Middle School Math Quiz: Derive Linear Equations Using Slope

20 questions · exam conditions

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Derive Linear Equations Using SlopeQuestion 1 of 20

A line passes through points $P(-2, 1)$ and $Q(3, -4)$ . To find where this line intersects the y-axis, Miguel uses the fact that slope is constant between any two points. What is the y-intercept of this line?

-1

1

-3

3

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Middle School Math Quiz

Middle School Math Quiz: Derive Linear Equations Using Slope

Practice Derive Linear Equations Using Slope in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Derive Linear Equations Using Slope, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

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.

All questions

Question 1

$-1$ (correct answer)
$1$
$-3$
$3$

Explanation: First find the slope:

m = \frac{-4-1}{3-(-2)} = \frac{-5}{5} = -1

. Using point-slope form with point

P(-2, 1)

y - 1 = -1(x - (-2))

, which simplifies to

y - 1 = -(x + 2)

, so

y = -x - 2 + 1 = -x - 1

. The y-intercept is

-1

. Choice B uses the y-coordinate of point P incorrectly. Choice C forgets to add 1 when solving for the y-intercept. Choice D uses the wrong sign and wrong calculation.

Question 2

A line passes through the points $(2,1)$ and $(6,9)$ . What is the slope $m$ of the line?

$m=\frac{9-1}{6-2}=2$ (correct answer)
$m=\frac{1-9}{2-6}=-2$
$m=\frac{6-2}{9-1}=\frac{1}{2}$
$m=\frac{9-1}{2-6}=-2$

Explanation: This question tests calculating constant slope between two points on a line, understanding any pair gives same m, with similar triangles explaining constancy, and deriving y=mx or y=mx+b. Slope m = (y₂ - y₁)/(x₂ - x₁) is constant for lines; similar triangles show equal angles (parallel sides) and proportional sides, like 6/2=12/4=3, proving constancy; y=mx for origin via m=y/x, y=mx+b using m from points and b from intercept. For (2,1) and (6,9), m = (9-1)/(6-2) = 8/4 = 2. This matches choice B with correct order and positive 2. Errors: wrong sign or order giving -2 in A/D, inverting to 1/2 in C. Process: (1) select points, compute m=(y₂-y₁)/(x₂-x₁), (2) verify if more points, (3) origin y=mx, else b from intercept or point plug-in, (4) equation. Similar triangles proof: shared angle, proportional sides, constant rise/run; mistakes: inverting, varying slope, incorrect form.

Question 3

A proportional relationship is graphed as a line through the origin. If similar triangles formed using this line show that when the horizontal leg increases by 5 units, the vertical leg increases by 8 units, which equation represents this relationship?

$y = 5x + 8$
$y = \frac{5}{8}x$
$y = 8x + 5$
$y = \frac{8}{5}x$ (correct answer)

Explanation: When you see a proportional relationship that passes through the origin, you're working with a direct variation that follows the form

y = kx

, where

k

is the constant of proportionality (also called the slope). The key insight here is understanding what "when the horizontal leg increases by 5 units, the vertical leg increases by 8 units" tells you about the slope. This describes the rise over run: for every 5 units you move horizontally (run), you move 8 units vertically (rise). Therefore, the slope is

\frac{\text{rise}}{\text{run}} = \frac{8}{5}

, making the equation

y = \frac{8}{5}x

. This confirms answer D is correct. Let's examine why the other options are wrong. Choice A (

y = 5x + 8

) has a y-intercept of 8, but proportional relationships must pass through the origin, so the y-intercept must be 0. Choice B (

y = \frac{5}{8}x

) flips the slope by putting the horizontal change (5) in the numerator and vertical change (8) in the denominator—this is backwards. Choice C (

y = 8x + 5

) makes two errors: it treats the vertical change as the slope coefficient (ignoring the horizontal change) and adds a y-intercept of 5, which again violates the "through the origin" requirement. Remember this pattern: when a problem gives you horizontal and vertical changes in a proportional relationship, the slope is always

\frac{\text{vertical change}}{\text{horizontal change}}

. Don't flip this fraction, and remember that proportional relationships never have y-intercepts other than zero.

Question 4

A line passes through points $A(0, -3)$ and $B(4, 5)$ . Using similar triangles, Marcus determines that the slope between any two points on this line is the same. What is the equation of this line?

