📝 Slope rise over run formula (14 MCQs)

📖 From Digital SAT Algebra • 4. Graphs • 14 questions available

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What is Slope rise over run formula?

Definition:
The slope rise over run formula $m = \frac{\text{rise}}{\text{run}}$ calculates the slope of a line, where rise is the vertical change between two points (difference in y), and run is the horizontal change (difference in x), and it provides a direct measure of the line's steepness and direction, and it is the basis for understanding linear equations, graphing, and many real-world applications.

Working:
To use the formula, identify any two points on the line, $(x_1, y_1)$ and $(x_2, y_2)$ , then calculate the differences: $\text{rise} = y_2 - y_1$ , $\text{run} = x_2 - x_1$ ; then the slope $m = \frac{y_2 - y_1}{x_2 - x_1}$ ; if rise is positive and run is positive, the slope is positive; if rise is negative (line falls), the slope is negative; if run is zero, the slope is undefined; and if rise is zero, the slope is zero; this formula is fundamental in algebra.

Example:
A simple example is finding the slope of a line through (1, 2) and (4, 8): rise = 8 - 2 = 6, run = 4 - 1 = 3, so slope $m = \frac{6}{3} = 2$ ; another example through (5, 5) and (7, 1): rise = 1 - 5 = -4, run = 7 - 5 = 2, so $m = \frac{-4}{2} = -2$ , demonstrating the formula.

Reason:
The rise over run formula is essential for calculating slopes, interpreting graphs, and solving problems in physics, engineering, and economics, making it a core concept in mathematics and its applications.

Easy

Medium

Hard

📝 All Slope rise over run formula MCQs

Q1. A line on a coordinate graph passes through $(2,3)$ and $(6,11)$ . Using $m=\frac{\text{rise}}{\text{run}}$ , what is the slope of the line?

2

✅

1/2

4

8

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The rise is $11-3=8$ , while the run is $6-2=4$ . Therefore, $m=\frac{8}{4}=2$ . The slope measures how much $y$ increases for every one-unit increase in $x$ .

Q2. A graph shows a line passing through $(-3,7)$ and $(1,-1)$ . What is the slope when the points are used in that order?

2

-2

✅

-1/2

1/2

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The vertical change is $-1-7=-8$ , and the horizontal change is $1-(-3)=4$ . Thus $m=\frac{-8}{4}=-2$ . The negative sign correctly indicates that the line falls as $x$ increases.

Q3. Two students calculate the slope between $(1,2)$ and $(5,10)$ . Student A uses rise $=8$ and run $=4$ . Student B uses rise $=4$ and run $=8$ . Which statement best evaluates their work?

A.Only Student A is correct because rise must always be larger than run ✅

B.Only Student B is correct because run should be larger than rise

C.Both are correct because each calculation gives the same slope

D.Neither is correct because slope must be positive

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Student A calculates $8/4=2$ , which is correct. Student B calculates $4/8=1/2$ , which reverses rise and run. Slope specifically requires vertical change divided by horizontal change.

Q4. A line rises 6 units while moving 3 units to the right. Another line rises 8 units while moving 4 units to the right. What can be concluded about their slopes?

A.The first line is steeper because its rise is smaller

B.The second line is steeper because its rise is larger

C.Both lines have the same slope ✅

D.Their slopes cannot be compared without the

y

-intercepts

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The first slope is $6/3=2$ , while the second is $8/4=2$ . Although their rises differ, the rise-to-run ratios are identical. Therefore, both lines have the same slope and rate of change.

Q5. A line passes through $(0,-4)$ , $(2,2)$ , and $(4,8)$ . A student says the slope changes because the $y$ -values become larger faster. What is the best response?

A.The student is correct because larger

y

-values always mean larger slope

B.The student is correct because slope depends only on

y

C.The student is incorrect because each 2-unit run produces the same 6-unit rise ✅

D.The student is incorrect because slope must always equal

1

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: From $(0,-4)$ to $(2,2)$ , the rise is $6$ and run is $2$ , giving $m=3$ . From $(2,2)$ to $(4,8)$ , the same ratio occurs. Thus the slope remains constant.

Q6. On a graph, moving from one marked point to another requires moving 5 units left and 10 units down. What is the slope represented by this movement?

