Digital SAT Math Practice Problems: Slope-Intercept Form
Slope-intercept form, y = mx + b, is the backbone of the SAT Math "Linear Relationships" domain and shows up in more questions than almost any other single skill on the exam. The College Board tests it from every angle: identifying the graph of a given equation, writing an equation from two points or a table of values, reading slope and y-intercept off a graph, interpreting what m and b mean in a real-world context, and comparing the slopes of several lines shown in the same coordinate plane. A shockingly large share of missed points on Digital SAT Math trace back to a shaky grip on this one topic, because it quietly sits underneath function notation questions, systems of equations, and real-world modeling problems as well.
The 31 problems below cover the full range of how slope-intercept form appears on the real test, including several graph-reading questions, constant-rate word problems (depreciation, tree growth, race times, billing rates), and function notation questions that hinge on finding m and b first. Every problem includes a complete step-by-step explanation, hidden by default so you can genuinely test yourself before checking your work. Work through them in order and pay close attention to the questions that ask you to interpret a slope or y-intercept in words — those are where careless errors cost the most points on test day. For unlimited additional practice, our free SAT Math QBank has over 2,500 more problems across every SAT Math topic, each with instant explanations.
1Identifying Graphs from Equations
Which of the following is the graph of the equation y = 3x + 1 in the xy-plane?
A)
B)
C)
D)
A) Graph A
B) Graph B
C) Graph C
D) Graph D
Answer Explanation
The equation y = 3x + 1 has a slope of 3 (steep and positive, rising quickly left to right) and a y-intercept of 1 (crossing the y-axis just above the origin).
Only one graph shows a steep, positive line crossing the y-axis slightly above the origin — not the origin itself, and not below it. The correct answer is A.
2Building a Linear Model from a Table
The table lists Gina's rollerblading speed x (mph) and her pulse f(x) (bpm): (4, 65), (8, 77), (12, 89). There is a linear relationship between Gina's speed and her pulse. Which of the following equations describes f(x)?
Speed (mph)
Pulse (bpm)
4
65
8
77
12
89
A) f(x) = 12x + 65
B) f(x) = 12x + 53
C) f(x) = 3x + 65
D) f(x) = 3x + 53
Answer Explanation
Find the slope using any two points: (77 − 65) / (8 − 4) = 12/4 = 3.
Use f(x) = 3x + b with the point (4, 65): 65 = 3(4) + b, so 65 = 12 + b, and b = 53. That gives f(x) = 3x + 53. The correct answer is D.
3Constant Rate of Change in Context
Milk production in a certain area dropped from 10 million cartons in 2000 to 2.7 million cartons in 2018. Assuming the milk production decreased at a constant rate, which linear function f best models the production, in millions of cartons, t years after 2000?
A) f(t) = 73/180 t + 10
B) f(t) = 27/180 t + 10
C) f(t) = −73/180 t + 10
D) f(t) = −27/180 t + 10
Answer Explanation
Find the slope using the two given points, (0, 10) and (18, 2.7): slope = (2.7 − 10) / (18 − 0) = −7.3/18 = −73/180.
The starting value in the year 2000 (t = 0) is 10 million cartons, so b = 10. That gives f(t) = −73/180 t + 10. The correct answer is C.
4Writing an Equation from Two Points
Which of the following is an equation of the line in the xy-plane that contains the points (1, 4) and (4, 16)?
A) y = 4x
B) y = x + 3
C) y = 3x + 4
D) y = (1/4)x
Answer Explanation
Find the slope: (16 − 4) / (4 − 1) = 12/3 = 4.
Use y = 4x + b with the point (1, 4): 4 = 4(1) + b, so b = 0. That gives y = 4x. The correct answer is A.
5Building a Linear Model from a Table
The table above shows the population of Hardlocke, Texas, for the years 1900 (711 people) and 1910 (687 people). If the relationship between population and year is linear, which function P models the population t years after 1900?
Year
Population
1900
711
1910
687
A) P(t) = 711 − 2.4t
B) P(t) = 711 − 24t
C) P(t) = 711 − 2.4(t − 1900)
D) P(t) = 711 − 24(t − 1900)
Answer Explanation
Find the rate of change per year: (687 − 711) / 10 = −24/10 = −2.4 people per year.
