The slope-intercept is the most “popular” form of a straight line. Many students find this useful because of its simplicity. One can easily describe the characteristics of the straight line even without seeing its graph because the slope and [latex]y[/latex]-intercept can easily be identified or read off from this form.
Slope-Intercept Form of the Equation of a Line
The linear equation written in the form
[latex]\large{y = mx + b}[/latex]
is in slope-intercept formwhere:
[latex]m[/latex] is the slope, and [latex]b[/latex] is the [latex]y[/latex]-intercept
![Slope-Intercept Form (1) Slope-Intercept Form (1)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2018/05/slope-intercept-form.png)
Quick notes:
- The slope [latex]m[/latex] measures how steep the line is with respect to the horizontal. Given two points [latex]\left( {{x_1},{y_1}} \right)[/latex] and [latex]\left( {{x_2},{y_2}} \right)[/latex] found in the line, the slope is computed as
![Slope-Intercept Form (2) Slope-Intercept Form (2)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex10.png)
- The [latex]y[/latex]-intercept [latex]b[/latex] is the point where the line crosses the [latex]y[/latex]-axis. Notice that in the graph below, the red dot is always found on the main vertical axis of the Cartesian plane. That is the basic characteristic of the [latex]y[/latex]-intercept.
![Slope-Intercept Form (3) Slope-Intercept Form (3)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2017/03/xyb.gif)
Let’s go over some examples of how to write the equation of a straight line in linear form [latex]y = mx + b[/latex].
Examples of Applying the Concept of Slope-Intercept Form of a Line
Example 1: Write the equation of the line in slope-intercept form with a slope of[latex] – \,5[/latex] and a [latex]y[/latex]-intercept of [latex]3[/latex].
The needed information to write the equation of the line in the form [latex]y = mx + b[/latex] are clearly given in the problem since
[latex]m = – \,5[/latex] (slope)
[latex]b = 3[/latex] ([latex]y[/latex]-intercept)
Substituting in [latex]y = mx + b[/latex], we obtain
![Slope-Intercept Form (4) Slope-Intercept Form (4)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex1a.png)
By having a negative slope, the line is decreasing/falling fromleft to right, and passing through the [latex]y[/latex]-axis at point [latex]\left( {0,3} \right)[/latex].
![Slope-Intercept Form (5) Slope-Intercept Form (5)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2017/03/ex1_g-1.gif)
Example 2: Write the equation of the line in slope-intercept form with a slope of [latex]7[/latex] and a [latex]y[/latex]-intercept of[latex] – \,4[/latex].
The slope is given as [latex]m = 7[/latex] and the [latex]y[/latex]-intercept as [latex]b = – \,4[/latex]. Substituting into the slope-intercept formula [latex]y = mx + b[/latex], we have
![Slope-Intercept Form (6) Slope-Intercept Form (6)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex2a.png)
The slope is positivethus the line is increasing or rising from left to right, but passing through the [latex]y[/latex]-axis at point [latex]\left( {0, – \,4} \right)[/latex].
![Slope-Intercept Form (7) Slope-Intercept Form (7)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2017/03/ex2_g-1.gif)
Example 3: Write the equation of the line in slope-intercept with a slope of [latex]9[/latex] and passing through the point [latex]\left( {0, – \,2} \right)[/latex].
This problem is slightly different from the previous two examplesbecause the [latex]y[/latex]-intercept [latex]b[/latex] is not given to us upfront. So our next goal is to somehow figure out the value of [latex]b[/latex] first.
However, if we examine the slope-intercept form, it shouldlead us to believe that we have enough information to solve for [latex]b[/latex].How?
![Slope-Intercept Form (8) Slope-Intercept Form (8)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex3a.png)
That means [latex]m = 9[/latex], and from the given point [latex]\left( {0, – \,2} \right)[/latex] we have [latex]x = 0[/latex] and [latex]y = – \,2[/latex].Let’s substitute these known values into the slope-intercept formula and solve for the missing value of [latex]b[/latex].
![Slope-Intercept Form (9) Slope-Intercept Form (9)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex3b.png)
Now it is possible to write the slope-intercept form as
![Slope-Intercept Form (10) Slope-Intercept Form (10)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex3c.png)
Example 4: Find the slope-intercept form of the line with a slope of[latex] – \,3[/latex] and passing through the point [latex]\left( { – 1,\,15} \right)[/latex].
Again, the value of [latex]y[/latex]-intercept [latex]b[/latex] is not directly provided to us. But we can utilize the given slope and a point to find it.
![Slope-Intercept Form (11) Slope-Intercept Form (11)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex4a.png)
Substitute the known values into the slope-intercept formula, and then solve for the unknown value of [latex]b[/latex].
![Slope-Intercept Form (12) Slope-Intercept Form (12)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex4b.png)
Back substitute the value of the slope and the solved value of the [latex]y[/latex]-intercept into [latex]y = mx + b[/latex].
