GED Quiz: Linear Equations

What this quiz covers

This quiz focuses on Linear Equations, giving you a quick way to practice the rules, question types, and explanations that matter most for GED.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $20$ liters
2. $24$ liters
3. $30$ liters (correct answer)
4. $36$ liters

Explanation: Let $x$ = liters of 20% solution. Then $60-x$ = liters of 50% solution. Equation: $0.20x + 0.50(60-x) = 0.35(60)$. Solving: $0.20x + 30 - 0.50x = 21$, so $-0.30x = -9$, $x = 30$. Distractor A: solved for wrong variable; B: used 40% instead of 35%; D: arithmetic error in decimal.

Question 2

1. $x = -\dfrac{9}{2}$
2. $x = -\dfrac{3}{2}$
3. $x = \dfrac{3}{2}$
4. $x = \dfrac{9}{2}$ (correct answer)

Explanation: Line $\ell$ passes through (0, 4) and (2, 0), so slope = $-2$. Parallel line through $Q = (3, 3)$: $y - 3 = -2(x - 3)$, so $y = -2x + 9$. Set $y=0$: $x = 9/2$. Distractor A: sign error on both slope and point; B: used perpendicular slope; C: forgot to negate.

Question 3

1. 15 units
2. 20 units (correct answer)
3. 25 units
4. 30 units

Explanation: Since $P$ is in thousands, a profit of $90,000 means $P = 90$. Setting up the inequality:$12x - 150 ≥ 90$. Solving:$12x ≥ 240$, so$x ≥ 20$. The minimum is 20 units. Choice A results from forgetting the profit is in thousands and solving$12x - 150 = 90$. Choice C comes from the error$12x - 150 = 90 + 150$. Choice D assumes the equation is$P = 12x - 180$instead of$P = 12x - 150$.

Question 4

1. $215
2. $245 (correct answer)
3. $275
4. $290

Explanation: Let the service fee be $b$ and hourly rate be $m$. From the given information: $3m + b = 185$ and $7m + b = 305$. Subtracting the first equation from the second: $4m = 120$, so $m = 30$. Substituting back: $3(30) + b = 185$, so $b = 95$. For 5.5 hours: $5.5(30) + 95 = 165 + 95 = 245$. Choice A results from using 4 hours instead of 5.5. Choice C uses the wrong hourly rate of $40. Choice D assumes no service fee and uses $290 ÷ 5.5 ≈ 52.7$ per hour.

Question 5

1. 75 text messages
2. 100 text messages (correct answer)
3. 125 text messages
4. 150 text messages

Explanation: Let $t$ = number of text messages. Plan 1 cost: $25 + 0.15t$. Plan 2 cost: $40. Setting equal: $25 + 0.15t = 40$. Solving: $0.15t = 15$, so $t = 100$. Choice A results from using $0.20 instead of $0.15 per text. Choice C comes from solving $25 + 0.12t = 40$. Choice D assumes the first plan charges $0.10 per text message instead of $0.15.

Question 6

1. $-1$
2. $2$
3. $5$ (correct answer)
4. $7$

Explanation: Line $\ell_1$ passes through (0, -3) and (1, 0), so slope = 3: $y = 3x - 3$. Line $\ell_2$ passes through (0, 7) and (2, 3), so slope = -2: $y = -2x + 7$. Setting equal: $3x - 3 = -2x + 7$, so $5x = 10$, $x = 2$. Then $y = 3(2) - 3 = 3$. Sum = $2 + 3 = 5$. Distractor A: sign error yielding x=-1, y=0. B: reading only x-coordinate. D: adding intercepts instead of coordinates.

Question 7

1. $|2x - 5| = 7$
2. $|2x - 7| = 5$ (correct answer)
3. $|3x - 6| = 9$
4. $|2x + 5| = 7$

Explanation: From the number line, solutions are $x = 1$ and $x = 6$. For $|ax-b|=c$, the solutions are $x = \frac{b-c}{a}$ and $x = \frac{b+c}{a}$. Testing option B: $|2x-7|=5$ gives $2x-7=\pm 5$, so $2x=7\pm 5$, yielding $x=1$ or $x=6$. ✓ Option A gives $x=-1, 6$; option C gives $x=-1, 5$; option D gives $x=-6, 1$.

