Tip: Rise over run — the change in y goes on top. When one of the points has x = 0, you can read b directly from it. Always test both given points in your final equation.
Question 2 of 36
Fill in
What is the slope of the line that passes through (1, 4) and (5, 12)?
m =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Slope = \dfrac{\text{change in } y}{\text{change in } x}. Subtract in the same order on top and bottom.
Step-by-step solution
Label the points: (x_1, y_1) = (1, 4) and (x_2, y_2) = (5, 12).
Change in y (rise): y_2 - y_1 = 12 - 4 = 8.
Change in x (run): x_2 - x_1 = 5 - 1 = 4.
Divide: m = \dfrac{8}{4} = 2.
Answer: m = 2
Check: Reverse the order: \dfrac{4 - 12}{1 - 5} = \dfrac{-8}{-4} = 2 ✓ Also, the line rises from left to right, so a positive slope makes sense.
Tip: Slope is the first step of writing any line through two points. With m = 2, the line here is y = 2x + 2 (since 4 = 2(1) + b gives b = 2).
Question 3 of 36
A line passes through the points (2, 3) and (4, 7).
Which equation represents the line in point-slope form?
Hint
Find the slope first. Then use y - y_1 = m(x - x_1) with one of the points.
(Using (4, 7) instead gives y - 7 = 2(x - 4) — the same line.)
Answer: y - 3 = 2(x - 2)
Check: Expand: y - 3 = 2x - 4, so y = 2x - 1. Test (4, 7): 2(4) - 1 = 7 ✓
Tip: Point-slope form uses minus signs, and each coordinate stays next to its own letter: y_1 with y, x_1 with x.
Question 4 of 36
Fill in
What is the slope of the line that passes through (-3, 8) and (5, -4)? Enter your answer as a decimal or a fraction.
m =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Slope = \dfrac{y_2 - y_1}{x_2 - x_1}. Subtract in the same order on the top and the bottom.
Step-by-step solution
Label the points: (x_1, y_1) = (-3, 8) and (x_2, y_2) = (5, -4).
Change in y (rise): y_2 - y_1 = -4 - 8 = -12.
Change in x (run): x_2 - x_1 = 5 - (-3) = 5 + 3 = 8.
Slope: m = \dfrac{-12}{8} = -\dfrac{3}{2} = -1.5.
Answer: m = -1.5 (or -\tfrac{3}{2})
Check: Swap the order: \dfrac{8 - (-4)}{-3 - 5} = \dfrac{12}{-8} = -1.5 ✓ Same answer either way. The line goes from upper left to lower right, so a negative slope makes sense.
Tip: Subtracting a negative (5 - (-3)) is the most common slip. Rewrite it as addition before you calculate.
Question 5 of 36
Find a point-slope equation for the straight line that contains both (-1, 6) and (3, -2).
Hint
Find the slope first. Then use either point in y - y_1 = m(x - x_1) and watch the signs.
Substitute (-2, 5) into y = -2x + b: 5 = -2(-2) + b = 4 + b.
Solve: b = 1.
Equation: y = -2x + 1.
Answer: y = -2x + 1
Check: Other point (4, -7): -2(4) + 1 = -8 + 1 = -7 ✓
Tip:4 - (-2) = 6 — subtracting a negative is adding. And a line that goes from a high point on the left, (-2, 5), to a low point on the right, (4, -7), must have a negative slope.
Question 8 of 36
The graph shows a line on the coordinate plane.
Which equation represents the line?
Hint
Use the two marked points to find the slope. Does the line go up or down from left to right? Where does it cross the y-axis?
Tip: On a graph, first check direction: falling left to right means a negative slope, which removes choices fast. Then count: from (-3, 4) to (3, 0) is 6 right and 4 down.
Question 9 of 36
Fill in
A phone plan charges a one-time activation fee plus the same amount each month. The total paid after 3 months is $145, and the total paid after 7 months is $285.
What is the activation fee, in dollars?
$
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Each sentence gives a point (\text{months}, \text{total}). Find the monthly rate first, then the value at 0 months.
Tip: The change in height (-2.25) happens over 3 hours, so divide by 3. A shrinking quantity always has a negative slope, and the starting height must be taller than 10 inches.
Question 11 of 36
Drag and drop
Drag each equation to the pair of points its line passes through. One equation is not used.
