Writing Linear Equations - Accuplacer Aaf

Line $k$ passes through points $(-2, 5)$ and $(6, -3)$ . Which equation represents a line parallel to line $k$ that has a $y$ -intercept of $7$ ?
Answer: $y = -x + 7$ When you encounter questions about parallel lines, remember that parallel lines have identical slopes but different y-intercepts. Your first step is always to find the slope of the given line. To find the slope of line $k$ through points $(-2, 5)$ and $(6, -3)$ , use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ . Substituting the coordinates: $m = \frac{-3 - 5}{6 - (-2)} = \frac{-8}{8} = -1$ . So line $k$ has a slope of $-1$ . Since you need a parallel line with a y-intercept of 7, the equation must have the same slope ( $-1$ ) but pass through point $(0, 7)$ . Using slope-intercept form $y = mx + b$ , this gives you $y = -1x + 7$ or $y = -x + 7$ . This matches choice C. Looking at the wrong answers: Choice A has slope $-\frac{1}{2}$ , which would make it neither parallel nor perpendicular to line $k$ . Choice B has slope $1$ , making it perpendicular to line $k$ (since $(-1) \times 1 = -1$ ), not parallel. Choice D has slope $\frac{1}{2}$ , which again is neither parallel nor perpendicular to the original line. For parallel line questions on the Accuplacer, always work systematically: find the original slope first, then match that slope with the given y-intercept. Don't get distracted by perpendicular slopes (negative reciprocals) or random slopes that might look appealing.
The monthly cost to run a factory is a linear function of the number of units produced. In May, 1,500 units were produced and the cost was $28,000. In June, 2,000 units were produced and the cost was $34,000. What is the equation for the cost $C$ as a function of the number of units $u$ ?
Answer: $C(u) = 12u + 10000$ The information gives two points: (1500, 28000) and (2000, 34000). First, calculate the slope (cost per unit): $m = \frac{34000 - 28000}{2000 - 1500} = \frac{6000}{500} = 12$ . This is the variable cost. Next, use the point-slope form with the point (1500, 28000): $C - 28000 = 12(u - 1500)$ . Distribute the 12: $C - 28000 = 12u - 18000$ . Add 28000 to both sides to find the equation in slope-intercept form: $C(u) = 12u + 10000$ . The fixed cost is $10,000.
A line has a slope of $\frac{2}{5}$ and passes through the point where $x = -10$ and $y = 6$ . If this line also passes through the point $(a, 16)$ , what is the value of $a$ ?
Answer: $a = 15$ When you encounter a problem about finding a point on a line given the slope and another point, you're working with the fundamental relationship between slope and coordinates. The slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ is your key tool. You know the line has slope $\frac{2}{5}$ and passes through $(-10, 6)$ and $(a, 16)$ . Using the slope formula: $\frac{2}{5} = \frac{16 - 6}{a - (-10)} = \frac{10}{a + 10}$ Cross-multiplying: $2(a + 10) = 5(10)$ , which gives you $2a + 20 = 50$ . Solving: $2a = 30$ , so $a = 15$ . This confirms answer C is correct. Let's examine why the other answers are wrong. Answer A ( $a = 35$ ) would give a slope of $\frac{10}{45} = \frac{2}{9}$ , which is too small. Answer B ( $a = 25$ ) yields a slope of $\frac{10}{35} = \frac{2}{7}$ , also incorrect. Answer D ( $a = 40$ ) produces a slope of $\frac{10}{50} = \frac{1}{5}$ , which is exactly half the required slope. These incorrect answers likely come from arithmetic errors in cross-multiplication or from mixing up the coordinate positions in the slope formula. Notice that answer D gives $\frac{1}{5}$ , which students might choose if they forgot to double-check their fraction arithmetic. Remember: always substitute your answer back into the slope formula to verify it gives the original slope. This catch-and-check approach prevents careless errors on coordinate geometry problems.
Line $L$ is defined by the equation $4x - 6y = 9$ . Which of the following is an equation for a line that is parallel to line $L$ and passes through the point $(-3, 5)$ ?
