Writing Linear Equations - Math 1

The points $(-3, 2)$ , $(1, 6)$ , and $(5, k)$ are collinear. What is the equation of the line containing these points?
Answer: $y = x + 5$ When you see three points described as collinear, you need to find the line they all lie on. Since you already have two complete points, you can find the slope and equation without even solving for k. Start by finding the slope using the two complete points $(-3, 2)$ and $(1, 6)$ . Using the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 2}{1 - (-3)} = \frac{4}{4} = 1$ Now use point-slope form with either point. Using $(1, 6)$ : $y - 6 = 1(x - 1)$ , which simplifies to $y = x + 5$ . You can verify this works with the other point: when $x = -3$ , $y = -3 + 5 = 2$ ✓ Looking at the wrong answers: Choice A ( $y = 2x + 5$ ) uses an incorrect slope of 2 instead of 1 - this might come from miscalculating $\frac{4}{4}$ as $\frac{4}{2}$ . Choice B ( $y = x + 2$ ) has the right slope but wrong y-intercept, likely from an error in the point-slope algebra. Choice D ( $y = 2x + 8$ ) combines both errors - wrong slope and wrong y-intercept. Study tip: For collinear point problems, you often don't need to find the missing coordinate first. Calculate the slope from the two complete points, then use point-slope form. Always double-check by substituting both known points back into your final equation.
Line $m$ is perpendicular to the line $3x + 4y = 8$ and passes through the point where this line intersects the y-axis. What is the equation of line $m$ ?
Answer: $y = \frac{4}{3}x + 2$ When you see a question about perpendicular lines, you need two key pieces of information: the slope relationship between perpendicular lines, and a point the new line passes through. First, find the slope of the given line $3x + 4y = 8$ . Convert to slope-intercept form by solving for $y$ : $4y = -3x + 8$ $y = -\frac{3}{4}x + 2$ The slope is $-\frac{3}{4}$ . Since perpendicular lines have slopes that are negative reciprocals of each other, line $m$ has slope $\frac{4}{3}$ . Next, find where the original line intersects the y-axis. From the equation $y = -\frac{3}{4}x + 2$ , when $x = 0$ , $y = 2$ . So the y-intercept is $(0, 2)$ . Since line $m$ passes through $(0, 2)$ with slope $\frac{4}{3}$ , its equation is $y = \frac{4}{3}x + 2$ . This matches choice C. Choice A has the correct slope but wrong y-intercept ( $-2$ instead of $2$ ). Choice B uses the original line's slope instead of the perpendicular slope. Choice D has the wrong slope ( $-\frac{4}{3}$ instead of $\frac{4}{3}$ ) — this would be perpendicular to a line with slope $\frac{3}{4}$ , not $-\frac{3}{4}$ . Study tip: Always remember that perpendicular slopes multiply to $-1$ . When finding a perpendicular slope, flip the fraction and change the sign. Also, carefully track your positive and negative signs throughout the problem.
A line passes through points $(2, -3)$ and $(6, 9)$ . If this line is shifted vertically so that it passes through the point $(4, 2)$ , what is the equation of the new line?
Answer: $y = 3x - 10$ First, find the slope of the original line: $m = \frac{9-(-3)}{6-2} = \frac{12}{4} = 3$ . The new line has the same slope but passes through $(4, 2)$ . Using point-slope form: $y - 2 = 3(x - 4)$ , which simplifies to $y = 3x - 10$ . Choice B results from forgetting to distribute the 3 in point-slope form. Choice C uses the y-coordinate as the y-intercept. Choice D comes from using the wrong point coordinates.
A rental car company charges a flat fee plus a rate per mile driven. The total cost for driving 100 miles is $85, and the total cost for driving 250 miles is $145. If someone drives exactly 200 miles, what would be their total cost?
Answer: The total cost would be $125 for this distance Let $C$ = total cost and $m$ = miles driven. We have points $(100, 85)$ and $(250, 145)$ . The rate per mile is $\frac{145-85}{250-100} = \frac{60}{150} = 0.4$ dollars per mile. Using point-slope form with $(100, 85)$ : $C - 85 = 0.4(m - 100) = 0.4m - 40$ , so $C = 0.4m + 45$ . When $m = 200$ : $C = 0.4(200) + 45 = 80 + 45 = 125$ . Choice B is correct.
The relationship between temperature $T$ (in degrees Fahrenheit) and the number of cricket chirps $c$ per minute is approximately linear. When the temperature is 68°F, there are 120 chirps per minute. When the temperature is 80°F, there are 180 chirps per minute. According to this model, at what temperature would you expect to hear exactly 200 chirps per minute?
Answer: The temperature would be approximately 84°F at this chirp rate We have points $(68, 120)$ and $(80, 180)$ where the first coordinate is temperature and second is chirps. The slope is $\frac{180-120}{80-68} = \frac{60}{12} = 5$ chirps per degree. Using point-slope form with $(68, 120)$ : $c - 120 = 5(T - 68)$ , so $c = 5T - 220$ . When $c = 200$ : $200 = 5T - 220$ , so $5T = 420$ and $T = 84$ °F. Choice A is correct.
