📝 Slope formula between two points (15 MCQs)

📖 From Digital SAT Algebra • 4. Graphs • 15 questions available

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What is Slope formula between two points?

Definition:
The slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$ , which calculates the slope of the line passing through them, representing the rate of change of y with respect to x, and it is the standard formula used in algebra, geometry, and calculus to determine line steepness and direction, and it is applicable to any straight line.

Working:
To use the formula, label the points as (x₁, y₁) and (x₂, y₂), then subtract the y-values to get the rise, subtract the x-values to get the run, and divide the rise by the run; it is important to maintain consistent order: $y_2 - y_1$ and $x_2 - x_1$ ; the result gives the slope, which can be positive, negative, zero, or undefined; for example, for points (2, 3) and (5, 11), $m = \frac{11 - 3}{5 - 2} = \frac{8}{3}$ , and this formula is used to determine the equation of a line and to analyze linear relationships.

Example:
A simple example is points (1, -2) and (4, 4): $m = \frac{4 - (-2)}{4 - 1} = \frac{6}{3} = 2$ ; another example is points (3, 5) and (3, 8): $m = \frac{8 - 5}{3 - 3} = \frac{3}{0}$ , which is undefined, indicating a vertical line, illustrating the use of the formula for any two points.

Reason:
The slope formula is a fundamental tool in mathematics, used to describe rates of change, graph lines, and solve problems in physics, economics, and engineering, and it is essential for understanding linear functions and their applications.

Easy

Medium

Hard

📝 All Slope formula between two points MCQs

Q1. Which expression correctly represents the slope of the line through $A(x_1,y_1)$ and $B(x_2,y_2)$ , assuming $x_1\ne x_2$ ?

\frac{x_2-x_1}{y_2-y_1}

\frac{y_2-y_1}{x_2-x_1}

✅

\frac{y_1+y_2}{x_1+x_2}

\frac{x_1+x_2}{y_1+y_2}

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The slope measures vertical change relative to horizontal change, so the numerator must represent the change in $y$ and the denominator must represent the corresponding change in $x$ . Reversing these changes gives the reciprocal rather than the slope.

Q2. The points $P(4,9)$ and $Q(10,3)$ lie on a line. What is its slope?

1

-1

-\frac{3}{2}

✅

\frac{3}{2}

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Using $m=\frac{y_2-y_1}{x_2-x_1}$ , the vertical change is $3-9=-6$ , while the horizontal change is $10-4=6$ . Therefore the slope is $-6/6=-1$ , indicating a decreasing line.

Q3. A student calculates the slope between $A(2,5)$ and $B(8,17)$ as $\frac{8-2}{17-5}=\frac12$ . What does this calculation actually represent?

A.The slope of the line

B.The negative of the slope

C.The reciprocal of the slope ✅

D.The

y

-intercept

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The student placed the horizontal change in the numerator and the vertical change in the denominator. This reverses the slope formula, producing $\frac{6}{12}=\frac12$ , which is the reciprocal of the actual slope $2$ .

Q4. Two points on a line are $C(-3,7)$ and $D(5,-1)$ . A second pair on the same line is $E(1,3)$ and $F(9,-5)$ . Which conclusion is best supported?

A.The two slopes are opposites

B.The two slopes are equal because both point pairs lie on the same line ✅

C.The second slope is twice the first

D.The slopes cannot be compared

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: For $C,D$ , the slope is $\frac{-1-7}{5-(-3)}=-1$ . For $E,F$ , it is $\frac{-5-3}{9-1}=-1$ . Equal slopes confirm that both pairs are consistent with the same line.

Q5. A line passes through $R(-6,2)$ and $S(3,8)$ . Without changing the order of subtraction within each difference, which setup correctly finds the slope?

\frac{2-8}{-6-3}

\frac{8-2}{3-(-6)}

✅

\frac{3-(-6)}{8-2}

\frac{-6-3}{2-8}

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Using the coordinates in the order $R$ to $S$ , the change in $y$ is $8-2=6$ and the change in $x$ is $3-(-6)=9$ . Thus the correct setup is $6/9=2/3$ .

Q6. A hiking trail is modeled by a straight line through locations represented by $A(2,450)$ and $B(8,750)$ , where $x$ is distance and $y$ is elevation. What does the slope mean?

A.Elevation decreases by 50 units per distance unit

B.Elevation increases by 50 units per distance unit ✅

C.Distance increases by 50 units per elevation unit

D.Elevation increases by 300 units per distance unit

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The slope is $\frac{750-450}{8-2}=\frac{300}{6}=50$ . Therefore, for every one-unit increase in horizontal distance, the modeled elevation increases by 50 units, giving a direct interpretation of the positive slope.

