📝 Slope of a line definition (14 MCQs)

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

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What is Slope of a line definition?

Definition:
The slope of a line is a measure of its steepness and direction, defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line, and it is typically denoted by the letter $m$ ; the slope indicates how much y changes for a unit change in x, and it is positive for lines sloping upward to the right, negative for lines sloping downward, zero for horizontal lines, and undefined for vertical lines.

Working:
Slope is calculated as $m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$ , using two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ ; a positive slope means the line rises from left to right, a negative slope means it falls, a zero slope means it is horizontal, and an undefined slope occurs when the line is vertical (division by zero); the slope is constant for any straight line, making it a characteristic property, and it is used in equations like $y = mx + b$ .

Example:
A simple example is the line passing through (1, 2) and (3, 6); the slope is $m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2$ , indicating a rise of 2 for every 1 unit run; another example is the line through (2, 5) and (4, 3), giving $m = \frac{3 - 5}{4 - 2} = \frac{-2}{2} = -1$ , a negative slope.

Reason:
Slope is a fundamental concept in algebra and calculus, used to describe rates of change, model linear relationships, and analyze data, making it essential for understanding functions, graphs, and real-world applications like speed and growth rates.

Easy

Medium

Hard

📝 All Slope of a line definition MCQs

Q1. A line passes through $(2,5)$ and $(6,13)$ . What does its slope tell you about the relationship between $x$ and $y$ ?

y

decreases by 2 units for every 1-unit increase in

x

y

increases by 2 units for every 1-unit increase in

x

✅

y

increases by 4 units for every 1-unit increase in

x

y

increases by 8 units for every 1-unit increase in

x

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The slope is $m=\frac{13-5}{6-2}=2$ . Therefore, $y$ increases by 2 units whenever $x$ increases by 1 unit. This interpretation is more meaningful than simply calculating a numerical slope because it describes the rate of change.

Q2. Which statement correctly describes a line with slope $-3$ ?

A.Moving 3 units right causes

y

to increase by 1 unit

B.Moving 1 unit right causes

y

to decrease by 3 units ✅

C.Moving 3 units right causes

y

to decrease by 1 unit

D.Moving 1 unit left causes

y

to decrease by 3 units

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A slope of $-3$ means the ratio of vertical change to horizontal change is $-3:1$ . Thus, for every 1-unit increase in $x$ , the corresponding $y$ -value decreases by 3 units. The negative sign indicates downward movement from left to right.

Q3. Two lines have slopes $\frac{2}{3}$ and $\frac{4}{6}$ . A student claims the second line is steeper because its numerator is larger. How should this claim be evaluated?

A.The claim is correct because 4 is greater than 2

B.The claim is correct because 6 is greater than 3

C.The claim is incorrect because both slopes have the same value ✅

D.The claim is incorrect because slopes cannot be fractions

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Since $\frac{4}{6}$ simplifies to $\frac{2}{3}$ , both lines have exactly the same slope. A larger numerator alone does not determine steepness. The entire ratio of vertical change to horizontal change must be considered.

Q4. A ramp rises 3 meters while extending 12 meters horizontally. Another ramp rises 2 meters over 8 meters horizontally. Which conclusion is justified?

A.The first ramp is steeper

B.The second ramp is steeper

C.Both ramps have the same steepness ✅

D.There is not enough information to compare them

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The first ramp has slope $\frac{3}{12}=\frac14$ , while the second has slope $\frac{2}{8}=\frac14$ . Because their rise-to-run ratios are equal, the ramps have the same slope and therefore the same steepness.

Q5. A student calculates the slope between $(1,4)$ and $(5,10)$ as $\frac{5-1}{10-4}$ . What is the main error?

A.The student subtracted the coordinates in the wrong order

B.The student divided horizontal change by vertical change ✅

C.The student should add the coordinates instead

D.The student used too many coordinates

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Slope is calculated as vertical change divided by horizontal change, so the correct expression is $\frac{10-4}{5-1}=\frac64=\frac32$ . The student's calculation reverses rise and run, producing the reciprocal instead of the slope.

Q6. A line rises 10 units when $x$ increases by 4 units. Later, it rises another 15 units when $x$ increases by 6 units. What can be concluded about the line?

A.Its slope changes because the rises are different

B.Its slope is constant at

\frac52

✅

C.Its slope is constant at

\frac23

D.Its slope cannot be determined

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The first slope is $\frac{10}{4}=\frac52$ , and the second is $\frac{15}{6}=\frac52$ . Since the vertical-to-horizontal ratio remains constant, the relationship is consistent with a straight line having slope $\frac52$ .