![Slope-Intercept Form (13) Slope-Intercept Form (13)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex4c.png)
Example 5: A line with the slope of[latex] – \,8[/latex] and passing through the point [latex]\left( { – \,4,\, – 1} \right)[/latex].
The given slope is [latex]m = – \,8[/latex] and from the given point [latex]\left( { – \,4,\, – 1} \right)[/latex], we have [latex]x = – \,4[/latex] and [latex]y = – \,1[/latex].Now, we are going to substitute the known values into the slope-intercept form of the lineto solve for [latex]b[/latex].
![Slope-Intercept Form (14) Slope-Intercept Form (14)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex5a.png)
Since [latex]m = – \,8[/latex] and [latex]b = – \,33[/latex],the slope-intercept form of the line becomes
![Slope-Intercept Form (15) Slope-Intercept Form (15)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex5b.png)
Example 6: Write the slope-intercept form of the line with a slope of[latex]{3 \over 5}[/latex] and through the point [latex]\left( {5,\, – 2} \right)[/latex].
We have a slope here that is notan integer, i.e. the denominator is other than positive or negative one,[latex] \pm 1[/latex]. In other words, we have a“true” fractional slope.
The procedure for solving this problem is very similar to examples #3, #4, and #5.But the main point of this example is to emphasize the algebraicsteps required on how to solve a linear equation involving fractions.
The known values ofthe problemare
- Given slope:
![Slope-Intercept Form (16) Slope-Intercept Form (16)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex6a.png)
- Given point:
![Slope-Intercept Form (17) Slope-Intercept Form (17)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex6b.png)
Plug thevalues into [latex]y = mx + b[/latex] and solve for [latex]b[/latex].
![Slope-Intercept Form (18) Slope-Intercept Form (18)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex6c.png)
As you can see the common factors of [latex]5[/latex] in the numerator and denominator nicely cancel each other out which greatly simplifies the process of solving for [latex]b[/latex].
Putting this togetherin the form [latex]y = mx + b[/latex]
![Slope-Intercept Form (19) Slope-Intercept Form (19)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex6d.png)
Example 7: Slope of[latex]{{\, – 3} \over 2}[/latex] and through the point [latex]\left( { – 1,\, – 1} \right)[/latex].
The given slope is [latex]m = {{\, – 3} \over 2}[/latex] and from the given point[latex]\left( { – 1,\, – 1} \right)[/latex],the values of [latex]x[/latex] and [latex]y[/latex] can easily be identified.
![Slope-Intercept Form (20) Slope-Intercept Form (20)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex7a.png)
Now plug in the known values into the slope-intercept form [latex]y = mx + b[/latex] to solve for [latex]b[/latex].
Make sure that when you add or subtract fractions, you generate a common denominator.
![Slope-Intercept Form (22) Slope-Intercept Form (22)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex7c.png)
After getting the value of [latex]b[/latex],we can now write the slope-intercept form of the line.
![Slope-Intercept Form (23) Slope-Intercept Form (23)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex7d.png)
Example 8: Slope of[latex] – \,6[/latex] and through the point [latex]\left( {{1 \over 2},{1 \over 3}} \right)[/latex].
The slope is given as [latex]m = – \,6[/latex] and from the point, we have [latex]x = {1 \over 2}[/latex] and [latex]y = {1 \over 3}[/latex].
Substitute the known values into [latex]y = mx + b[/latex]. Then solve the missing value of [latex]b[/latex].
![Slope-Intercept Form (24) Slope-Intercept Form (24)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex8a.png)
Therefore, the slope-intercept form of the line is
![Slope-Intercept Form (25) Slope-Intercept Form (25)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex8b.png)
Example 9: Slope of[latex]{{\,7} \over 3}[/latex] andthrough the point[latex]\left( {{{ – \,2} \over 5},{5 \over 2}} \right)[/latex].
Identifying the known values
- Given slope:
![Slope-Intercept Form (26) Slope-Intercept Form (26)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex9a.png)
- Given point:
![Slope-Intercept Form (27) Slope-Intercept Form (27)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex9b.png)
The setup to find [latex]b[/latex] becomes
![Slope-Intercept Form (28) Slope-Intercept Form (28)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex9c.png)
That makes the slope-intercept form of the line as
![Slope-Intercept Form (29) Slope-Intercept Form (29)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex9d.png)
Example 10: A line passing through the given two points [latex]\left( {4,\,5} \right)[/latex] and [latex]\left( {0,\,3} \right)[/latex].
In this problem, we are not provided with both the slope [latex]m[/latex] and [latex]y[/latex]-intercept [latex]b[/latex].However, weshould realize that the slope is easily calculated when two points are known using the Slope Formula.
Slope Formula
The slope, [latex]m[/latex], of a line passing through two arbitrary points [latex]\left( {{x_1},{y_1}} \right)[/latex] and [latex]\left( {{x_2},{y_2}} \right)[/latex] is calculated as follows…
![Slope-Intercept Form (30) Slope-Intercept Form (30)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex10.png)
If we let [latex]\left( {4,\,5} \right)[/latex] be the first point, then [latex]\left( {0,\,3} \right)[/latex] must be the second.
