Slope & Unit Rate — Challenge
Slope is a rate of change wearing a geometric coat: how much y moves for every one step in x.
What these problems cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Rate — A ratio of two different units, like miles per hour or dollars per pound.
- Unit rate — A rate with denominator 1 — how much for exactly one.
- Proportional — Two quantities with a constant ratio: yx = k always.
- Slope — How much y changes for each 1 that x increases.
- Rise — The vertical change, y2 - y1.
- Run — The horizontal change, x2 - x1.
Problem 1
What is the slope of the line through (0,0) and (2,6)?
- 2
- 3
- 4
- 6
- 13
Worked answer
3
Slope is the change in y divided by the change in x.
The change in y is 6 and the change in x is 2, so the slope is 62 = 3.
Dividing the other way round gives 13, which is the reciprocal and not the slope.
Problem 2
What is the slope of the line through (1,5) and (3,1)?
- -2
- -1
- 2
- 3
- -3
Worked answer
-2
The change in y is 1-5 = -4 and the change in x is 3-1 = 2.
So the slope is -42 = -2.
The slope is negative because y falls as x rises.
Problem 3
What is the slope of the line 2x+3y=6?
- -23
- 23
- -32
- 2
- -2
Worked answer
-23
Rearrange into the form y = mx+b, where m is the slope.
3y = -2x+6, so y = -23x+2.
The slope is therefore -23. Reading the coefficient of x in the original equation would wrongly give 2.
Problem 4
What is the slope of the line y=5x-1?
- -1
- 1
- 5
- -5
- 4
Worked answer
5
The equation is already in the form y = mx+b.
The slope is the number multiplying x, which is 5.
The -1 is the y-intercept, not the slope.
Problem 5
What is the slope of the line through (-2,4) and (2,4)?
- 0
- 1
- 2
- Undefined
- -1
Worked answer
0
The change in y is 4-4 = 0, while the change in x is 2-(-2) = 4.
So the slope is 04 = 0: the line is horizontal.
A slope is undefined only when the change in x is zero, which would make the line vertical.
Problem 6
A line has slope 3 and passes through (1,2). What is its y-intercept?
- -1
- 1
- 2
- 3
- 5
Worked answer
-1
Start from y = 3x+b and substitute the point: 2 = 3(1)+b.
So b = 2-3 = -1.
The line is y = 3x-1, and checking x=1 gives y=2 as required.
Problem 7
What is the slope of a vertical line?
- 0
- 1
- Undefined
- -1
- 10
Worked answer
Undefined
On a vertical line every point has the same x value, so the change in x is 0.
Slope would require dividing by that 0, which is not a number, so the slope is undefined.
A horizontal line is the opposite case: its change in y is 0, giving a slope of exactly 0.
Problem 8
What is the slope of any line parallel to y=4x+1?
- -4
- 14
- -14
- 4
- 1
Worked answer
4
Parallel lines never meet, which means they rise at exactly the same rate.
So a parallel line has the same slope, 4.
The value -14 would be the slope of a perpendicular line instead.