$y = 2x - 3$ (correct answer)
$y = 2x + 3$
$y = -2x - 3$
$y = \frac{1}{2}x - 3$

Explanation: First, calculate the slope:

m = \frac{5-(-3)}{4-0} = \frac{8}{4} = 2

. Since the line passes through the y-axis at point

(0, -3)

, the y-intercept is

b = -3

. Therefore, the equation is

y = 2x - 3

. Choice B has the wrong sign for the y-intercept. Choice C has the wrong sign for the slope. Choice D uses the reciprocal of the correct slope.

Question 5

A line passes through the origin and the point $(4,12)$ . Which equation matches the line?

$y=12x+4$
$y=3x$ (correct answer)
$y=\dfrac{1}{3}x$
$y=4x+12$

Explanation: This question tests that slope is constant on a line, verifiable with any two points giving the same m, explained via similar triangles, and deriving y=mx for origin-passing lines or y=mx+b otherwise. Slope m=(y₂-y₁)/(x₂-x₁) stays constant, a key line property; similar triangles with different sizes on the line share angles (parallel sides) and have proportional sides, e.g., ratios 6/2=12/4=3 remain equal. For points (0,0) and (4,12), m=(12-0)/(4-0)=12/4=3. Since it passes through the origin, the equation is y=3x. Mistakes include calculating slope as 1/3 by inverting or choosing y=mx+b form unnecessarily. Steps: (1) compute m from points, (2) confirm with another pair if available, (3) for origin lines use y=mx, else find b from intercept or point substitution, (4) form equation. Similar triangles prove constancy through proportional rise/run from shared angles; errors like claiming varying slopes or wrong forms for origin lines are common.

Question 6

A straight line passes through the three points $(0,4)$ , $(2,8)$ , and $(4,12)$ . Which equation represents this line?

$y=2x+4$ (correct answer)
$y=2x-4$
$y=4x+2$
$y=\frac{1}{2}x+4$

Explanation: This question tests deriving a line equation from points, ensuring constant slope where any two give same m, using similar triangles for constancy, and forming y=mx+b. Slope m=(y₂-y₁)/(x₂-x₁) constant; similar triangles with equal angles (parallel sides), proportional like 6/2=12/4=3; y=mx+b with m from points, b from (0,b). Using (0,4),(2,8), m=(8-4)/(2-0)=4/2=2, b=4; check (4,12): 2*4+4=12. Equation y=2x+4 matches A. Errors: swapping to 4x+2 in B, -4 in C, 1/2 in D. Steps: (1) compute m from 2 points, (2) verify with third, (3) non-origin find b from intercept, (4) equation. Similar triangles: shared angle, proportionality, constant rise/run; mistakes: inverting, variation claim, wrong form.

Question 7

A line passes through the points $(0,2)$ and $(3,8)$ . What is the equation of the line in the form $y=mx+b$ ?

$y=2x-2$
$y=\frac{1}{2}x+2$
$y=2x+2$ (correct answer)
$y=3x+2$

Explanation: This question tests your understanding of deriving the equation of a line using its constant slope, where any two points give the same m, and incorporating similar triangles to explain constancy while forming y=mx for origin lines or y=mx+b for others. The slope m = (y₂ - y₁)/(x₂ - x₁) is constant for any point pairs on the line, a key feature of straight lines; similar triangles demonstrate this as triangles on the same line share angles from parallel sides and have proportional sides, like 2× rise and 2× run yielding 6/2 = 12/4 = 3, confirming constant slope; for y=mx through the origin, m = y/x from any point, or for y=mx+b, find m from two points and b from the y-intercept (0,b). Here, using points (0,2) and (3,8), m = (8-2)/(3-0) = 6/3 = 2, and since it crosses y at 2, b=2. This gives the equation y=2x+2, matching choice B. Errors include wrong slope like 3 in A or 1/2 in D, or incorrect b like -2 in C. The process is: (1) pick 2 points to calculate m = (y₂ - y₁)/(x₂ - x₁), (2) verify if needed with another pair, (3) for non-origin lines, find b by noting y-intercept or solving y=mx+b with a point, (4) write the equation. Similar triangles show proof via shared angles and proportional sides ensuring constant rise/run; common mistakes are inverting rise/run, assuming varying slope, or using y=mx for non-origin lines.

Question 8

A line passes through $(2,7)$ and $(6,15)$ . What is the slope $m$ of the line?