Since t already represents the number of years after 1900 (not the calendar year itself), the starting population of 711 needs no further adjustment: P(t) = 711 − 2.4t. The correct answer is A.
6Comparing Slopes of Several Lines
In the xy-plane, lines k, ℓ, and p are shown. Which of the following lists the slopes from least to greatest?
Lines k, ℓ, and p in the xy-plane
A) The slope of p, the slope of ℓ, the slope of k
B) The slope of ℓ, the slope of p, the slope of k
C) The slope of k, the slope of ℓ, the slope of p
D) The slope of p, the slope of k, the slope of ℓ
Answer Explanation
Line p slants down from upper left to lower right, so it has the only negative slope — making it the least. Lines k and ℓ both slant up from lower left to upper right (positive slopes), but line k is much steeper than line ℓ, so k has the greater slope.
From least to greatest: the slope of p, then the slope of ℓ, then the slope of k. The correct answer is A.
7Reading Slope from a Graph
The graph of y = mx + b, where m and b are constants, is shown in the xy-plane, passing through the points (−2, 6) and (−1, 3). What is the value of m?
Graph of y = mx + b
Answer Explanation
Use the two labeled points on the line to find the slope: m = (3 − 6) / (−1 − (−2)) = −3/1 = −3.
Check: the line y = −3x passes through (−2, 6) since −3(−2) = 6, and through (−1, 3) since −3(−1) = 3. So m = −3.
8Interpreting a Rate in Context
The equation p = 14.7 + 15.4d approximates the pressure p, in pounds per square inch, exerted on a peanut at a depth of d inches below the surface of a chocolate syrup vat. What is the increase in depth necessary to increase the pressure by 1 pound per square inch?
A) 1/14.7 in
B) 1/15.4 in
C) 14.7 in
D) 15.4 in
Answer Explanation
The coefficient 15.4 is the slope of the equation: pressure increases by 15.4 psi for every 1 inch increase in depth. To find how much depth is needed for a 1 psi increase, take the reciprocal of the rate.
1 psi ÷ 15.4 psi/in = 1/15.4 inches. The correct answer is B.
9Finding a Constant from a Graph
The graph of the linear function f is shown in the xy-plane. The slope of the graph of the linear function g is 3 times the slope of the graph of f. If the graph of g passes through the point (0, −8), what is the value of g(6)?
Graph of the linear function f
A) −8
B) −6
C) −2
D) 10
Answer Explanation
From the graph, f has a y-intercept of 1 and a slope of 1/3 (it rises gently — up 1 for every 3 units to the right).
The slope of g is 3 times f's slope: 3 × 1/3 = 1. Since g passes through (0, −8), g(x) = x − 8. So g(6) = 6 − 8 = −2. The correct answer is C.
10Solving for a Constant Using Slope
A line in the xy-plane passes through the origin and has a slope of 1/10. Which of the following points lies on the line?
A) (20, 2)
B) (10, 10)
C) (1, 10)
D) (0, 10)
Answer Explanation
A line through the origin with slope 1/10 has the equation y = (1/10)x.
Test point (20, 2): (1/10)(20) = 2, which matches. The correct answer is A.
11Finding y-intercept from Two Points
The function f is linear, f(1) = 20, and f(4) = 11. If f(x) = mx + b, where m and b are constants, what is the value of b?
A) 61/3
B) 23
C) 17
D) 34/3
Answer Explanation
Find the slope: m = (11 − 20) / (4 − 1) = −9/3 = −3.
Use f(1) = 20: 20 = −3(1) + b, so b = 23. The correct answer is B.
12Reading Slope to Find a Ratio
In the xy-plane, a point (not shown) with coordinates (s, t) lies on the graph of the linear function f, whose graph passes through the origin and the point (4, 2). If s and t are positive integers, what is the ratio of t to s?
Graph of the linear function f
A) 1 to 2
B) 2 to 3
C) 2 to 1
D) 3 to 2
Answer Explanation
Since the graph passes through the origin, f(x) = mx where m is the slope. Using the point (4, 2): m = 2/4 = 1/2.
For any point (s, t) on this line, t = (1/2)s, so t/s = 1/2, meaning the ratio of t to s is 1 to 2. The correct answer is A.