If the linear pattern continues, which equation represents the bacteria population $P$ after $t$ hours?
Answer: $P = 30t + 90$ When you encounter a problem about linear growth with two data points, you're looking to find the equation of a line in the form $P = mt + b$ , where $m$ is the rate of change (slope) and $b$ is the y-intercept. First, calculate the slope using the two given points: (2, 150) and (5, 240). The slope is $m = \frac{240 - 150}{5 - 2} = \frac{90}{3} = 30$ bacteria per hour. This means the population increases by 30 bacteria each hour. Now use the point-slope form with either data point. Using (2, 150): $P - 150 = 30(t - 2)$ . Expanding: $P - 150 = 30t - 60$ , so $P = 30t + 90$ . You can verify this works with the second point: $P = 30(5) + 90 = 240$ ✓ Looking at the wrong answers: Choice A gives a slope of 45, which doesn't match our calculated rate of 30 bacteria per hour. Choice B has the correct slope of 30, but using the y-intercept of 150 would mean 150 bacteria at time zero—however, plugging in $t = 2$ gives $P = 210$ , not the required 150. Choice C has an inflated slope of 90 and would predict far too many bacteria at any given time. The key strategy here is always to calculate the slope first from your two points, then use point-slope form to find the equation. Don't be tempted by answer choices that simply use one of the given data points as the y-intercept—the y-intercept represents the population at $t = 0$ , not at your given time points.
A line has equation $y = mx + b$ where $m$ and $b$ are constants. If the line passes through $(-2, 7)$ and has the same slope as the line connecting $(0, 3)$ and $(4, 11)$ , what is the value of $m + b$ ?
Answer: $13$ When you encounter a linear equation problem involving a point and slope information, you need to find both the slope $m$ and y-intercept $b$ to determine the line's equation. First, find the slope using the two given points $(0, 3)$ and $(4, 11)$ . The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$ , so $m = \frac{11 - 3}{4 - 0} = \frac{8}{4} = 2$ . Now you know the line has slope $m = 2$ and passes through $(-2, 7)$ . Substitute these values into $y = mx + b$ : $7 = 2(-2) + b$ , which gives $7 = -4 + b$ , so $b = 11$ . Therefore, $m + b = 2 + 11 = 13$ . Looking at the wrong answers: Choice A ( $11$ ) is just the value of $b$ alone—you might get this if you forgot to add $m$ . Choice B ( $9$ ) could result from a sign error when solving for $b$ , perhaps calculating $7 = -4 + b$ as $b = 7 - 4 = 3$ , then getting $m + b = 2 + 3 = 5$ ... actually, this likely comes from miscalculating the slope as $\frac{8}{4} = 1$ instead of $2$ , giving $m + b = 1 + 8 = 9$ . Choice D ( $15$ ) might result from an error in the slope calculation, perhaps getting $m = 4$ instead of $2$ . Remember: always double-check your slope calculation using the slope formula, and verify your final equation by substituting the given point back into $y = mx + b$ .
Which equation represents the total cost $C$ (in dollars) for driving $m$ miles?
Answer: $C = 0.5m + 20$ This problem tests your ability to create a linear equation from two data points, which is a fundamental skill in algebra. When you see a scenario involving a base fee plus a rate per mile (or per unit), you're dealing with a linear relationship where the total cost equals the fixed cost plus the variable cost. To find the equation, you need to determine both the rate per mile (slope) and the base fee (y-intercept). Using the two given points: (50, 45) and (120, 80), calculate the slope: $\text{slope} = \frac{80 - 45}{120 - 50} = \frac{35}{70} = 0.5$ dollars per mile. Now use either point to find the base fee. With (50, 45): $45 = 0.5(50) + b$ , so $45 = 25 + b$ , which gives $b = 20$ . Therefore, $C = 0.5m + 20$ . Choice A ( $C = 1.4m + 15$ ) uses an incorrect slope calculation—likely from dividing incorrectly or mixing up the coordinate values. Choice B ( $C = 0.5m + 45$ ) correctly identifies the slope but mistakenly uses one of the total costs (45) as the base fee instead of calculating the y-intercept. Choice C ( $C = 35m + 0.5$ ) appears to use the difference in costs (35) as the slope and the correct slope (0.5) as the y-intercept—a complete reversal of the correct values. When solving linear word problems, always identify your two points clearly, calculate slope carefully using the slope formula, then substitute back to find the y-intercept. Double-check by testing both original points in your final equation.
A line passes through the point $(4, -1)$ and has the same y-intercept as the line $2x - 3y = 12$ . What is the equation of this line?