$m=2$
$m=\frac{4}{8}=\frac{1}{2}$
$m=\frac{1}{2}$
$m=\frac{8}{4}=2$ (correct answer)

Explanation: This question tests understanding that the slope is constant on a straight line, meaning any two points give the same

m

, and can be explained using similar triangles, while deriving equations like

y=mx

for lines through the origin or

y=mx+b

otherwise. The slope

m

is calculated as

(y_2 - y_1)/(x_2 - x_1)

and remains constant for any pair of points on the line, which is a defining feature of straight lines; similar triangles on the same line have equal angles due to parallel sides and proportional sides, such as a larger triangle with twice the rise and run yielding the same ratio like

6/2 = 12/4 = 3

, proving constant slope; for derivation,

y=mx

uses

m = y/x

for origin lines, or

y=mx+b

finds

m

from points and

b

from the y-intercept. Specifically, for

(2,7)

and

(6,15)

m = \frac{15-7}{6-2} = \frac{8}{4} = 2

. This matches choice C, showing the correct fraction and value. Errors include just

1/2

without fraction in A, just 2 in B, or inverting to

4/8=1/2

in D. To calculate, (1) pick the two points, compute

m = (y_2 - y_1)/(x_2 - x_1)

, (2) no more points to verify, (3) not deriving equation here. Similar triangles would apply if more segments, proving constant ratio; common mistakes: inverting rise/run, claiming varies, wrong form.

Question 9

A straight line passes through the three points $(0,4)$ , $(2,8)$ , and $(4,12)$ . Which equation represents this line?

$y=\frac{1}{2}x+4$
$y=4x+2$
$y=2x+4$ (correct answer)
$y=2x-4$

Question 10

Look at the diagram showing how similar triangles prove that slope is constant on a line. The line passes through points $A(2, 5)$ , $B(6, 11)$ , and $C(8, 14)$ . What is the complete equation of this line?

$y = \frac{3}{2}x + 2$ (correct answer)
$y = \frac{2}{3}x + \frac{11}{3}$
$y = \frac{3}{2}x - 2$
$y = \frac{3}{2}x + \frac{11}{2}$

Explanation: Calculate slope using any two points:

m = \frac{11-5}{6-2} = \frac{6}{4} = \frac{3}{2}

. Using point

A(2,5)

in point-slope form:

y - 5 = \frac{3}{2}(x - 2)

, which gives

y - 5 = \frac{3}{2}x - 3

, so

y = \frac{3}{2}x + 2

. Choice B uses the reciprocal slope. Choice C has the wrong sign for the y-intercept. Choice D makes an error in calculating the y-intercept.

Question 11

Refer to the coordinate plane. Triangle $PQR$ and triangle $STU$ are similar, where both triangles have one vertex on the origin and demonstrate that the slope is constant along line $\ell$ . If the slope of line $\ell$ is $\frac{3}{4}$ , what is the equation of line $\ell$ ?

$y = \frac{3}{4}x$ (correct answer)
$y = \frac{4}{3}x$
$y = \frac{3}{4}x + 1$
$y = \frac{3}{4}x - 1$

Explanation: Since the line passes through the origin and has slope

\frac{3}{4}

, the equation is

y = \frac{3}{4}x

. When a line passes through the origin, there is no y-intercept term (

b = 0

). Choice B uses the reciprocal of the slope. Choices C and D incorrectly add a y-intercept when the line passes through the origin.

Question 12

Examine the coordinate plane where line $m$ demonstrates constant slope through similar triangles. Points $D(1, -2)$ and $E(4, 4)$ lie on line $m$ . Based on the similar triangle principle, what is the equation of line $m$ ?

$y = 2x - 4$ (correct answer)
$y = 2x + 4$
$y = -2x + 4$
$y = \frac{1}{2}x - \frac{5}{2}$

Explanation: Calculate the slope:

m = \frac{4-(-2)}{4-1} = \frac{6}{3} = 2

. Using point

D(1,-2)

y - (-2) = 2(x - 1)

, so

y + 2 = 2x - 2

, which gives

y = 2x - 4

. Choice B has the wrong sign for the y-intercept. Choice C has the wrong sign for the slope. Choice D uses an incorrect slope calculation and y-intercept.

Question 13

Two similar triangles are used to prove that slope is constant on a line. The larger triangle has a horizontal leg of 12 and a vertical leg of 9. The smaller triangle has a horizontal leg of 4. If the line intersects the y-axis at $(0, -5)$ , what is the equation of the line?