13Interpreting a Rate in Context
The equation s = 32.2t can be used to approximate the speed s, in feet per second, of an object t seconds after being dropped into a free fall. Which of the following is the best interpretation of the number 32.2 in this context?
A) The speed, in ft/s, of the object when it hits the ground
B) The initial speed, in ft/s, of the object when it is dropped
C) The speed, in ft/s, of the object t seconds after it is dropped
D) The increase in speed, in ft/s, of the object for each second after it is dropped
Answer Explanation
In the equation s = 32.2t, the number 32.2 is the slope — the constant that multiplies t. A slope represents the rate of change of the output per unit of input.
Since t is measured in seconds, 32.2 represents how much the speed increases for each additional second the object falls. The correct answer is D.
14Applying Slope-Intercept Form to a Table
The table above shows selected values of the linear function f. The function can be written in the form f(x) = ax + b, where a and b are constants. What is the value of a + b? (x = 4, f(x) = 15; x = 8, f(x) = 7)
x
f(x)
4
15
8
7
Answer Explanation
Find the slope: a = (7 − 15) / (8 − 4) = −8/4 = −2.
Use f(4) = 15: 15 = −2(4) + b, so 15 = −8 + b, and b = 23. Then a + b = −2 + 23 = 21.
15Interpreting a Rate in Context
The equation A = 205 + 24.14m can be used to estimate the body surface area A, in square inches, of a child with mass m, in pounds, where 7 ≤ m ≤ 67. Which statement is consistent with the equation?
A) For each increase of 205 pounds in mass, A increases by approximately 1 square inch.
B) For each increase of 1 pound in mass, A increases by approximately 205 square inches.
C) For each increase of 1 pound in mass, A increases by approximately 24.14 square inches.
D) For each increase of 24.14 pounds in mass, A increases by approximately 1 square inch.
Answer Explanation
The coefficient 24.14 is the slope of the equation, representing the rate of change of A per unit increase in m.
Since m is measured in pounds, 24.14 means A increases by approximately 24.14 square inches for every 1-pound increase in mass. The correct answer is C.
16Using Slope to Find an Unknown Coordinate
The table above shows the coordinates of three points on a line in the xy-plane, where k and n are constants: (1, −2), (k, 10), (5, n). If the slope of the line is 6, what is the value of n − k?
x
y
1
−2
k
10
5
n
Answer Explanation
Use the slope formula between (1, −2) and (5, n): (n − (−2)) / (5 − 1) = 6, so (n + 2)/4 = 6, giving n + 2 = 24 and n = 22.
Use the slope formula between (1, −2) and (k, 10): (10 − (−2)) / (k − 1) = 6, so 12/(k − 1) = 6, giving k − 1 = 2 and k = 3. So n − k = 22 − 3 = 19.
17Writing an Equation from Rate and Starting Value
The graph in the xy-plane of the linear function f contains the point (1, 6). For every increase of 3 units in x, f(x) increases by 4 units. Which of the following equations defines the function?
A) f(x) = −4/3 x + 10
B) f(x) = −3/4 x + 21/4
C) f(x) = 3/4 x + 2
D) f(x) = 4/3 x + 14/3
Answer Explanation
An increase of 4 units in f(x) for every 3 units in x means the slope is 4/3.
Use f(x) = (4/3)x + b with the point (1, 6): 6 = (4/3)(1) + b, so b = 6 − 4/3 = 14/3. That gives f(x) = (4/3)x + 14/3. The correct answer is D.
18Applying a Rate to Find a Head Start
The graph above shows the positions of Tom and Mahalia during a race. Tom and Mahalia each rollerbladed at a constant rate, and Mahalia was given a head start to shorten the distance she needed to skate. Tom finished the race in 13 seconds, and Mahalia finished the race in 11 seconds. According to the graph, Mahalia was given a head start of how many yards?
Distance (yards) vs. time (seconds) for Tom and Mahalia
A) 3
B) 10
C) 15
D) 20
Answer Explanation
Read each skater's starting distance directly from the graph — the value of their line at time t = 0. Tom's line starts at 0 yards (no head start), while Mahalia's line starts higher on the distance axis, since she began the race already partway to the finish.