Q7. A delivery company models travel distance $d$ against time $t$ . A driver travels from 40 km at hour 2 to 100 km at hour 5. What does the slope represent?

A.The driver's starting distance

B.The total distance traveled

C.The average rate of change of distance with respect to time ✅

D.The number of hours needed to travel 100 km

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The slope is $\frac{100-40}{5-2}=20$ km per hour. In this model, slope represents the average rate at which distance changes with time, so it describes the driver's average speed during that interval.

Q8. A temperature model changes from $18^\circ C$ at 2 p.m. to $12^\circ C$ at 5 p.m. If the change is linear, what does the slope indicate?

A.Temperature increases by

2^\circ C

per hour

B.Temperature decreases by

2^\circ C

per hour ✅

C.Temperature decreases by

6^\circ C

per hour

D.Temperature remains constant

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The slope is $\frac{12-18}{5-2}=\frac{-6}{3}=-2$ . Thus, temperature decreases at an average rate of $2^\circ C$ per hour. The negative slope is important because it indicates that temperature falls as time increases.

Q9. A student says a line through $(2,3)$ and $(6,11)$ has slope $4$ because $11-3=8$ and $6-2=4$ , then adds the two changes. What is the correct reasoning?

A.The slope is

12

because the changes should be added

B.The slope is

2

because the changes should be divided ✅

C.The slope is

4

because only the vertical change matters

D.The slope is

\frac12

because horizontal change must be divided by vertical change

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Slope compares the two changes by division: $m=\frac{11-3}{6-2}=\frac84=2$ . Adding the changes has no meaning for slope. Both vertical and horizontal changes are necessary to determine the rate of change.

Q10. A graph shows a line passing through $(-4,-2)$ and $(2,4)$ . Without finding an equation, which statement best describes the line?

A.It falls 1 unit for every 1 unit moved right

B.It rises 1 unit for every 1 unit moved right ✅

C.It rises 6 units for every 2 units moved right

D.It has zero slope because both coordinates change

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The slope is $\frac{4-(-2)}{2-(-4)}=\frac66=1$ . Therefore, moving one unit to the right corresponds to moving one unit upward. The graph represents a line with a positive slope and equal rise and run.

Q11. A line has slope $0.4$ . Another line has slope $-0.4$ . Which comparison is correct?

A.Both lines rise at the same rate

B.Both lines fall at the same rate

C.They have equal steepness but opposite directions ✅

D.The first line is twice as steep as the second

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The slopes have equal absolute value, $0.4$ , so their steepness is the same. However, their signs differ, meaning one rises from left to right while the other falls. Slope magnitude measures steepness, while sign indicates direction.

Q12. A line connects $A(3,7)$ to $B(9,19)$ . Another line connects $C(0,1)$ to $D(6,13)$ . A student concludes the first line is steeper because its points have larger coordinates. What is the best conclusion?

A.The first line is steeper

B.The second line is steeper

C.Both lines have the same slope ✅

D.The slopes cannot be compared

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: For the first line, $m=\frac{19-7}{9-3}=\frac{12}{6}=2$ . For the second, $m=\frac{13-1}{6-0}=\frac{12}{6}=2$ . Coordinate size does not determine steepness; the rate of vertical change relative to horizontal change does.

Q13. A hiking trail rises 120 m over a horizontal distance of 800 m. A map uses a different scale, showing the same trail as a 3 cm rise over 20 cm horizontally. Which statement is correct?

A.The map changes the actual slope

B.The actual trail slope is

0.15

, and the map preserves this ratio ✅

C.The actual trail slope is

6.67

, while the map slope is

0.15

D.The map cannot represent slope using scaled measurements

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The actual slope is $\frac{120}{800}=0.15$ , while the map gives $\frac{3}{20}=0.15$ . Because both vertical and horizontal dimensions are scaled proportionally, the slope ratio remains unchanged. This demonstrates why proportional maps can preserve steepness.

Q14. Consider all straight lines passing through $P(2,5)$ . One line rises 9 units while running 3 units; another falls 6 units while running 2 units. What surprising relationship can be concluded?

A.The first line is steeper

B.The second line is steeper

C.Both have the same slope ✅

D.Both have slope zero

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The first line has slope $\frac{9}{3}=3$ , while the second has slope $\frac{-6}{2}=-3$ . Their absolute slopes are equal, so they have the same steepness but opposite directions. The common point does not affect this comparison.

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