$y = \frac{3}{4}x + 5$
$y = \frac{4}{3}x - 5$
$y = \frac{3}{4}x - 5$ (correct answer)
$y = 3x - 5$

Explanation: When you see similar triangles used to find slope, remember that similar triangles have proportional sides, which is exactly why slope stays constant along a line. First, find the vertical leg of the smaller triangle using the proportion from similar triangles. Since the horizontal legs are 12 and 4, the ratio is

\frac{4}{12} = \frac{1}{3}

. Therefore, the smaller triangle's vertical leg is

9 \times \frac{1}{3} = 3

. Now calculate the slope using either triangle:

m = \frac{\text{rise}}{\text{run}} = \frac{9}{12} = \frac{3}{4}

m = \frac{3}{4}

. Both give the same result, proving slope is constant. Since the line intersects the y-axis at

(0, -5)

, the y-intercept is

-5

. Using slope-intercept form

y = mx + b

y = \frac{3}{4}x + (-5) = \frac{3}{4}x - 5

. Choice A (

y = \frac{3}{4}x + 5

) has the correct slope but wrong y-intercept sign—this traps students who forget the negative sign from

(0, -5)

. Choice B (

y = \frac{4}{3}x - 5

) flips the slope fraction, a common error when students mix up rise over run. Choice D (

y = 3x - 5

) incorrectly uses just the numerator as the slope, ignoring the denominator entirely. The correct answer is C. Study tip: Always double-check that your slope fraction matches rise over run, and be extra careful with positive/negative signs in the y-intercept. Similar triangles will always give you the same slope—use this as a way to verify your work.

Question 14

A line passes through the three points $(0,1)$ , $(2,5)$ , and $(4,9)$ . Which choice correctly shows that the slope is constant and gives the equation of the line?

Slopes: $\frac{5-1}{2-0}=2$ and $\frac{9-5}{4-2}=2$ ; equation: $y=2x-1$ .
Slopes: $\frac{5-1}{2-0}=2$ and $\frac{9-5}{4-2}=2$ ; equation: $y=2x+1$ . (correct answer)
Slopes: $\frac{5-1}{2-0}=4$ and $\frac{9-5}{4-2}=2$ ; equation: $y=4x+1$ .
Slopes: $\frac{2-0}{5-1}=\frac{1}{2}$ and $\frac{4-2}{9-5}=\frac{1}{2}$ ; equation: $y=\frac{1}{2}x+1$ .

Explanation: This question tests understanding that the slope is constant on a straight line, meaning any two points give the same m, and can be explained using similar triangles, while deriving equations like y=mx for lines through the origin or y=mx+b otherwise. The slope m is calculated as (y₂ - y₁)/(x₂ - x₁) and remains constant for any pair of points on the line, which is a defining feature of straight lines; similar triangles on the same line have equal angles due to parallel sides and proportional sides, such as a larger triangle with twice the rise and run yielding the same ratio like 6/2 = 12/4 = 3, proving constant slope; for derivation, y=mx uses m = y/x for origin lines, or y=mx+b finds m from points and b from the y-intercept. Specifically, for (0,1), (2,5), (4,9), slopes (5-1)/(2-0)=4/2=2 and (9-5)/(4-2)=4/2=2, and using (0,1) b=1 so y=2x+1. This matches choice A. Errors include inverting to 1/2 in B, wrong first slope 4 in C, and wrong b=-1 in D. To verify and derive, (1) pick two points for m, (2) verify with another pair, (3) not through origin so solve for b using a point, (4) write y=mx+b. Similar triangles prove constancy as they share angles with parallel sides, making sides proportional and rise/run ratios equal; common mistakes include inverting rise/run, claiming slope varies, or wrong sign for b.

Question 15

A line goes through $(0,0)$ , $(2,8)$ , and $(5,20)$ . Which equation matches the line, and why?

$y=5x$ , because the $x$ -values increase to 5 at the last point.
$y=\frac{1}{4}x$ , because slope is run over rise: $\frac{2}{8}=\frac{5}{20}=\frac{1}{4}$ .
$y=4x$ , because $\frac{8}{2}=4$ and $\frac{20}{5}=4$ , so the slope is constant and the line passes through the origin. (correct answer)
$y=4x+8$ , because the line uses the point $(2,8)$ as the intercept.

