1: Find the dimensions of the large rectangle. Length = difference in x-coordinates: 17 - 2 = 15. Width = difference in y-coordinates: 12 - 7 = 5. Area of large rectangle = 15 * 5 = 75 square units. Step 2: Find the dimensions of the small rectangle. Length = difference in x-coordinates: 12 - 7 = 5. Width = difference in y-coordinates: 12 - 9 = 3. Area of small rectangle = 5 * 3 = 15 square units. Step 3: Write an expression for the shaded area: 15*5 - 5*3 = 75 - 15 = 60. Step 4: Factor the expression: 15*5 - 5*3 = 5(15 - 3) = 5(12) = 60. The answer is: expanded form is 75 - 15, factored form is 5(15 - 3), and the area is 60 square units.

4. (3/4 × 8/9) ÷ (2/3) = ? Answer: 1 Solution: First, compute 3/4 × 8/9. Multiply the numerators: 3 × 8 = 24 Multiply the denominators: 4 × 9 = 36 So, 3/4 × 8/9 = 24/36 Simplify 24/36. The greatest common divisor of 24 and 36 is 12.
   Full step-by-step solution

   Let's solve step-by-step. Step 1: First, compute 3/4 × 8/9. Multiply the numerators: 3 × 8 = 24 Multiply the denominators: 4 × 9 = 36 So, 3/4 × 8/9 = 24/36 Step 2: Simplify 24/36. The greatest common divisor of 24 and 36 is 12. Divide numerator and denominator by 12: 24 ÷ 12 = 2 36 ÷ 12 = 3 So, 24/36 = 2/3 Step 3: Now the expression is (2/3) ÷ (2/3). Dividing by a fraction is the same as multiplying by its reciprocal. So, (2/3) ÷ (2/3) = (2/3) × (3/2) Step 4: Multiply numerators: 2 × 3 = 6 Multiply denominators: 3 × 2 = 6 So, 6/6 = 1 Final answer: 1

5. Hana is designing a rectangular community garden. The length of the garden is 8 meters more than 4 times its width. Hana writes the expression 4w + 8 to represent the length, where w is the width in meters. She then realizes she needs to write an expression for the total perimeter of the garden. Her friend Matiu suggests the expression 2(4w + 8 + w) for the perimeter. Hana thinks the perimeter can also be written as 10w + 16. Are the expressions 2(4w + 8 + w) and 10w + 16 equivalent? Show your work to prove whether or not they are equivalent. Answer: Yes, they are equivalent. Solution: Start with Matiu's expression: 2(4w + 8 + w). Combine like terms inside the parentheses: 4w + w = 5w. So inside we have 5w + 8.
   Full step-by-step solution

   Step 1: Start with Matiu's expression: 2(4w + 8 + w). Step 2: Combine like terms inside the parentheses: 4w + w = 5w. So inside we have 5w + 8. Step 3: The expression becomes 2(5w + 8). Step 4: Distribute the 2: 2 * 5w = 10w and 2 * 8 = 16. Step 5: This gives 10w + 16, which is exactly Hana's expression. Step 6: Since we transformed 2(4w + 8 + w) into 10w + 16 using algebraic rules, the two expressions are equivalent. The answer is Yes, they are equivalent.

6. Rewrite 6(2x + 1) + 3(4x + 2) in simplest form. Answer: 24x + 12 Solution: Distribute 6 into (2x + 1): 6 × 2x = 12x, 6 × 1 = 6, so 6(2x + 1) = 12x + 6. Distribute 3 into (4x + 2): 3 × 4x = 12x, 3 × 2 = 6, so 3(4x + 2) = 12x + 6.
   Full step-by-step solution

   Step 1: Distribute 6 into (2x + 1): 6 × 2x = 12x, 6 × 1 = 6, so 6(2x + 1) = 12x + 6. Step 2: Distribute 3 into (4x + 2): 3 × 4x = 12x, 3 × 2 = 6, so 3(4x + 2) = 12x + 6. Step 3: Add the results: (12x + 6) + (12x + 6) = 12x + 12x + 6 + 6 = 24x + 12. The simplest form is 24x + 12.

7. Rewrite 8(3x + 7) - 2(5x - 9) in simplest form. Answer: 14x + 74 Solution: Distribute 8 into (3x + 7): 8 * 3x = 24x, 8 * 7 = 56, so 8(3x + 7) = 24x + 56. Distribute -2 into (5x - 9): -2 * 5x = -10x, -2 * (-9) = 18, so -2(5x - 9) = -10x + 18.
   Full step-by-step solution

   Step 1: Distribute 8 into (3x + 7): 8 * 3x = 24x, 8 * 7 = 56, so 8(3x + 7) = 24x + 56. Step 2: Distribute -2 into (5x - 9): -2 * 5x = -10x, -2 * (-9) = 18, so -2(5x - 9) = -10x + 18. Step 3: Combine the two results: (24x + 56) + (-10x + 18) = 24x - 10x + 56 + 18. Step 4: Combine like terms: 24x - 10x = 14x, and 56 + 18 = 74. Step 5: The simplified expression is 14x + 74. The answer is 14x + 74.

8. Rewrite 3(5x + 7) - 2(3x - 5) in simplest form. Answer: 9x + 31 Solution: Expand the first group: 3(5x + 7) = 3*5x + 3*7 = 15x + 21 Expand the second group: -2(3x - 5) = -2*3x + (-2)*(-5) = -6x + 10 Combine the expanded forms: (15x + 21) + (-6x + 10) Combine like terms: 15x - 6x = 9x and 21 + 10 = 31 The simplest form is 9x + 31.
   Full step-by-step solution

   Step 1: Expand the first group: 3(5x + 7) = 3*5x + 3*7 = 15x + 21 Step 2: Expand the second group: -2(3x - 5) = -2*3x + (-2)*(-5) = -6x + 10 Step 3: Combine the expanded forms: (15x + 21) + (-6x + 10) Step 4: Combine like terms: 15x - 6x = 9x and 21 + 10 = 31 Step 5: The simplest form is 9x + 31. The answer is 9x + 31.