(0, 2) and (1, 5)
(1, 1) and (3, -3)
(-2, 0) and (2, 2)
Tap a tile, then tap a box to place it.
Hint
For each pair, find the slope \dfrac{y_2 - y_1}{x_2 - x_1}, then find b. Check both points in your chosen equation.
Step-by-step solution
(0, 2) and (1, 5):m = \dfrac{5 - 2}{1 - 0} = 3; (0, 2) is on the y-axis so b = 2 → y = 3x + 2.
(1, 1) and (3, -3):m = \dfrac{-3 - 1}{3 - 1} = \dfrac{-4}{2} = -2; 1 = -2(1) + b so b = 3 → y = -2x + 3.
(-2, 0) and (2, 2):m = \dfrac{2 - 0}{2 - (-2)} = \dfrac{2}{4} = \dfrac{1}{2}; 2 = \tfrac{1}{2}(2) + b so b = 1 → y = \tfrac{1}{2}x + 1.
Not used:y = \tfrac{1}{3}x + 2 — that slope is run over rise for the first pair.
Answer:y = 3x + 2 → first box; y = -2x + 3 → second box; y = \tfrac{1}{2}x + 1 → third box.
Tip: Checking the second point of each pair is the fastest way to confirm a match.
Question 12 of 36
A moving-truck company charges a daily fee plus a charge for each mile driven. A one-day rental driven 40 miles costs $59.80. A one-day rental driven 100 miles costs $89.80.
Which equation gives the cost C, in dollars, of a one-day rental driven m miles?
Hint
Use the two situations as points (m, C). Find the cost per mile, then the daily fee.
Step-by-step solution
Points: (40, 59.80) and (100, 89.80).
Rate: \dfrac{89.80 - 59.80}{100 - 40} = \dfrac{30}{60} = 0.5 dollar per mile.
Find b: 59.80 = 0.5(40) + b = 20 + b, so b = 39.80.
Equation: C = 0.5m + 39.80.
Answer: C = 0.5m + 39.80 ($0.50 per mile plus a $39.80 daily fee)
Check:0.5(100) + 39.80 = 50 + 39.80 = 89.80 ✓
Tip: Don't divide one total by its miles (59.80 \div 40) — that only works when there is no fee. Using two points separates the per-mile charge from the fixed fee.
Question 13 of 36
The points (3, 1) and (9, 5) both lie on one straight line. Which slope-intercept equation describes it?
Hint
Find the slope as a fraction in lowest terms, then substitute one point to find b.
Tip: The extra 30 guests cost an extra $750 — that difference gives the per-guest price. Then remove the guest charges from either total to reveal the fixed fee.
Question 17 of 36
Drag and drop
Drag each pair of points to the equation of the line that passes through both points. One pair is not used.
y = 2x - 1
y = -x + 4
y = \tfrac{1}{2}x + 3
Tap a tile, then tap a box to place it.
Hint
Substitute both points of a pair into an equation. A pair belongs to a line only if both points work.
Step-by-step solution
(1, 1) and (4, 7): slope \dfrac{7 - 1}{4 - 1} = 2; 1 = 2(1) + b gives b = -1 → y = 2x - 1.
(-1, 5) and (3, 1): slope \dfrac{1 - 5}{3 - (-1)} = \dfrac{-4}{4} = -1; 5 = -1(-1) + b gives b = 4 → y = -x + 4.
(-2, 2) and (4, 5): slope \dfrac{5 - 2}{4 - (-2)} = \dfrac{3}{6} = \tfrac{1}{2}; 2 = \tfrac{1}{2}(-2) + b gives b = 3 → y = \tfrac{1}{2}x + 3.
Not used: (1, 1) and (7, 4). Its slope is \tfrac{3}{6} = \tfrac{1}{2}, the same as the third line, but 1 = \tfrac{1}{2}(1) + b gives b = \tfrac{1}{2}, not 3.
Tip: Matching the slope is not enough — parallel lines share a slope. Always confirm the intercept too.
Question 18 of 36
A hiker climbs a mountain trail at a steady rate. After 2 hours she is at an elevation of 5,300 feet. After 5 hours she is at an elevation of 6,350 feet.
Which equation gives her elevation E, in feet, after t hours of hiking?
Hint
Find the climbing rate (feet per hour) from the two data points, then find the elevation at t = 0.
