Slope & the Forms of a Line — Print
Three ways to write the same line. Each one is best at answering a different question.
What the practice 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.
Here is a line on a grid. Describe it to someone on the phone.
You could say: "it passes through (3, 5) and climbs 5 for every 6 across."
Or: "it crosses the y-axis at 2.5 and has slope 56."
Or: "5x - 6y = -15."
All three describe the same pencil line. Nothing on the grid changed — only the sentence.
So "which form should I use?" is really "what am I about to do?" Graphing wants y = mx + b. A point and a slope want point-slope, which needs no algebra at all. Hunting intercepts wants Ax + By = C.
Choosing a form
Given a point and a slope, use point-slope. Nothing to solve: substitute and you are done.
Given two points, find the slope, then use point-slope with either point. Both give the same line.
Rearrange to y = mx + b at the end if the question wants it, or leave it if it does not.
Point-slope is underrated
y - y1 = m(x - x1) takes no algebra to set up. Write it first, then tidy. Trying to jump straight to y = mx + b means solving for b, which is one more place to slip.
Intercepts from standard form
Set y = 0 to find the x-intercept. Set x = 0 to find the y-intercept.
For 3x + 4y = 24: at y = 0, x = 8; at x = 0, y = 6. Plot (8, 0) and (0, 6) and draw the line.
This is the fastest way to graph a line in standard form — faster than rearranging.
Zero the other one
For the x-intercept, make y zero. For the y-intercept, make x zero. The names sound backwards, so say it deliberately each time.
Parallel and perpendicular
Parallel lines have equal slopes. To find a line parallel to y = 3x + 1 through a given point, keep m = 3 and use point-slope.
Perpendicular slopes are negative reciprocals: flip the fraction and change the sign. The perpendicular to m = 23 is m = -32.
Check: 23 × (-32) = -1 ✓
Both changes, not one
The perpendicular of 4 is -14: flip and negate. Doing only one of the two is the standard error.
Horizontal and vertical
Horizontal: slope 0, equation y = k. Vertical: slope undefined, equation x = k.
They are perpendicular to each other, but the negative-reciprocal rule does not apply — you cannot take the reciprocal of zero.
Just remember the pair: y = 3 is flat, x = 3 is upright.
Worked examples
Point and slope
Write the equation of the line with slope -2 through (3, 7), in slope-intercept form.
- Point-slope: y - 7 = -2(x - 3) Substitute straight in.
- Distribute: y - 7 = -2x + 6 The negative hits both terms.
- Add 7: y = -2x + 13 Check: -2(3) + 13 = 7 ✓
y = -2x + 13
Two points
Find the equation of the line through (-2, 5) and (4, -7).
- m = -7 - 54 - (-2) = -126 = -2 Slope first.
- Point-slope with (-2, 5): y - 5 = -2(x + 2) x - (-2) is x + 2.
- y - 5 = -2x - 4, so y = -2x + 1 Check the other point: -2(4) + 1 = -7 ✓
y = -2x + 1
Perpendicular
Find the line perpendicular to y = 34x - 2 passing through (6, 1).
- Given slope 34, the perpendicular slope is -43. Flip and negate.
- Point-slope: y - 1 = -43(x - 6) Use the given point.
- y - 1 = -43x + 8 -43 × -6 = 8.
- y = -43x + 9 Check: 34 × -43 = -1 ✓
y = -43x + 9
Words to know
- Slope-intercept form
- y = mx + b. Best for graphing.
- 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.
Mistakes to watch for
Using x + x1 in point-slope form. The formula is y - y1 = m(x - x1), so a point at x1 = -3 gives x - (-3) = x + 3. The sign flips because of the subtraction, not in spite of it.
Taking the reciprocal without changing the sign. A perpendicular slope needs both moves: flip the fraction and negate. The perpendicular of 4 is -14, and doing only one of the two is the standard error.
Setting x = 0 when hunting the x-intercept. To find where it crosses the x-axis you set y to zero, because that is what being on the x-axis means. The names sound backwards, so say it deliberately.
Subtracting coordinates in opposite orders. y2 - y1x1 - x2 gives the slope with the wrong sign. Pick an order and use it top and bottom.
Descartes published the coordinate plane in 1637 — as an appendix to a philosophy book, Discourse on the Method.
The mathematics was an illustration of his argument about how to reason well. The appendix outlived the book.
The story that he thought of it while watching a fly on his ceiling is charming and almost certainly invented.
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