Read each skater's starting distance directly from the graph — the value of their line at time t = 0. Tom's line starts at 0 yards (no head start) and reaches 50 yards at t = 13 seconds, matching his finish time. Mahalia's line starts at 10 yards and reaches 50 yards at t = 11 seconds, matching hers.
The head start is Mahalia's starting distance minus Tom's: 10 − 0 = 10 yards. The correct answer is B.
19Comparing Slope, Y-Intercept, and X-Intercept
The graph of the function f is shown in the xy-plane. The function f is defined by the equation f(x) = (a/b)x + c for positive constants a, b, and c, where a/b is a fraction in lowest terms. Which of the following orders a, b, and c from least to greatest?
Graph of the function f
A) a < b < c
B) a < c < b
C) b < c < a
D) c < a < b
Answer Explanation
From the graph, the y-intercept c is a small positive value (the line crosses the y-axis close to the origin). The slope a/b is a gentle, shallow positive fraction — the line rises slowly, meaning a (the numerator) is smaller than b (the denominator).
Reading the graph's rise-over-run gives a slope of roughly 2/5, and a y-intercept around 1. That puts c (1) as the smallest, a (2) next, and b (5) the largest: c < a < b. The correct answer is D.
20Writing an Equation from Two Points
In the xy-plane, the points (−1, 4) and (2, −1) lie on the graph of which of the following linear functions?
A) f(x) = −5/3 x + 7/3
B) f(x) = −3/5 x + 1
C) f(x) = 1/2 x − 2
D) f(x) = 5x + 9
Answer Explanation
Find the slope: (−1 − 4) / (2 − (−1)) = −5/3.
Use point-slope form with (−1, 4): y − 4 = −5/3(x + 1), so y = −5/3 x − 5/3 + 4 = −5/3 x + 7/3. The correct answer is A.
21Comparing Rates from a Verbal Description
A line in the xy-plane has a slope of 0. Which of the following could be the equation of the line?
A) x = 0
B) y = x
C) y = 0
D) y = −x
Answer Explanation
A slope of 0 means the line is perfectly horizontal — y never changes as x changes.
The equation y = 0 describes a horizontal line lying exactly on the x-axis, which has a slope of 0. The correct answer is C.
22Finding Slope in Terms of Variables
The line y = kx + 5, where k is a constant, is graphed in the xy-plane. If the line contains the point (c, d), where c ≠ 0 and d ≠ 0, what is the slope of the line in terms of c and d?
A) (d − 5) / c
B) (c − 5) / d
C) (4 − d) / c
D) (4 − c) / d
Answer Explanation
Since (c, d) lies on the line, substitute it into the equation: d = kc + 5.
Solve for k, the slope: d − 5 = kc, so k = (d − 5)/c. The correct answer is A.
23Writing an Equation with a Given Slope and Point
Which of the following is an equation of the line in the xy-plane that has slope −5 and passes through the point (0, 2)?
A) y = −5x − 2
B) y = −5x + 2
C) y = −5(x + 2)
D) y = −5(x − 3)
Answer Explanation
The point (0, 2) is the y-intercept, since it has an x-coordinate of 0. So b = 2 in y = mx + b.
With slope m = −5, the equation is y = −5x + 2. The correct answer is B.
24Applying a Rate Difference in Context
The equation C = 85h + 100 gives the amount C, in dollars, an HVAC technician charges for a job that takes h hours. Ms. Porter and Mr. Green each hired this technician. The technician worked 3 hours longer on Ms. Porter's job than on Mr. Green's job. How much more did the technician charge Ms. Porter than Mr. Green?
A) $85
B) $100
C) $170
D) $255
Answer Explanation
Only the number of hours worked differs between the two jobs, and each extra hour adds $85 to the cost (the slope of the equation). The flat $100 fee is the same for both and cancels out in the difference.
3 extra hours × $85 per hour = $255. The correct answer is D.
25Interpreting the Meaning of a Constant
A snake had a length of 10 inches when it was hatched. The equation 3n + 10 = 46 can be used to find how many months n it took the snake to reach a length of 46 inches. Which of the following is the best interpretation of the number 3 in this context?
