Linear Equations - GED Math

Refer to the graph, which shows the distance (in miles) traveled by two cyclists as a function of time (in hours). Cyclist A started at time 0, while Cyclist B started at a later time. Use the graph to determine how many hours after Cyclist A starts does Cyclist B catch up.
Answer: $4$ hours 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.
The diagram below represents a mixture problem. A chemist combines a solution that is $20\%$ acid with a solution that is $50\%$ acid to produce $60$ liters of a solution that is $35\%$ acid. Based on the diagram, how many liters of the $20\%$ solution are used?
Answer: $30$ liters 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.
The chart below displays ticket pricing at a theater: adult tickets cost $12 and child tickets cost $8. On a certain night, the theater sold a total of $200$ tickets for total revenue of $\$2{,}120$ . Use the chart to find how many MORE adult tickets than child tickets were sold.
Answer: $60$ 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 = $130 - 70 = 60$ .
A cell phone plan charges $25 per month plus $0.15 for each text message sent. Another plan charges $40 per month with unlimited texting. After how many text messages will the two plans cost the same amount?
Answer: 100 text messages 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.
The coordinate plane below shows a line $\ell$ and a point $Q$ not on the line. Based on the graph, write the equation of the line through $Q$ parallel to $\ell$ , and find its $x$ -intercept.
Answer: $x = \dfrac{9}{2}$ 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.
Solve the equation $5(x-2)=3x+6$ for $x$ .
Answer: $x=8$ 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.
A company's profit $P$ (in thousands of dollars) is related to the number of units sold $x$ by the equation $P = 12x - 150$ . If the company needs a profit of at least $90,000, what is the minimum number of units they must sell?
Answer: 20 units 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 $P = 12x - 150$ .
For what value of $k$ will the equation $3x + k = 7x - 12$ have the solution $x = 5$ ?
Answer: $k = 8$ 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.
A plumber charges a service call fee plus an hourly rate. After working 3 hours, the total cost is $185. After working 7 hours, the total cost is $305. If the plumber worked for 5.5 hours, what would be the total cost?
Answer: $245 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.
The number line below shows the solution to a linear equation of the form $|ax - b| = c$ where $a, b, c$ are positive integers with $a > 1$ . Based on the number line, which equation has exactly this solution set?
Answer: $|2x - 7| = 5$ 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 $x=-6, 1$ .
The coordinate plane below shows two lines, $\ell_1$ and $\ell_2$ , that intersect at point $P$ . Based on the graph shown, what is the sum of the $x$ - and $y$ -coordinates of $P$ ?
Answer: $5$ 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.
Use the system of equations displayed in the coordinate plane below to determine the value of $2x - y$ at the point of intersection.
Answer: $4$ 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.

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