Step-by-step solution
Points (t, E): (2, 5300) and (5, 6350).
Slope: m = \dfrac{6350 - 5300}{5 - 2} = \dfrac{1050}{3} = 350 feet per hour.
Find b: 5300 = 350(2) + b = 700 + b, so b = 4600. She started at 4,600 feet.
Find b: 9 = -2(-2) + b = 4 + b, so b = 5. The line is y = -2x + 5.
The x-intercept has y = 0: 0 = -2x + 5.
Solve: 2x = 5, so x = 2.5.
Answer: x = 2.5
Check:-2(2.5) + 5 = -5 + 5 = 0 ✓. Also, the line goes from y = 9 down to y = -3, so it must cross y = 0 between x = -2 and x = 4 — and 2.5 is in that range ✓.
Tip:y-intercept: set x = 0. x-intercept: set y = 0. Mixing these up gives the answer 5 instead of 2.5.
Question 20 of 36
Fill in
A line passes through the points (-4, 7) and (6, -8).
What is the y-coordinate of the point where the line crosses the y-axis?
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Find the slope, then substitute one point into y = mx + b and solve for b.
Substitute (-4, 7) into y = -1.5x + b:
7 = -1.5(-4) + b = 6 + b
Solve: b = 7 - 6 = 1.
The line is y = -1.5x + 1; it crosses the y-axis at (0, 1).
Answer: 1
Check:(6, -8): -1.5(6) + 1 = -9 + 1 = -8 ✓
Tip:-1.5 \times (-4) = +6. When both factors are negative the product is positive — sign errors here would give b = 13.
Question 21 of 36
The points (1, 5), (4, 11) and (k, 21) all lie on the same line.
What is the value of k?
Hint
Use the two complete points to write the equation of the line. Then substitute y = 21.
Step-by-step solution
Slope from the complete points: m = \dfrac{11 - 5}{4 - 1} = \dfrac{6}{3} = 2.
Find b: 5 = 2(1) + b, so b = 3. Line: y = 2x + 3.
The third point has y = 21 and x = k: 21 = 2k + 3.
Subtract 3: 18 = 2k. Divide by 2: k = 9.
Answer: k = 9
Check: Slope from (4, 11) to (9, 21): \dfrac{21 - 11}{9 - 4} = \dfrac{10}{5} = 2 ✓ Same slope, so the points are on one line.
Tip: "Same line" means "same slope between any two of the points." You can also solve \dfrac{21 - 11}{k - 4} = 2 directly: 10 = 2(k - 4), k = 9.
Question 22 of 36
A straight line contains (-3, 10) and (1, 2). Further along the same line, the y-coordinate reaches -8 at the point (k, -8).
Find k.
Hint
Find the slope from the two complete points. The third point must give the same slope.
Step-by-step solution
Slope from (-3, 10) to (1, 2): m = \dfrac{2 - 10}{1 - (-3)} = \dfrac{-8}{4} = -2.
The slope from (1, 2) to (k, -8) must also be -2:
\dfrac{-8 - 2}{k - 1} = -2
Multiply both sides by (k - 1): -10 = -2(k - 1).
Divide by -2: k - 1 = 5, so k = 6.
Answer: k = 6
Check: The line is y = -2x + 4 (since 2 = -2(1) + b gives b = 4). At x = 6: -12 + 4 = -8 ✓.
Tip: Think in steps: y must fall 10 more units from (1, 2). At 2 units down per 1 unit right, that is 5 units right: 1 + 5 = 6.
Question 23 of 36
Select all
A new freezer cools down at a steady rate after it is switched on. 30 minutes after it is switched on, the temperature inside is 52^\circ\text{F}. 75 minutes after it is switched on, the temperature is 25^\circ\text{F}.
Let T be the temperature in ^\circ\text{F}t minutes after the freezer is switched on. Select all statements that are true.
Select all that apply.
Hint
Find the slope (degrees per minute) and the temperature at t = 0. Then test each statement.
Step-by-step solution
Points (t, T): (30, 52) and (75, 25).
Slope: m = \dfrac{25 - 52}{75 - 30} = \dfrac{-27}{45} = -0.6. The temperature drops 0.6°F per minute. ✓
Find b: 52 = -0.6(30) + b = -18 + b, so b = 70. The temperature was 70°F when it was switched on. ✓ Model: T = -0.6t + 70.
