Equivalent Algebraic Expressions: practice solutions, Grade 7 – download the PDF

1 Parts of an expression ★★★

1. The terms are \(7x\), \(-4y\), and \(9\).
2. The coefficient of \(x\) is \(7\) and the coefficient of \(y\) is \(-4\) (the sign belongs to the term).
3. The constant term is \(9\).

2 Combine like terms: one variable ★★★

1. \(6n + 4n = 10n\).
2. \(11p - 7p = 4p\).
3. \(3k + k = 3k + 1k = 4k\).
4. \(8t - 8t = 0\).

3 Simplify with constants ★★★

1. \(4x + 2x = 6x\) and \(7 + 1 = 8\), so \(6x + 8\).
2. \(9a - 3a = 6a\) and \(5 - 2 = 3\), so \(6a + 3\).
3. \(6b - b = 5b\) and \(10 - 4 = 6\), so \(5b + 6\).

4 Apply the distributive property ★★★

1. \(3x + 15\).
2. \(6y - 12\).
3. \(8a + 2\).
4. \(15 - 5m\).

5 True or false? Test with a number ★★★

1. False. At \(x = 2\): \(4 \cdot 2 + 3 = 11\) but \(7 \cdot 2 = 14\).
2. True. Distributing gives \(2x + 2\).
3. False. At \(x = 1\): \(5 - 1 = 4\), not \(5\). Actually \(5x - x = 4x\).
4. False. At \(x = 3\): \(3 + 3 = 6\) but \(3^2 = 9\). Actually \(x + x = 2x\).

6 Factor a common number ★★★

1. GCF is 4: \(4(x + 2)\).
2. GCF is 3: \(3(3y - 2)\).
3. GCF is 5: \(5(2a + 3)\).
4. GCF is 7: \(7(m + 1)\).

7 Phone plan ★★★

1. The cost is \(25 + 3g\) dollars.
2. \(25 + 3 \cdot 5 = 25 + 15 = 40\). The cost is 40 dollars.

8 Several variables ★★★

1. \(5x + 2x = 7x\) and \(-3y + 7y = 4y\), so \(7x + 4y\).
2. \(-4a - 6a = -10a\) and \(9 - 12 = -3\), so \(-10a - 3\).
3. \(2.5k - 0.5k = 2k\) and \(4 - 1.5 = 2.5\), so \(2k + 2.5\).

9 Fraction coefficients ★★★

1. \(\dfrac{1}{2} + \dfrac{3}{2} = 2\), so \(2x\).
2. \(\dfrac{2}{3} + \dfrac{1}{3} = 1\), so \(y - 5\).
3. \(\dfrac{3}{4} - \dfrac{1}{4} = \dfrac{2}{4} = \dfrac{1}{2}\), so \(\dfrac{1}{2}a\).

10 Distribute with negatives and decimals ★★★

1. \(-2x - 12\).
2. \(-12y + 15\), because \(-3 \cdot (-5) = 15\).
3. \(3a + 5\).
4. \(-m + 8\): the minus sign changes both signs.

11 Expand, then combine ★★★

1. \(3x + 12 + 2x = 5x + 12\).
2. \(10y - 5 - 4y = 6y - 5\).
3. \(2a + 6 + 3a - 3 = 5a + 3\).
4. \(4m + 8 - 2m - 10 = 2m - 2\).

12 Factor with several terms ★★★

1. \(6(2x + 3)\).
2. \(4(2y - 5)\).
3. \(5(3a + 5b)\).
4. \(3(2m - 3n + 4)\). Each check gives back the original expression.

13 Add and subtract expressions ★★★

1. \(5x - 3\).
2. \(4a + 4\).
3. \(6x + 7 - 2x - 3 = 4x + 4\).
4. \(8m + 2 - 10m + 6 = -2m + 8\).

14 Garden bed ★★★

1. Area \(= 4(x + 7) = 4x + 28\) square meters.
2. Perimeter \(= 2 \cdot 4 + 2(x + 7) = 8 + 2x + 14 = 2x + 22\) meters.
3. For \(x = 5\): area \(= 4 \cdot 12 = 48\) m\(^2\) and perimeter \(= 2 \cdot 5 + 22 = 32\) m.

15 Which are equivalent? ★★★

Expanding gives \(2(3x - 4) = 6x - 8\).

1. Equivalent.
2. Not equivalent: the second term should be \(2 \cdot 4 = 8\).
3. Equivalent: \(2 \cdot 3x = 6x\).
4. Not equivalent: \(2 \cdot 3x\) is \(6x\), not \(5x\).

Answer: (a) and (c).

16 Find the mistake ★★★

1. Sam multiplied only the first term. Correct: \(4x + 12\).
2. Lena subtracted 2 from 7 first, which breaks the order of operations (multiplication comes before subtraction). Correct: \(7 - 2x - 2 = 5 - 2x\).

17 Theater tickets ★★★

1. A child ticket costs \(a - 5\). Total \(= 2a + 3(a - 5) = 2a + 3a - 15 = 5a - 15\).
2. \(5 \cdot 12 - 15 = 45\). Check: \(2 \cdot 12 + 3 \cdot 7 = 24 + 21 = 45\). The family pays 45 dollars.

18 Perimeter of a triangle ★★★

1. \((2x + 3) + (x + 5) + (3x - 1) = 6x + 7\) centimeters.
2. \(6 \cdot 4 + 7 = 31\). Check: the sides are 11, 9, and 11, and \(11 + 9 + 11 = 31\). The perimeter is 31 cm.

19 Stickers ★★★

1. \((5n + 12) - (2n + 5) = 5n + 12 - 2n - 5 = 3n + 7\).
2. \(3 \cdot 6 + 7 = 25\). Check: \(42 - 17 = 25\). She has 25 stickers left.

20 Fully factored? ★★★

1. \(2(9x + 12) = 18x + 24\) and \(6(3x + 4) = 18x + 24\).
2. Ben’s. The GCF of 18 and 24 is 6. Inside Ava’s parentheses, \(9x + 12\) still has the common factor 3.
3. The GCF of 12, 30, and 18 is 6: \(6(2a - 5b + 3)\).

21 Prove equivalence ★★★

1. \(3(2x + 1) - (x - 4) = 6x + 3 - x + 4 = 5x + 7\).
2. \(6x - 2 + 4x + 8 = 10x + 6\). The GCF is 2, so \(10x + 6 = 2(5x + 3)\).

22 Two gyms ★★★

1. \(12(m + 1) + 8 = 12m + 12 + 8 = 12m + 20\), the same as \(20 + 12m\).
2. You pay 12 dollars for each of \(m\) months plus one extra month, and then an 8-dollar fee.
3. \(12 \cdot 6 + 20 = 92\). Both gyms cost 92 dollars for 6 months.

23 L-shaped patio ★★★

1. Area \(= 10(x + 8) - 3x = 10x + 80 - 3x = 7x + 80\) square feet.
2. \(7 \cdot 5 + 80 = 115\). Check: \(10 \cdot 13 - 15 = 130 - 15 = 115\). The area is 115 ft\(^2\).

24 Find the missing number ★★★

1. The left side is \(6x + 4k\), so \(4k = 20\) and \(k = 5\).
2. The left side is \(5x + 3k\), so \(3k = 12\) and \(k = 4\).