Slope & the Forms of a Line — Practice
Three ways to write the same line. Each one is best at answering a different question.
What these questions cover
These questions use 12 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Reciprocal — The flip of a fraction. The reciprocal of 35 is 53.
- Integer — A whole number, positive, negative, or zero.
- Coefficient — The number multiplying a variable — the 3 in 3n.
- 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.
- Distribute — Multiply a bracket out: 3(x + 4) = 3x + 12.
- 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.
- Point-slope form — y - y1 = m(x - x1). Best when you know a point and the slope.
- Standard form — Ax + By = C. Best for finding intercepts fast.
- Parallel — Same slope, different intercept.
- Perpendicular — Slopes multiply to -1 — negative reciprocals.
Problem 1
Write the equation of the line with slope 3 through (2, 5), in point-slope form.
Worked answer
y - 5 = 3(x - 2)
y - 5 = 3(x - 2)
Problem 2
Write y - 4 = 2(x - 1) in slope-intercept form.
Worked answer
y = 2x + 2
y - 4 = 2x - 2
y = 2x + 2
Problem 3
Find the x-intercept of 3x + 4y = 24.
Worked answer
8
3x = 24
x = 8
Problem 4
Find the y-intercept of 3x + 4y = 24.
Worked answer
6
4y = 24
y = 6
Problem 5
A line parallel to y = 5x - 2 has what slope?
Worked answer
5
m = 5
Problem 6
A line perpendicular to y = 4x + 1 has what slope?
Worked answer
-1/4
Reciprocal of 4 is 14
Negated: -14
Problem 7
A line perpendicular to y = -25x + 3 has what slope?
Worked answer
5/2
Reciprocal: -52
Negated: 52
Problem 8
Find the equation of the line through (1, 3) and (4, 12).
Worked answer
y = 3x
m = 12 - 34 - 1 = 3
3 = 3(1) + b, so b = 0
y = 3x
Problem 9
Write the equation of the line with slope -2 through (-3, 8), in slope-intercept form.
Worked answer
y = -2x + 2
y - 8 = -2(x + 3)
y - 8 = -2x - 6
y = -2x + 2
Problem 10
What is the equation of the horizontal line through (7, -3)?
Worked answer
y = -3
y = -3
Problem 11
What is the equation of the vertical line through (7, -3)?
Worked answer
x = 7
x = 7
Problem 12
Which line is parallel to 2x + 3y = 9?
- y = -23x + 1
- y = 32x + 1
- y = 23x + 1
- y = -32x + 1
Worked answer
y = -23x + 1
3y = -2x + 9, so y = -23x + 3
Parallel needs the same slope -23
Problem 13
Find the line perpendicular to y = 34x - 5 through (8, 2).
Worked answer
y = -4x/3 + 38/3
m = -43
y - 2 = -43(x - 8)
y = -43x + 323 + 2 = -43x + 383
Problem 14
A line through (-2, 7) and (3, -3) meets the y-axis where?
Worked answer
3
m = -3 - 73 + 2 = -2
7 = -2(-2) + b, so b = 3
At (0, 3)
Problem 15
Write y = 23x - 4 in standard form Ax + By = C with integer coefficients.
Worked answer
2x - 3y = 12
3y = 2x - 12
2x - 3y = 12
Problem 16
Are y = 3x + 1 and 2x + 6y = 5 perpendicular? Answer yes or no.
Worked answer
yes
Second: 6y = -2x + 5, so y = -13x + 56
3 × (-13) = -1
So yes