Answer: $y = \frac{2}{3}x + 7$ First, find the slope of line $L$ by rewriting its equation in slope-intercept form ( $y = mx+b$ ). $4x - 6y = 9 \implies -6y = -4x + 9 \implies y = \frac{-4}{-6}x + \frac{9}{-6} \implies y = \frac{2}{3}x - \frac{3}{2}$ . The slope of line $L$ is $\frac{2}{3}$ . A parallel line must have the same slope. Now, use the point-slope form with the point $(-3, 5)$ and the slope $\frac{2}{3}$ : $y - 5 = \frac{2}{3}(x - (-3))$ . This simplifies to $y - 5 = \frac{2}{3}(x + 3) \implies y - 5 = \frac{2}{3}x + 2$ . Adding 5 to both sides gives $y = \frac{2}{3}x + 7$ .
A line passes through the point $(-6, 2)$ and has a slope of $\frac{2}{3}$ . Which of the following is the equation of the line in standard form $Ax + By = C$ , where $A, B$ , and $C$ are integers with $A > 0$ ?
Answer: $2x - 3y = -18$ Start with the point-slope form: $y - y_1 = m(x - x_1)$ . Substituting the given point and slope gives $y - 2 = \frac{2}{3}(x - (-6))$ , which is $y - 2 = \frac{2}{3}(x + 6)$ . To eliminate the fraction, multiply the entire equation by 3: $3(y - 2) = 2(x + 6)$ . Distribute on both sides: $3y - 6 = 2x + 12$ . Rearrange into standard form $Ax + By = C$ by moving the x-term to the left and the constant to the right: $-2x + 3y = 18$ . Since the convention is for $A$ to be positive, multiply the entire equation by -1 to get $2x - 3y = -18$ .
A cell phone plan charges a monthly fee plus a rate per minute of usage. In January, 120 minutes of usage cost $45 total. In February, 200 minutes of usage cost $65 total. What equation represents the total monthly cost $C$ for $m$ minutes of usage?
Answer: $C = 0.25m + 25$ Using points (120, 45) and (200, 65), the rate per minute is (65 - 45)/(200 - 120) = 20/80 = 0.25 dollars per minute. Using point-slope form with (120, 45): C - 45 = 0.25(m - 120), which gives C = 0.25m - 30 + 45 = 0.25m + 25. The monthly fee is $25 and the rate is $0.25 per minute. Choice A uses the correct rate but calculates the monthly fee as $15 instead of $25. Choice C uses an incorrect rate of $0.30. Choice D uses an incorrect rate of $0.20 and wrong monthly fee.
The temperature of a cooling object decreases linearly over time. At $t = 3$ minutes, the temperature is $85°C$ , and the temperature decreases at a rate of $12°C$ per minute. Which equation gives the temperature $T$ (in degrees Celsius) as a function of time $t$ (in minutes)?
Answer: $T = 121 - 12t$ Given that the temperature decreases at 12°C per minute, the slope is -12. Using point-slope form with (3, 85): T - 85 = -12(t - 3), which expands to T - 85 = -12t + 36, so T = -12t + 121 or T = 121 - 12t. Choice A is in point-slope form but wasn't simplified to slope-intercept form. Choice C incorrectly calculates the y-intercept as 49. Choice D incorrectly calculates the y-intercept as 97.
A linear function $g(x)$ satisfies the conditions $g(-2) = 5$ and $g(4) = -7$ . Which of the following defines $g(x)$ ?
Answer: $g(x) = -2x + 1$ The given conditions correspond to the points $(-2, 5)$ and $(4, -7)$ . First, find the slope of the line passing through these points: $m = \frac{-7 - 5}{4 - (-2)} = \frac{-12}{6} = -2$ . Next, use the point-slope form with the point $(-2, 5)$ : $y - 5 = -2(x - (-2))$ . This simplifies to $y - 5 = -2(x + 2) \implies y - 5 = -2x - 4$ . Adding 5 to both sides gives $y = -2x + 1$ . Thus, $g(x) = -2x + 1$ .
What is the equation of the perpendicular bisector of the line segment with endpoints $A(-3, 8)$ and $B(5, -2)$ ?