Explanation: This question tests understanding that the slope is constant on a straight line, meaning any two points give the same m, and can be explained using similar triangles, while deriving equations like y=mx for lines through the origin or y=mx+b otherwise. The slope m is calculated as (y₂ - y₁)/(x₂ - x₁) and remains constant for any pair of points on the line, which is a defining feature of straight lines; similar triangles on the same line have equal angles due to parallel sides and proportional sides, such as a larger triangle with twice the rise and run yielding the same ratio like 6/2 = 12/4 = 3, proving constant slope; for derivation, y=mx uses m = y/x for origin lines, or y=mx+b finds m from points and b from the y-intercept. Specifically, for (0,0), (2,8), (5,20), slopes 8/2=4 and 20/5=4, through origin so y=4x. This matches choice A, explaining constancy and origin. Errors: adding b=8 in B, inverting to 1/4 in C, wrong m=5 in D. To derive, (1) calculate m with pairs, (2) verify equality, (3) through origin so y=mx, (4) write equation. Similar triangles prove constant ratio; mistakes: adding b when origin, inverting, wrong m based on x-value.

Question 16

A line goes through the origin and the point $(4,12)$ . Which equation matches the line?

$y=3x$ (correct answer)
$y=4x+12$
$y=12x$
$y=\frac{1}{3}x$

Explanation: This question tests understanding of constant slope on a line through the origin, where any two points yield the same m, using similar triangles for explanation, and deriving y=mx specifically for origin-passing lines. Slope m = (y₂ - y₁)/(x₂ - x₁) stays constant across point pairs, defining straight lines; similar triangles illustrate this with equal angles from parallel sides and proportional dimensions, such as a doubled triangle giving 6/2 = 12/4 = 3, proving constancy; for origin lines, y=mx with m = y/x from any point, or generally y=mx+b with b=0. For (0,0) and (4,12), m = (12-0)/(4-0) = 12/4 = 3. Thus, the equation is y=3x, as in choice C. Mistakes include using y=12x in A, adding b=12 in B despite origin, or inverting to 1/3 in D. Steps: (1) choose 2 points, compute m = (y₂ - y₁)/(x₂ - x₁), (2) check another pair if available, (3) since through origin, use y=mx, (4) write equation. Similar triangles confirm via shared angles and proportionality, keeping rise/run constant; errors like inverting rise/run, claiming variable slope, or applying y=mx+b unnecessarily.

Question 17

Line $L_1$ passes through $(0,2)$ and $(4,10)$ . Line $L_2$ passes through $(0,0)$ and $(3,9)$ . Which statement is true?

They are equally steep because both lines rise by 8 from their first point to their second point.
They are equally steep because both lines have a $y$ -intercept of 2.
$L_2$ is steeper because its slope is 3 and $L_1$ has slope 2. (correct answer)
$L_1$ is steeper because its slope is $\frac{10-2}{4-0}=2$ and $L_2$ has slope $\frac{9-0}{3-0}=3$ .

Explanation: This question tests understanding that the slope is constant on a straight line, meaning any two points give the same m, and can be explained using similar triangles, while deriving equations like y=mx for lines through the origin or y=mx+b otherwise. The slope m is calculated as (y₂ - y₁)/(x₂ - x₁) and remains constant for any pair of points on the line, which is a defining feature of straight lines; similar triangles on the same line have equal angles due to parallel sides and proportional sides, such as a larger triangle with twice the rise and run yielding the same ratio like 6/2 = 12/4 = 3, proving constant slope; for derivation, y=mx uses m = y/x for origin lines, or y=mx+b finds m from points and b from the y-intercept. Specifically, L1 slope (10-2)/(4-0)=8/4=2, L2 (9-0)/(3-0)=9/3=3, so L2 steeper since |3| > |2|. This matches choice B. Errors: backwards in A claiming L1 steeper, wrong equal due to b in C, equal rise but different run in D. To compare, (1) calculate m for each, (2) compare magnitudes, (3) steeper has larger |m|. Similar triangles explain each line's constant slope; mistakes: miscomparing m values, confusing with b or rise alone, claiming equality wrongly.

Question 18

A student says the points $(2,5)$ , $(4,9)$ , and $(6,13)$ lie on one straight line. Which calculation correctly checks that the slope is the same between the first two points and the last two points?