2

✅

-2

1/2

-1/2

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Moving left 5 means the run is $-5$ , and moving down 10 means the rise is $-10$ . Therefore $m=\frac{-10}{-5}=2$ . Both changes are negative, so their ratio is positive.

Q7. A wheelchair ramp rises 3 feet over a horizontal distance of 24 feet. If the graph uses horizontal distance as $x$ and height as $y$ , what slope represents the ramp?

8

3/24

✅

24/3

-3/24

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The vertical change is $3$ feet and the horizontal change is $24$ feet. Thus $m=\frac{3}{24}=\frac18$ . This positive slope models the ramp's increase in height as horizontal distance increases.

Q8. A line on a graph passes through $(-4,-2)$ and $(2,7)$ . A student computes the slope as $\frac{7-(-2)}{-4-2}=-\frac32$ . What error did the student make?

A.The student used the wrong

y

-values

B.The student subtracted the coordinates in inconsistent order ✅

C.The student forgot to multiply the coordinates

D.The student should divide run by rise

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The numerator uses the change from the first point to the second, $7-(-2)=9$ . The denominator must use the same order, $2-(-4)=6$ . Thus $m=9/6=3/2$ , not $-3/2$ .

Q9. A graph shows a line through $(3,8)$ and $(7,0)$ . Another student chooses the movement from $(7,0)$ to $(3,8)$ and obtains $m=\frac{8}{-4}$ . Is the result valid?

A.No, because points must always be used from left to right

B.No, because a slope cannot be negative when

y

is positive

C.Yes, because reversing both the rise and run preserves the slope ✅

D.Yes, but only for horizontal lines

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: From $(3,8)$ to $(7,0)$ , the slope is $(-8)/4=-2$ . Reversing direction gives rise $8$ and run $-4$ , which also gives $-2$ . Consistent reversal preserves the slope.

Q10. A line on a graph passes through $(1,5)$ and $(4,11)$ . A student says the slope is $6$ because the $y$ -value increases by 6. Which reasoning correctly identifies the slope?

A.The slope is

6

because only vertical change matters

B.The slope is

3

because the run is ignored

C.The slope is

2

because

m=\frac{6}{3}

✅

D.The slope is

1/2

because

m=\frac{3}{6}

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The change in $y$ is $11-5=6$ , but the change in $x$ is $4-1=3$ . Since slope is rise divided by run, $m=\frac63=2$ . Ignoring the run produces an incorrect rate.

Q11. A graph shows Line P moving 4 units upward for every 2 units right, while Line Q moves 6 units downward for every 3 units right. Which comparison is correct?

A.Line P has slope

2

, and Line Q has slope

-2

✅

B.Both lines have slope

2

C.Line P has slope

-2

, and Line Q has slope

2

D.Both lines have slope

-2

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: For Line P, $m=4/2=2$ . For Line Q, the rise is $-6$ and run is $3$ , so $m=-6/3=-2$ . They have equal steepness but opposite directions of change.

Q12. A line rises 12 units while running 8 units. A second line rises 9 units while running 6 units. A third line rises 5 units while running 4 units. Which line has the greatest slope?

A.The first line

B.The second line

C.The third line

D.The first and second lines tie ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: The first slope is $12/8=1.5$ , the second is $9/6=1.5$ , and the third is $5/4=1.25$ . Therefore, the first and second lines tie for the greatest slope.

Q13. A line has slope $-3$ . Starting at a point on the line, a student moves 4 units to the right. Which vertical movement must return the student to the line?

A.Move 12 units up

B.Move 12 units down ✅

C.Move 7 units down

D.Move 4 units up

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A slope of $-3$ means $m=\frac{\text{rise}}{\text{run}}=-3$ . With a run of $4$ , the rise must be $-12$ . Therefore, the student must move 12 units downward.

Q14. Three marked points on a straight line are $A=(-2,9)$ , $B=(1,3)$ , and $C=(5,-5)$ . Which conclusion about the slope is strongest?

A.The slope from A to B differs from the slope from B to C

B.The slope is

2

because the line moves downward

C.The slope is

-2

because both intervals give the same rise-to-run ratio ✅

D.The slope is

-1/2

because run is greater than rise

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: From A to B, the rise is $3-9=-6$ and run is $1-(-2)=3$ , giving $-2$ . From B to C, the rise is $-5-3=-8$ and run is $5-1=4$ , again giving $-2$ . The consistent ratio confirms the slope.

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