Answer: $y = \frac{3}{4}x - 4$ When you encounter a problem about finding a line equation with specific conditions, break it down into parts: identify what you know and what you need to find. First, find the y-intercept of the given line $2x - 3y = 12$ . Convert to slope-intercept form by solving for y: $-3y = -2x + 12$ , so $y = \frac{2}{3}x - 4$ . The y-intercept is -4. Now you need a line that passes through $(4, -1)$ and has y-intercept -4. Using the slope-intercept form $y = mx + b$ , you know $b = -4$ , so your equation looks like $y = mx - 4$ . To find the slope $m$ , substitute the point $(4, -1)$ : $-1 = m(4) - 4$ . Solving: $-1 + 4 = 4m$ , so $3 = 4m$ , which gives $m = \frac{3}{4}$ . Therefore, the equation is $y = \frac{3}{4}x - 4$ , which is answer D. Looking at the wrong answers: A has the wrong slope ( $-\frac{3}{4}$ instead of $\frac{3}{4}$ ) and wrong y-intercept (4 instead of -4). B has the slope from the original line ( $\frac{2}{3}$ ) rather than calculating the new slope needed. C has the correct slope but uses the wrong y-intercept (4 instead of -4), possibly from misreading the original equation. Remember: when a line must pass through a specific point and have a given y-intercept, always use the point to find the slope after identifying the y-intercept. Don't assume the slopes will be the same between different lines.
Line $L$ has slope $\frac{2}{3}$ and passes through the point where the lines $y = 5x - 7$ and $x = 2$ intersect. What is the equation of line $L$ ?
Answer: $y = \frac{2}{3}x + \frac{5}{3}$ First, find the intersection point. When $x = 2$ , $y = 5(2) - 7 = 3$ , so the point is $(2, 3)$ . Using point-slope form with slope $\frac{2}{3}$ : $y - 3 = \frac{2}{3}(x - 2) = \frac{2}{3}x - \frac{4}{3}$ , so $y = \frac{2}{3}x - \frac{4}{3} + 3 = \frac{2}{3}x + \frac{5}{3}$ . Choice B results from using 3 directly as the y-intercept. Choice C comes from a sign error when solving for the y-intercept. Choice D uses -3 as the y-intercept.
Which equation represents the temperature $T$ (in °F) as a function of time $h$ hours after 6 AM?
Answer: $T = 4h + 58$ The temperature rises from 58°F to 74°F in 4 hours (6 AM to 10 AM), so the rate is $\frac{74-58}{4} = 4$ degrees per hour. At $h = 0$ (6 AM), $T = 58$ , so the equation is $T = 4h + 58$ . Choice B uses the total temperature change as the rate. Choice C uses 74 as the y-intercept instead of the starting temperature. Choice D uses half the correct rate.
A line passes through points $(-2, 7)$ and $(1, -5)$ . What is the equation of the line that is parallel to this line and passes through the origin?
Answer: $y = -4x$ When you encounter parallel line problems, 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 the line through $(-2, 7)$ and $(1, -5)$ , use the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-5 - 7}{1 - (-2)} = \frac{-12}{3} = -4$ . Since parallel lines have the same slope, your target line also has slope $-4$ . The line you're looking for passes through the origin $(0, 0)$ with slope $-4$ . Using point-slope form: $y - 0 = -4(x - 0)$ , which simplifies to $y = -4x$ . This confirms that choice B is correct. Let's examine why the other answers are wrong. Choice A gives $y = -4x + 3$ , which has the correct slope but the wrong y-intercept—this line doesn't pass through the origin since when $x = 0$ , $y = 3$ . Choice C offers $y = 4x$ , which passes through the origin but has slope $4$ instead of $-4$ , making it perpendicular to the original line, not parallel. Choice D presents $y = -\frac{1}{4}x$ , which passes through the origin but has the wrong slope—this would actually be perpendicular to the original line since $(-4) \times (-\frac{1}{4}) = 1$ . Remember: for parallel line problems, match the slope exactly, then use the given point to find the y-intercept. Lines through the origin always have y-intercept zero.
The line passing through points $A(2, k)$ and $B(6, 3k-1)$ has slope $\frac{1}{2}$ . What is the equation of this line?
Answer: $y = \frac{1}{2}x + \frac{1}{2}$ First, find $k$ using the slope formula: $\frac{3k-1-k}{6-2} = \frac{1}{2}$ , so $\frac{2k-1}{4} = \frac{1}{2}$ . Cross-multiplying: $2(2k-1) = 4$ , so $4k-2 = 4$ , giving $k = \frac{3}{2}$ . Point $A$ is $(2, \frac{3}{2})$ . Using point-slope form: $y - \frac{3}{2} = \frac{1}{2}(x - 2)$ , which gives $y = \frac{1}{2}x + \frac{1}{2}$ . Choice A results from a sign error in the calculation. Choice C uses $k = 1$ instead of the correct value. Choice D uses $k = 2$ .

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