$m_{(2,5)(4,9)}=\frac{9-5}{4-2}=4$ and $m_{(4,9)(6,13)}=\frac{13-9}{6-4}=4$ (correct answer)
$m_{(2,5)(4,9)}=\frac{9-5}{4-2}=2$ and $m_{(4,9)(6,13)}=\frac{13-9}{6-4}=2$
$m_{(2,5)(4,9)}=\frac{4-2}{9-5}=\frac{1}{2}$ and $m_{(4,9)(6,13)}=\frac{6-4}{13-9}=\frac{1}{2}$
$m_{(2,5)(4,9)}=\frac{9-5}{4-2}=2$ and $m_{(4,9)(6,13)}=\frac{13-9}{6-4}=4$

Explanation: This question tests verifying constant slope on a line, where any two points give the same m, using similar triangles for explanation, and linking to deriving y=mx or y=mx+b equations. Slope m = (y₂ - y₁)/(x₂ - x₁) remains constant, defining lines; similar triangles prove this with equal angles from parallel sides and proportional sides, e.g., 6/2 = 12/4 = 3, showing constant slope; derive y=mx for origin via m=y/x, or y=mx+b with m from points and b from (0,b). For points (2,5),(4,9),(6,13), m between first two: (9-5)/(4-2)=4/2=2, between last two: (13-9)/(6-4)=4/2=2. This confirms constant slope as in choice A. Common errors: inverting to 1/2 in B, stating 4 without dividing in C, or miscalculating second as 4 in D. Steps: (1) pick 2 points, calculate m, (2) verify with another pair, (3) if origin y=mx, else find b via intercept or solving with point, (4) equation. Similar triangles: shared angle, proportional sides from parallelism, constant ratio; mistakes: inverting rise/run, claiming variation, wrong form for origin/non-origin.

Question 19

A line passes through the points $(1,4)$ and $(5,12)$ . Another point on the same line is $(3,8)$ . Which choice correctly uses slopes to show that all three points are on one line?

$\frac{5-1}{12-4}=2$ and $\frac{3-1}{8-4}=2$ , so the slopes match. (correct answer)
$\frac{12-4}{5-1}=\frac{1}{2}$ and $\frac{8-4}{3-1}=\frac{1}{2}$ , so the slopes match.
$\frac{12-4}{5-1}=2$ and $\frac{8-4}{3-1}=2$ , so the slopes match.
$\frac{12-4}{5-1}=2$ and $\frac{8-4}{3-1}=4$ , so the slopes match.

Explanation: This question tests verifying collinearity by constant slope, where any pairs give same m, using similar triangles to explain, and relating to y=mx+b derivation. Slope m=(y₂-y₁)/(x₂-x₁) constant on lines; similar triangles prove via equal angles (parallel), proportional sides like 6/2=12/4=3; derive y=mx+b with m from points, b from intercept. Slopes: (12-4)/(5-1)=8/4=2, (8-4)/(3-1)=4/2=2. Matches A showing same slope. Errors: inverting to 1/2 in B, misstating second as 4 in C, inverting formula in D. Process: (1) calculate m for pairs, (2) verify equality, (3) find b if needed via point or intercept, (4) equation optional here. Similar triangles: shared angle, proportional sides, constant ratio; mistakes: inverting rise/run, claiming variation, wrong form.

Question 20

A line goes through the points $(0,2)$ and $(3,8)$ . What is the equation of the line in the form $y=mx+b$ ?

$y=3x+2$
$y=2x-2$
$y=\dfrac{1}{2}x+2$
$y=2x+2$ (correct answer)

Explanation: This question tests understanding that the slope of a straight line is constant, meaning any two points give the same m, and we can use similar triangles to explain this while deriving y=mx for origin lines or y=mx+b for others. Slope m=(y₂-y₁)/(x₂-x₁) is constant for any points on the line; similar triangles on the line have equal angles from parallel sides and proportional dimensions, like 6/2=12/4=3, showing constant slope. Here, using points (0,2) and (3,8), m=(8-2)/(3-0)=6/3=2, and since it intersects y-axis at (0,2), b=2. The correct equation is y=2x+2. Errors include using wrong slope like 3 by miscalculating differences or inverting to 1/2. The process is: (1) calculate m from two points, (2) verify if needed, (3) for non-origin line, find b from y-intercept or by solving y=mx+b with a point, (4) write the equation. Similar triangles confirm constant slope via shared angles and proportional rise/run; common mistakes are claiming variable slope or using y=mx form incorrectly for lines not through origin.