When is T = 10? 10 = -0.6t + 70 \Rightarrow -0.6t = -60 \Rightarrow t = 100 minutes. ✓
The false statements:
"52°F at the start" uses a data value as the intercept.
T = 0.6t + 70 has the wrong sign: the freezer cools, it does not warm.
"Drops about 1.67°F per minute" is run over rise (45 \div 27).
Answer: it drops 0.6°F per minute; it started at 70°F; it reaches 10°F after 100 minutes.
Check:-0.6(75) + 70 = -45 + 70 = 25 ✓.
Tip: Units help: the slope is °F per minute, so the temperature change goes on top and the time change goes on the bottom.
Question 24 of 36
Drop-down
The table shows points on a line. The x-values go up by 3 each time.
x
-2
1
4
7
y
11
5
-1
-7
Select the values that complete the equation of the line.
y =x
Choose an answer from each menu.
Hint
The y-values change by the same amount each step, but each step in x is 3, not 1. The table has no x = 0 column, so you must calculate b.
Step-by-step solution
From one column to the next, x increases by 3 and y decreases by 6 (for example 11 \to 5).
Slope: m = \dfrac{-6}{3} = -2.
Find b with (1, 5): 5 = -2(1) + b = -2 + b, so b = 7.
Tip: When the x-values do not go up by 1, divide the change in y by the change in x. Using -6 as the slope is the most common mistake with tables like this.
Question 25 of 36
Select all
Several equations are listed below. Select all of them whose graph contains both (-4, 1) and (2, 4).
Select all that apply.
Hint
Find the slope. Point-slope form can use either point, and slope-intercept form is also allowed.
Point-slope with (-4, 1): y - 1 = \tfrac{1}{2}(x + 4) ✓.
Point-slope with (2, 4): y - 4 = \tfrac{1}{2}(x - 2) ✓.
Slope-intercept: 1 = \tfrac{1}{2}(-4) + b = -2 + b, so b = 3: y = \tfrac{1}{2}x + 3 ✓.
The wrong ones:
y - 1 = \tfrac{1}{2}(x - 4) uses the point (4, 1) — sign error.
y - 4 = 2(x - 2) uses run over rise for the slope.
y = \tfrac{1}{2}x + 1 uses a y-coordinate as the intercept.
Answer: y - 1 = \tfrac{1}{2}(x + 4), y - 4 = \tfrac{1}{2}(x - 2) and y = \tfrac{1}{2}x + 3.
Check: Simplify the two point-slope equations: y = \tfrac{1}{2}x + 2 + 1 = \tfrac{1}{2}x + 3 and y = \tfrac{1}{2}x - 1 + 4 = \tfrac{1}{2}x + 3 ✓. All three are the same line.
Tip: In select-all items, expect more than one correct form. Converting each to y = mx + b is the quickest way to compare them.
Question 26 of 36
Look at the rising line and its two labeled points.
Which point-slope equation describes this line?
Hint
Find the slope from the two marked points. The answer choices are in point-slope form — check the signs inside the parentheses.
Point-slope form with (-3, -4): y - (-4) = \tfrac{2}{3}\big(x - (-3)\big), which is
y + 4 = \tfrac{2}{3}(x + 3)
Answer: y + 4 = \tfrac{2}{3}(x + 3)
Check: Point (3, 0): left 0 + 4 = 4; right \tfrac{2}{3}(6) = 4 ✓. In slope-intercept form, y = \tfrac{2}{3}x + 2 - 4 = \tfrac{2}{3}x - 2, which matches the graph crossing at (0, -2) ✓.
Tip: Both coordinates of (-3, -4) are negative, so both signs in point-slope form become plus: (y + 4) and (x + 3).
Question 27 of 36
An electric company charges a monthly service fee plus a fixed price for each kilowatt-hour (kWh) used. One month a customer used 400 kWh and paid $68.60. The next month she used 650 kWh and paid $102.85.
What is the monthly service fee?
Hint
Find the price per kWh from the two months. Then remove the energy charges from either bill.
Step-by-step solution
Points (k, C): (400, 68.60) and (650, 102.85).
Price per kWh (slope): m = \dfrac{102.85 - 68.60}{650 - 400} = \dfrac{34.25}{250} = 0.137 dollars per kWh.