Q7. A delivery route is represented by points $A(1,120)$ and $B(5,80)$ , where $x$ is time and $y$ is remaining distance. What is the slope and its interpretation?

10

; remaining distance increases

-10

; remaining distance decreases by 10 units per time unit ✅

-40

; remaining distance decreases by 40 units per time unit

40

; remaining distance increases

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The slope is $\frac{80-120}{5-1}=-40/4=-10$ . The negative value means remaining distance decreases as time increases, at a modeled rate of 10 distance units for each time unit.

Q8. A student finds the slope between $M(-4,-2)$ and $N(2,10)$ as $\frac{-2-10}{-4-2}=2$ . Another student gets $-2$ . Who is correct?

A.Only the first student

B.Only the second student

C.Both students, because changing point order changes the slope

D.Neither student; the slope is

3

✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The correct slope is $\frac{10-(-2)}{2-(-4)}=\frac{12}{6}=2$ . Reversing both differences gives the same positive result. The second student's negative value likely came from reversing only one difference.

Q9. A line segment connects $P(-2,6)$ to $Q(4,-3)$ . Another segment connects $R(1,2)$ to $S(7,-7)$ . Which comparison is correct?

A.The first segment is steeper because its slope is more negative

B.The second segment is steeper because its slope is positive

C.Both segments have the same slope and therefore the same steepness ✅

D.The slopes cannot be determined from the coordinates

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The first slope is $\frac{-3-6}{4-(-2)}=-9/6=-3/2$ . The second is $\frac{-7-2}{7-1}=-9/6=-3/2$ . Equal slopes indicate equal direction and steepness.

Q10. On a coordinate graph, a line passes through the labeled points $(-5,4)$ and $(3,-8)$ . Which statement best describes the line?

A.It rises 3 units for every 2 units right

B.It falls 3 units for every 2 units right ✅

C.It falls 2 units for every 3 units right

D.It rises 2 units for every 3 units right

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The slope is $\frac{-8-4}{3-(-5)}=-12/8=-3/2$ . Thus moving two units to the right corresponds to a three-unit decrease in $y$ , so the line falls at that rate.

Q11. A graph shows two marked points $A(-1,-4)$ and $B(5,5)$ . A student says the slope is $9/6$ because the vertical change is 9 and horizontal change is 6. Is the reasoning correct?

A.Yes, because both changes are positive ✅

B.No, because the vertical change is actually

-9

C.No, because the horizontal change is actually

-6

D.Yes, but the slope should be simplified to

6/9

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: From $A$ to $B$ , $y$ changes from $-4$ to $5$ , giving $9$ , while $x$ changes from $-1$ to $5$ , giving $6$ . Therefore $9/6=3/2$ , so the student's reasoning is correct.

Q12. A line segment joins $A(2,-7)$ and $B(2,9)$ . What happens when the slope formula is applied?

A.It gives

0

B.It gives

1

C.It requires division by zero, so the slope is undefined ✅

D.It gives

-16

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The horizontal change is $2-2=0$ , while the vertical change is $9-(-7)=16$ . Since the slope formula requires division by the horizontal change, division by zero makes the slope undefined.

Q13. A model uses points $A(0,5)$ and $B(4,13)$ . A revised model uses $C(0,5)$ and $D(8,21)$ . Which statement correctly compares their slopes?

A.Both slopes are

2

✅

B.The revised model has twice the slope of the original

C.The revised model has half the slope of the original

D.The revised model has slope

1

, while the original has slope

4

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The original slope is $\frac{13-5}{4-0}=2$ . The revised slope is $\frac{21-5}{8-0}=16/8=2$ . Although both coordinates changed, the proportional relationship between vertical and horizontal changes remained the same.

Q14. Four points are $A(0,1)$ , $B(2,5)$ , $C(5,11)$ , and $D(8,17)$ . Which pair of points provides the strongest evidence that all four points may lie on one straight line?

A

and

B

only

B

and

C

only

C.Comparing the slopes

AB

BC

, and

CD

✅

D.Comparing only the

x

-coordinates

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The slopes are $AB=4/2=2$ , $BC=6/3=2$ , and $CD=6/3=2$ . Since consecutive segments have identical slopes, the points follow one constant rate of change and therefore support a single straight-line model.

Q15. Suppose $P(a,b)$ and $Q(a+4,b-12)$ are two points on a line. Without knowing $a$ or $b$ , what can be concluded about the slope?

A.It is

3

B.It is

-3

✅

C.It is

-\frac13

D.It depends on the values of

a

and

b

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The horizontal change is $(a+4)-a=4$ , and the vertical change is $(b-12)-b=-12$ . Therefore the slope is always $-12/4=-3$ , regardless of the starting coordinates.

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