Answer: $y = \frac{4}{5}x + \frac{11}{5}$ The perpendicular bisector passes through the midpoint of the segment and has a slope that is the negative reciprocal of the segment's slope. First, find the midpoint: $M = (\frac{-3+5}{2}, \frac{8+(-2)}{2}) = (1, 3)$ . Next, find the slope of segment AB: $m_{AB} = \frac{-2 - 8}{5 - (-3)} = \frac{-10}{8} = -\frac{5}{4}$ . The perpendicular slope is $m_{\perp} = \frac{4}{5}$ . Finally, use the point-slope form with the midpoint $(1, 3)$ and the perpendicular slope: $y - 3 = \frac{4}{5}(x - 1) \implies y - 3 = \frac{4}{5}x - \frac{4}{5} \implies y = \frac{4}{5}x + \frac{11}{5}$ .
Which of the following equations represents the total cost, $C$ , as a function of the number of shirts, $n$ ?
Answer: $C = 7n + 50$ The problem provides two points on a linear function: (20, 190) and (50, 400). First, calculate the slope (price per shirt): $m = \frac{400 - 190}{50 - 20} = \frac{210}{30} = 7$ . Then, use the point-slope form with one of the points, such as (20, 190): $C - 190 = 7(n - 20)$ . Distributing the 7 gives $C - 190 = 7n - 140$ . Adding 190 to both sides yields $C = 7n + 50$ . This means the setup fee is $50 and each shirt costs $7.
What is the equation of the line that passes through the x-intercept of $3x - 5y = 15$ and the y-intercept of $y = 2x + 8$ ?
Answer: $y = -\frac{8}{5}x + 8$ First, find the required intercepts. For $3x - 5y = 15$ , the x-intercept occurs when $y = 0$ , so $3x = 15 \implies x = 5$ . The point is $(5, 0)$ . For $y = 2x + 8$ , the y-intercept occurs when $x = 0$ , so $y = 8$ . The point is $(0, 8)$ . Now, find the equation of the line passing through $(5, 0)$ and $(0, 8)$ . The y-intercept is $b = 8$ . The slope is $m = \frac{8 - 0}{0 - 5} = -\frac{8}{5}$ . Using the slope-intercept form, the equation is $y = -\frac{8}{5}x + 8$ .
What is the equation of the line that passes through the point $(-4, 1)$ and is perpendicular to the line with the equation $2x + 5y = 10$ ?
Answer: $y = \frac{5}{2}x + 11$ First, find the slope of the given line by converting its equation to slope-intercept form ( $y = mx + b$ ). $2x + 5y = 10 \implies 5y = -2x + 10 \implies y = -\frac{2}{5}x + 2$ . The slope is $-\frac{2}{5}$ . A line perpendicular to this has a slope that is the negative reciprocal, which is $\frac{5}{2}$ . Now, use the point-slope form with the point $(-4, 1)$ and the new slope: $y - 1 = \frac{5}{2}(x - (-4))$ . This simplifies to $y - 1 = \frac{5}{2}(x + 4) \implies y - 1 = \frac{5}{2}x + 10 \implies y = \frac{5}{2}x + 11$ .
A line passes through the origin and the point of intersection of the lines $y = 2x - 1$ and $x + y = 5$ . What is the equation of this line?
Answer: $y = \frac{3}{2}x$ First, find the point of intersection by solving the system of equations. Substitute the first equation into the second: $x + (2x - 1) = 5$ . This simplifies to $3x - 1 = 5$ , so $3x = 6$ , and $x = 2$ . Substitute $x = 2$ back into the first equation to find y: $y = 2(2) - 1 = 3$ . The point of intersection is $(2, 3)$ . The line in question passes through this point and the origin $(0, 0)$ . The y-intercept is 0. The slope is $m = \frac{3 - 0}{2 - 0} = \frac{3}{2}$ . Therefore, the equation of the line is $y = \frac{3}{2}x$ .
Which of the following is an equation of the line passing through the points $(\frac{1}{2}, -1)$ and $(-\frac{1}{3}, \frac{2}{3})$ ?
Answer: $y = -2x$ First, calculate the slope $m$ using the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ . $m = \frac{\frac{2}{3} - (-1)}{-\frac{1}{3} - \frac{1}{2}} = \frac{\frac{2}{3} + 1}{-\frac{2}{6} - \frac{3}{6}} = \frac{\frac{5}{3}}{-\frac{5}{6}}$ . To divide by a fraction, multiply by its reciprocal: $m = \frac{5}{3} \cdot (-\frac{6}{5}) = -\frac{30}{15} = -2$ . Now use the point-slope form with the point $(\frac{1}{2}, -1)$ : $y - (-1) = -2(x - \frac{1}{2})$ . This simplifies to $y + 1 = -2x + 1$ . Subtracting 1 from both sides gives $y = -2x$ .