Energy charge for 400 kWh: 0.137 \times 400 = 54.80.
Substitute (4, 5) into y = 0.6x + b: 5 = 0.6(4) + b = 2.4 + b.
Subtract: b = 5 - 2.4 = 2.6.
Answer: 2.6 (or \tfrac{13}{5})
Check: Use the other point (-6, -1): 0.6(-6) + 2.6 = -3.6 + 2.6 = -1 ✓.
Tip: The y-intercept of a line through two points does not have to be a whole number. Converting the slope to a decimal (\tfrac{3}{5} = 0.6) makes the rest easy on the calculator.
Question 30 of 36
A line passes through the points (-2, -7) and (3, 8).
Which point also lies on this line?
Hint
Write the equation of the line, then test each answer choice in it.
Tip: Large numbers make careful setup even more important. Write the points with their units, find the slope, and let the calculator do the arithmetic.
Question 32 of 36
Fill in
A line passes through the points (-4, 5) and (6, k). The line crosses the y-axis at (0, 7).
What is the value of k?
k =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
The y-intercept is a second complete point. Use it with (-4, 5) to find the slope.
Step-by-step solution
Two complete points are known: (-4, 5) and the y-intercept (0, 7).
The point (6, k) is on the line: k = \tfrac{1}{2}(6) + 7 = 3 + 7 = 10.
Answer: k = 10
Check: Slope from (-4, 5) to (6, 10): \dfrac{10 - 5}{6 - (-4)} = \dfrac{5}{10} = \dfrac{1}{2} ✓.
Tip: A y-intercept given as a point (0, b) is just another point on the line. Once you see two complete points, the problem becomes a normal two-point problem.
Question 33 of 36
Select all
A print shop charges a setup fee plus a fixed price per flyer. An order of 200 flyers costs $46. An order of 500 flyers costs $85.
Let C be the cost in dollars of an order of n flyers. Select all statements that are true.
Select all that apply.
Hint
Find the price per flyer from the two orders, then the setup fee. Be suspicious of any statement that just doubles or divides a total.
Step-by-step solution
Points (n, C): (200, 46) and (500, 85).
Price per flyer: m = \dfrac{85 - 46}{500 - 200} = \dfrac{39}{300} = 0.13 dollars. ✓
Setup fee: 46 = 0.13(200) + b = 26 + b, so b = 20. ✓ Model: C = 0.13n + 20.
Tip: Uneven gaps in a table are a trap: the drop from week 2 to week 5 is $315, but that covers 3 weeks. Always divide by the change in x.
Question 35 of 36
A kayak rental costs a fixed fee plus an hourly rate. Renting for 2 hours costs $38 and renting for 5 hours costs $77. Tomás wrote an equation for the cost C of renting for h hours.
In b = y_1 - m x_1, the x-value and the y-value must come from the same point.
Step-by-step solution
Step 1 is correct: (hours, cost) pairs.
Step 2 is correct: $39 more for 3 more hours is $13 per hour.
Step 3 is wrong: Tomás used y = 77 from the 5-hour point but x = 2 from the 2-hour point. He must use one point:
b = 38 - 13(2) = 12 \quad\text{or}\quad b = 77 - 13(5) = 12
The correct model is C = 13h + 12 (fixed fee $12).
Answer: Step 3 mixes coordinates from different points; the fee is $12.
Check:13(2) + 12 = 38 ✓ and 13(5) + 12 = 77 ✓. Tomás's model gives 13(2) + 51 = 77 for 2 hours, which is wrong.
Tip: After finding b, test both data points. A correct line must reproduce every given value.
Question 36 of 36
Fill in
A straight line contains the points (-2.5, 4) and (1.5, -6). What is its slope m? Enter your answer as a decimal or a fraction.
m =
Type your answer in the box. Fractions like 3/4 are fine.
Hint
Use the slope formula with the decimals. Be careful subtracting a negative decimal.
Check: Reverse the order: \dfrac{4 - (-6)}{-2.5 - 1.5} = \dfrac{10}{-4} = -2.5 ✓. The y-values fall as x rises, so the slope is negative ✓.
Tip: Decimal coordinates follow exactly the same rules. On the calculator, use the (-) key for negative numbers and put the whole top and bottom in parentheses.
