Solving Linear Equations: math test solutions, Grade 8 – download the PDF

1 Warm-up: one and two steps / 4 pts

1. \(2x = 16\), so \(x = 8\). (1 pt)
2. \(-3y = -15\), so \(y = 5\). (1 pt)
3. \(\dfrac{m}{5} = -3\), so \(m = -15\). (1 pt)
4. \(7 = 2p - 6 + 1 = 2p - 5\), so \(2p = 12\) and \(p = 6\). (1 pt)

2 Variables on both sides / 4 pts

1. \(4x + 1 = 17\), so \(4x = 16\) and \(x = 4\). Check: \(25 = 25\). (2 pts)
2. \(10n - 5 = 9n + 6\), so \(n = 11\). Check: \(5 \cdot 21 = 105\) and \(3 \cdot 35 = 105\). (2 pts)

3 Fractions and decimals / 3 pts

1. Multiply by 6: \(4x - 3 = x + 6\) (1 pt), so \(3x = 9\) and \(x = 3\) (1 pt).
2. \(0.5x = 2.5\), so \(x = 5\). (1 pt)

4 How many solutions? / 3 pts

1. Right side: \(5x - 5 + 3 = 5x - 2\). The statement \(5x - 2 = 5x - 2\) is always true: infinitely many solutions. (1 pt)
2. \(6x + 3 = 6x + 5\) gives \(3 = 5\), false: no solution. (1 pt)
3. \(2x = -8\), so \(x = -4\): exactly one solution. (1 pt)

5 Print shop / 3 pts

With \(p\) pages: \(6 + 0.20p = 0.35p\) (1 pt).

\(6 = 0.15p\), so \(p = 40\) (1 pt).

The cost is \(0.35 \cdot 40 = 14\) dollars. For 40 pages both shops charge $14. (1 pt)

6 Consecutive even integers / 3 pts

Let the integers be \(n\), \(n + 2\), \(n + 4\): \(3n + 6 = 90\) (1 pt).

\(3n = 84\), so \(n = 28\) (1 pt).

The integers are 28, 30, and 32, and \(28 + 30 + 32 = 90\). (1 pt)