What is the equation of the line that passes through the point $(-7, 4)$ and is perpendicular to the y-axis?
Answer: $y = 4$ The y-axis is a vertical line. A line perpendicular to a vertical line must be a horizontal line. The equation of any horizontal line is of the form $y = k$ , where $k$ is a constant equal to the y-coordinate of every point on the line. Since the line must pass through $(-7, 4)$ , its y-coordinate must be 4. Therefore, the equation of the line is $y = 4$ .
Which equation gives the value $V(t)$ of the item, where $t$ is the number of years after 2000?
Answer: $V(t) = -30t + 2800$ First, determine the slope (rate of change). The value decreases by $150 over 5 years, so the slope is $m = \frac{\Delta V}{\Delta t} = \frac{-150}{5} = -30$ dollars per year. Next, identify a point on the line. The year 2010 is $t = 10$ years after 2000, and the value was $2,500. So, we have the point $(10, 2500)$ . Using the point-slope form $V - V_1 = m(t - t_1)$ , we get $V - 2500 = -30(t - 10)$ . Distributing gives $V - 2500 = -30t + 300$ . Adding 2500 to both sides yields $V(t) = -30t + 2800$ .
Which equation models the volume of water $V$ , in gallons, in the tank after $t$ minutes?
Answer: $V(t) = 25t$ The problem requires finding the rate of change of volume (gallons per minute), which is the slope of the linear function. The rate of height change is 2.5 inches/minute. The conversion factor is 10 gallons/inch. To find the rate in gallons per minute, multiply these values: Rate = (2.5 inches/minute) × (10 gallons/inch) = 25 gallons/minute. Since the tank starts empty, the initial value (y-intercept) is 0. Therefore, the equation is $V(t) = 25t$ .
A community garden sells small plots, $s$ , for $45 and large plots, $l$ , for $75. The garden's total revenue from selling all available plots is $9,000. Which of the following equations, in simplified standard form, represents this relationship?
Answer: $3s + 5l = 600$ The total revenue is the sum of the revenue from small plots and large plots. Revenue from small plots is $45s$ and revenue from large plots is $75l$ . The total revenue equation is $45s + 75l = 9000$ . To simplify this equation, find the greatest common divisor (GCD) of 45, 75, and 9000. The GCD of 45 and 75 is 15. Since 9000 is also divisible by 15 (9000 / 15 = 600), we can divide the entire equation by 15. This gives $\frac{45s}{15} + \frac{75l}{15} = \frac{9000}{15}$ , which simplifies to $3s + 5l = 600$ .
A line passes through $C(-3, 5)$ and is perpendicular to the line that passes through $A(7, -1)$ and $B(4, 8)$ . What is the equation of this line?
Answer: $y = \frac{1}{3}x + 6$ First, find the slope of the line through points A and B: $m_{AB} = \frac{8 - (-1)}{4 - 7} = \frac{9}{-3} = -3$ . The line we want is perpendicular to this, so its slope is the negative reciprocal of -3, which is $m_{\perp} = \frac{1}{3}$ . Now, use the point-slope form with the point $C(-3, 5)$ and the perpendicular slope: $y - 5 = \frac{1}{3}(x - (-3))$ . This simplifies to $y - 5 = \frac{1}{3}(x + 3) \implies y - 5 = \frac{1}{3}x + 1$ . Adding 5 to both sides gives $y = \frac{1}{3}x + 6$ .
A line is parallel to $y = 3x - 7$ and has the same y-intercept as the line $4x - 2y = 12$ . What is the equation of this line?
Answer: $y = 3x - 6$ A line parallel to $y = 3x - 7$ must have the same slope, so $m = 3$ . To find the y-intercept of $4x - 2y = 12$ , set $x = 0$ : $4(0) - 2y = 12 \implies -2y = 12 \implies y = -6$ . So, the y-intercept is $b = -6$ . Combining the slope $m = 3$ and y-intercept $b = -6$ gives the equation $y = 3x - 6$ .

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