Linear Functions & y = mx + b — Challenge
Two numbers describe a whole line: where it starts, and how fast it changes.
What these problems cover
These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Slope — How much y changes for each 1 that x increases.
- y-intercept — The output when x = 0 — where the line crosses the vertical axis.
- Slope-intercept form — y = mx + b, with m the slope and b the y-intercept.
- x-intercept — The x value when y = 0 — where it crosses the x-axis.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
If y=3x+2, what is y when x=4?
- 9
- 11
- 14
- 18
- 20
Worked answer
14
Substitute x=4, doing the multiplication before the addition.
y = 3(4)+2 = 12+2 = 14.
Problem 2
What is the equation of the line with slope -2 and y-intercept 5?
- y=-2x+5
- y=2x+5
- y=5x-2
- y=-2x-5
- y=-5x+2
Worked answer
y=-2x+5
A line is written y = mx+b, with m the slope and b the y-intercept.
Here m=-2 and b=5, giving y = -2x+5.
Swapping the two numbers gives y=5x-2, a quite different line.
Problem 3
A line passes through (0,4) and (2,10). What is its slope?
- 2
- 3
- 4
- 6
- 7
Worked answer
3
The change in y is 10-4 = 6 and the change in x is 2-0 = 2.
The slope is 62 = 3.
Since the first point is on the y-axis, the line is y = 3x+4.
Problem 4
What is the x-intercept of the line y=-x+7?
- -7
- 0
- 1
- 7
- -1
Worked answer
7
The x-intercept is where the line meets the x-axis, so set y=0.
0 = -x+7 gives x = 7.
The y-intercept is 7 as well here, but that is a coincidence of this particular line.
Problem 5
A phone plan charges $20 a month plus $0.10 for each minute of calls. What is the cost in dollars for a month with 150 minutes?
- $30
- $32
- $35
- $50
- $170
Worked answer
$35
The relationship is linear: cost = 20+0.10m, where the 20 is a fixed charge and the 0.10 is the rate.
With m=150: 0.10· 150 = 15 dollars of calls.
The total is 20+15 = 35 dollars.
Problem 6
Which point lies on the line y=4x-3?
- (2,5)
- (1,-3)
- (0,3)
- (3,2)
- (5,2)
Worked answer
(2,5)
Test each point by substituting its x value and comparing with its y value.
For (2,5): 4(2)-3 = 5, which matches.
For (1,-3): 4(1)-3 = 1, not -3, so that point is not on the line.
Problem 7
What is the slope of the line y=7?
- 0
- 7
- 1
- Undefined
- -7
Worked answer
0
This equation says y is 7 no matter what x is, so the line is horizontal.
A horizontal line has no rise, so its slope is 0.
Writing it as y = 0x+7 makes the slope visible.
Problem 8
What is the slope of the line 2y=6x+8?
- 2
- 3
- 4
- 6
- 8
Worked answer
3
The equation is not yet in the form y = mx+b, because of the 2 on the left.
Divide every term by 2: y = 3x+4.
The slope is 3. Reading 6 straight off the unrearranged equation is the trap here.