Slope Intercept Form: y = mx + b & Examples
Slope Intercept Form: y = mx + b & Examples
TL;DR
Slope intercept form is the equation of a straight line written as y=mx+b, where m is the slope and b is the y-intercept. This article covers how to read m and b straight off the equation, how to find them from points or a graph, how to convert other forms, and six worked examples.
Why Every Straight Line Comes Down to Just Two Numbers
There are infinitely many straight lines you could draw on a grid, yet any one of them is fixed completely by answering two questions: how steeply does it tilt, and where does it cross the y-axis? Slope intercept form is nothing more than those two answers written in a fixed order, which is why a single short equation can pin down a line exactly.
Once you can see how the two numbers carry the whole line, reading, finding, and converting the form all become the same small idea applied three ways.
What Is Slope Intercept Form?
Slope intercept form is the equation of a straight line written in this exact shape:
y=mx+b.
Each symbol has a fixed job:
| Symbol | Meaning | Units |
|---|---|---|
| y | the vertical position (y-coordinate) | same as the y-axis |
| x | the horizontal position (x-coordinate) | same as the x-axis |
| m | the slope: how much y changes per 1-unit change in x (rise over run) | y-units per x-unit |
| b | the y-intercept: the y-value where the line crosses the y-axis, at x=0 | same as the y-axis |
The form is named slope intercept because both the slope m and the y-intercept b sit in plain view in the equation, with no algebra needed to dig them out. Read y=2x+3 and you can say at once: the slope is 2 and the line crosses the y-axis at (0,3).
A quick read of signs, since negatives drive most of the confusion here. In y=−12x−4, the slope is m=−12, so the line falls half a unit for every unit right, and the y-intercept is b=−4, so it crosses the y-axis at (0,−4).
How Do You Find Slope and Y-Intercept From Two Points?
That question comes up more than any other, so here is the method directly. Given two points (x1,y1) and (x2,y2) on the line, the slope is the change in y over the change in x:
m=y2−y1x2−x1.
This is rise over run: y on top, x on the bottom, never the other way around. Once you have m, find b by substituting either point into y=mx+b and solving for b. From a graph instead of points, read b off where the line meets the y-axis, then pick any two clear lattice points to compute m.
How to Convert Other Forms Into Slope Intercept Form
Lines often arrive in standard form Ax+By=C or point-slope form y−y1=m(x−x1). To reach slope intercept form, isolate y on the left.
From the standard form 3x+2y=12:
2y=−3x+12; ⇒ y=−32x+6,
so m=−32 and b=6.
From the point-slope form y−5=4(x−2):
y=4(x−2)+5=4x−8+5=4x−3,
so m=4 and b=−3.
Examples of Slope Intercept Form
Example 1: Identify the slope and y-intercept of y=−3x+7.
By inspection, m=−3 and b=7. The line falls 3 units for every unit right and crosses the y-axis at (0,7).
Final answer: m=−3, b=7.
Example 2: Find the slope intercept form of the line through (2,1) and (5,7).
A common first move is to write the slope as (m=\frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 1}{5 - 2} = \frac{6}{3} = 2.)
Now substitute (2,1) to find b: 1=2(2)+b, so b=1−4=−3, giving y=2x−3.
Final answer: y=2x−3.
Example 3: Convert 4x−2y=6 into slope intercept form and state m and b.
Isolate y:
−2y=−4x+6; ⇒ y=2x−3.
Final answer: y=2x−3, so m=2 and b=−3.
Example 4: Write the equation of the line with slope m=−23 passing through (3,1).
Substitute into y=mx+b: 1=−23(3)+b, so 1=−2+b and b=3.
Final answer: y=−23x+3.
Example 5. Is y=5 in slope intercept form, and if so what are m and b?
Yes. Write it as y=0x+5, a horizontal line with slope m=0 and y-intercept b=5. The line is flat, so y never changes as x moves.
Final answer: m=0, b=5.
Example 6: A taxi charges a fixed 3.00$ pickup fee plus 1.50 per kilometre. Write the fare as a function of distance x in slope intercept form, and find the fare for a 6 km trip.
The per-kilometre rate is the slope and the pickup fee is the y-intercept, so y=1.5x+3. For x=6: y=1.5(6)+3=12.
Final answer: y=1.5x+3; a 6 km trip costs $12.00.
Where Slope Intercept Form Shows Up
Slope intercept form is the entry point to almost every linear model people actually use, because so many real quantities grow at a steady rate from a fixed starting value.
- Cost models. Total cost equals a per-unit rate times quantity plus a fixed cost, which is y=mx+b.
- Uniform motion. Distance equals speed times time plus starting position, d=vt+d0, with speed as the slope and starting position as the intercept.
- Linear regression. Fitting a trend line of the form y=mx+b to data is the most widely used statistical method.
- Unit conversion. Celsius to Fahrenheit is F=95C+32; a slope intercept line with a non-zero intercept.
The coordinate framework that lets every line become an equation traces back to René Descartes and his 1637 work that married algebra to geometry.
Where Students Trip Up on Slope Intercept Form
Mistake 1: Inverting rise and run in the slope formula
Correct way: y-difference over x-difference. Rise over run, always.
Mistake 2: Treating the y-intercept value as an x-coordinate
Correct way: A y-intercept sits at (0,b), on the y-axis where x=0.
Mistake 3: Reading off a standard-form equation without isolating y
Correct way: Solve for y first: get y=mx+b.
Key Takeaways
- Slope intercept form is y=mx+b, with m the slope and b the y-intercept.
- The slope from two points is m=y2−y1x2−x1.
- Convert standard or point-slope form by isolating y on the left.
- The most common slip is inverting rise and run; write the slope formula in full before substituting.
Practice These Problems to Solidify Your Understanding
- Identify the slope and y-intercept of y=−12x+6.
- Find the slope intercept form of the line through (1,3) and (4,9).
- Convert 5x−2y=10 into slope intercept form.
Answer to Question 1: m=−12, b=6.
Answer to Question 2: y=2x+1.
Answer to Question 3: y=\frac{5}{2}x−5.