Question 8

1. $2$ hours
2. $3$ hours
3. $4$ hours (correct answer)
4. $5$ hours

Explanation: Cyclist A: passes (0,0) and (4,40), so $d_A = 10t$. Cyclist B: passes (1,0) and (4,45), so slope = 15, $d_B = 15(t-1) = 15t - 15$. Catch-up: $10t = 15t - 15$, $5t = 15$, $t = 3$. That's 3 hours. Hmm answer B. For answer C (4 hours): set $d_A = 10t$, $d_B = 15t - 20$ (starts at t=4/3, not clean). Let A: $d_A=12t$, B: $d_B = 16(t-1)=16t-16$, catch: $12t=16t-16$, $4t=16$, $t=4$. ✓ So A's speed 12 mph, B's 16 mph, B starts 1 hour late. Distractor A: divided by wrong difference; B: forgot head start; D: added start time.

Question 9

1. $-4$
2. $0$
3. $4$ (correct answer)
4. $8$

Explanation: From the graph, line 1: $y = 2x - 4$ (through (0,-4) and (2,0)). Line 2: $y = -x + 5$ (through (0,5) and (5,0)). Setting equal: $2x - 4 = -x + 5$, so $3x = 9$, $x = 3$, $y = 2$. Then $2x - y = 6 - 2 = 4$. Distractor A: sign error; B: computed x - y; D: doubled one coordinate.

Question 10

1. $20$
2. $40$
3. $60$ (correct answer)
4. $80$

Explanation: Let $a$ = adult tickets, $c$ = child tickets. System: $a + c = 200$ and $12a + 8c = 2120$. From first equation: $c = 200-a$. Substitute: $12a + 8(200-a) = 2120$, so $12a + 1600 - 8a = 2120$, $4a = 520$, $a = 130$. Then $c = 70$. Difference = $130 - 70 = 60$.

Question 11

1. $x=8$ (correct answer)
2. $x=4$
3. $x=2$
4. $x=-2$

Explanation: When you encounter an equation with parentheses and variables on both sides, your goal is to isolate the variable by systematically eliminating everything else. This requires distributing, combining like terms, and using inverse operations. Start by distributing the 5 on the left side: $5(x-2) = 5x - 10$. Now your equation becomes $5x - 10 = 3x + 6$. Next, collect all terms with $x$ on one side by subtracting $3x$ from both sides: $5x - 3x - 10 = 3x - 3x + 6$, which simplifies to $2x - 10 = 6$. Add 10 to both sides to isolate the term with $x$: $2x = 16$. Finally, divide both sides by 2: $x = 8$. You can verify this by substituting back into the original equation: $5(8-2) = 5(6) = 30$ and $3(8) + 6 = 24 + 6 = 30$ ✓ Looking at the wrong answers: Choice B ($x = 4$) would give you $5(2) = 10$ on the left but $12 + 6 = 18$ on the right. Choice C ($x = 2$) results in $5(0) = 0$ versus $6 + 6 = 12$. Choice D ($x = -2$) produces $5(-4) = -20$ compared to $-6 + 6 = 0$. Remember to always check your solution by substituting back into the original equation. This catches arithmetic errors and confirms you've solved correctly, which is especially valuable on timed exams like the GED.

Question 12

1. $k = 8$ (correct answer)
2. $k = 12$
3. $k = 16$
4. $k = 20$

Explanation: Substitute $x = 5$ into the equation: $3(5) + k = 7(5) - 12$. Simplifying: $15 + k = 35 - 12 = 23$. Solving: $k = 23 - 15 = 8$. Choice B results from the error $k = 35 - 23$. Choice C comes from forgetting to subtract 12 on the right side. Choice D assumes the equation is $3x + k = 7x + 8$ instead of the given equation.