A) The number of months it took the snake to triple its length
B) The length, in inches, of the snake when it was 1 month old
C) The average number of months it takes similar snakes to grow 46 inches
D) The average number of inches that the snake grew per month
Answer Explanation
In the equation, 10 is the snake's starting length and 46 is its final length. The number 3 is being multiplied by n (the number of months), which makes it the rate of change — how much the snake's length increases each month.
Since length is measured in inches and n is measured in months, 3 represents inches grown per month. The correct answer is D.
26Using a Table to Find a Function Value
The table above shows selected values for the function h: (0, 1), (1, 3), (3, 7). In the xy-plane, the graph of y = h(x) is a line. What is the value of h(6)?
x
h(x)
0
1
1
3
3
7
A) 9
B) 11
C) 13
D) 14
Answer Explanation
Find the slope using (0, 1) and (1, 3): (3 − 1) / (1 − 0) = 2. Check with (3, 7): 1 + 2(3) = 7, which confirms the pattern.
So h(x) = 2x + 1. Then h(6) = 2(6) + 1 = 13. The correct answer is C.
27Finding an Equation with a Constant
For the linear function f, the table above gives some values of x and their corresponding values f(x), where c is a constant: (0, c), (1, −2c), (2, −5c). Which of the following equations defines f?
x
f(x)
0
c
1
−2c
2
−5c
A) f(x) = −5cx − 5c
B) f(x) = −3cx + c
C) f(x) = −cx + c
D) f(x) = cx + c
Answer Explanation
The y-intercept is f(0) = c, so b = c. Find the slope using f(0) and f(1): slope = (−2c − c) / (1 − 0) = −3c.
That gives f(x) = −3cx + c. Check with f(2): −3c(2) + c = −6c + c = −5c, which matches. The correct answer is B.
28Using a Table to Find a Function Value
Some values of the linear function f are shown in the table above: (0, −2), (3, 7), (7, 19). What is the value of f(5)?
x
f(x)
0
−2
3
7
7
19
A) 10
B) 13
C) 14
D) 15
Answer Explanation
Find the slope using (0, −2) and (3, 7): (7 − (−2)) / (3 − 0) = 9/3 = 3.
So f(x) = 3x − 2. Then f(5) = 3(5) − 2 = 15 − 2 = 13. The correct answer is B.
29Constant-Rate Depreciation
A business owner purchased a vehicle valued at $90,000. The value of the vehicle depreciates by the same amount each year so that after 5 years the value will be $50,000. Which equation gives the value v, in dollars, of the vehicle t years after it was purchased, for 0 ≤ t ≤ 5?
A) v = 50,000 − 8,000t
B) v = 90,000 − 50,000t
C) v = 90,000 − 8,000t
D) v = 90,000 + 8,000t
Answer Explanation
Total depreciation over 5 years is $90,000 − $50,000 = $40,000, so the yearly rate is $40,000 ÷ 5 = $8,000 per year.
Starting value is $90,000, decreasing by $8,000 each year: v = 90,000 − 8,000t. The correct answer is C.
30Reading a Graph to Find a Slope-Intercept Model
The graph above displays the total cost C, in dollars, of renting a jet ski for h hours, passing through the points (0, 2), (1, 6), (2, 10), (3, 14), (4, 18), and (5, 22). Which of the following represents the relationship between h and C?
Total cost C (dollars) of renting a jet ski for h hours
A) h = 4C
B) C = h + 2
C) C = 2h
D) C = 4h + 2
Answer Explanation
Find the slope using consecutive points: (6 − 2) / (1 − 0) = 4. The line crosses the C-axis at 2, so the y-intercept is 2.
That gives C = 4h + 2. The correct answer is D.
31Interpreting the Y-Intercept in Context
The graph above displays the total cost C, in dollars, of renting a jet ski for h hours. What does the C-intercept represent in the graph?
Total cost C (dollars) of renting a jet ski for h hours
A) The total number of jet skis rented
B) The total number of hours the jet ski is rented
C) The increase in cost to rent the jet ski for each additional hour
D) The initial cost of renting the jet ski
Answer Explanation
The C-intercept is the value of C when h = 0 — before any hours of rental time have passed.
That represents the flat starting fee charged just to rent the jet ski, before any hourly charges apply. The correct answer is D.
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