Labeling the components of each point should help in identifying the correct values that would be substituted into the slope formula.
![Slope-Intercept Form (31) Slope-Intercept Form (31)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex10a.png)
Based on the labeling above, now we know that
![Slope-Intercept Form (32) Slope-Intercept Form (32)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex10b.png)
Next, write the slope formula, plug in the known valuesand simplify.
![Slope-Intercept Form (33) Slope-Intercept Form (33)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex10d.png)
Great! We found the slope to be [latex]m = {{\,1} \over 2}\,[/latex].The only missing piece of the puzzle is to determine the [latex]y[/latex]-intercept. Use theslope that we found, together with ANY of the two given points. In this exercise, I will show you that we should arrive at the same value of the [latex]y[/latex]-intercept regardless of which point is selected for the calculation.
Finding the [latex]y[/latex]-intercept
- Using the first point [latex]\left( {4,\,5} \right)[/latex].
![Slope-Intercept Form (34) Slope-Intercept Form (34)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex10e.png)
![Slope-Intercept Form (35) Slope-Intercept Form (35)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex10f.png)
- Using the second point [latex]\left( {0,\,3} \right)[/latex].
![Slope-Intercept Form (36) Slope-Intercept Form (36)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex10g.png)
![Slope-Intercept Form (37) Slope-Intercept Form (37)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex10h.png)
Indeed, the [latex]y[/latex]-intercepts come out the same in both calculations.We can now write the linear equation in slope-intercept form.
![Slope-Intercept Form (38) Slope-Intercept Form (38)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex10i.png)
Below is the graph of the line passing through the given two points.
![Slope-Intercept Form (39) Slope-Intercept Form (39)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2017/03/ex10_g.gif)
Example 11: A line passing through the given two points [latex]\left( { – \,7,\,4} \right)[/latex] and [latex]\left( { – \,2,\,19} \right)[/latex].
Let’s solve this step by step.
- Step 1: Assign which point is the first and second, and then label its components.
![Slope-Intercept Form (40) Slope-Intercept Form (40)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex11a.png)
- Step 2: Substitute the known values into the slope formula, and simplifyif necessary.
![Slope-Intercept Form (41) Slope-Intercept Form (41)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex11b.png)
- Step 3: Pick any of the two given points. Suppose we pick the point [latex]\left( { – \,7,\,4} \right)[/latex].That means [latex]x = – \,7[/latex] and [latex]y = 4[/latex].Using the calculated value of slope in step 2, we can now find the [latex]y[/latex]-intercept [latex]b[/latex].
![Slope-Intercept Form (42) Slope-Intercept Form (42)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex11c.png)
- Step 4: Putting them together in [latex]y = mx + b[/latex] form, since [latex]m = 3[/latex] and [latex]b = 25[/latex],we have the slope-intercept form of the line as
![Slope-Intercept Form (43) Slope-Intercept Form (43)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex11d.png)
- Step 5:Using a graphing utility, show that the solved linear equation in slope-intercept form passes through the two points.
![Slope-Intercept Form (44) Slope-Intercept Form (44)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2017/03/ex11_g.gif)
Example 12: A line passing through the given two points [latex]\left( { – \,6,\, – \,3} \right)[/latex] and [latex]\left( { – \,7,\, – 1} \right)[/latex].
- Find the slope
![Slope-Intercept Form (45) Slope-Intercept Form (45)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex11e.png)
![Slope-Intercept Form (46) Slope-Intercept Form (46)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex12b.png)
- Pick any of the two given points. Suppose, we chose the second point which is
![Slope-Intercept Form (47) Slope-Intercept Form (47)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex12c.png)
Substitute known values in the slope-intercept form [latex]y = mx + b[/latex] to solve for [latex]b[/latex].
![Slope-Intercept Form (48) Slope-Intercept Form (48)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex12d.png)
- Putting them together. Since [latex]m = – \,2[/latex] and [latex]b = – \,15[/latex],the slope-intercept form of the line is
![Slope-Intercept Form (49) Slope-Intercept Form (49)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex12e.png)
- This is the graph of the line showing that it passes both ofthe two points.
![Slope-Intercept Form (50) Slope-Intercept Form (50)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2017/03/ex12_g.gif)
Example 13: A line passing through the given two points [latex]\left( {5,\, – \,2} \right)[/latex] and [latex]\left( { – \,2,\,5} \right)[/latex].
- Determine the slope from the given two points
![Slope-Intercept Form (51) Slope-Intercept Form (51)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex13a.png)
![Slope-Intercept Form (52) Slope-Intercept Form (52)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex13b.png)
- Pick any of the two given points. Let’s say we chose the first one, [latex]\left( {5,\, – \,2} \right)[/latex]. That means [latex]x = 5[/latex], and [latex]y = – \,2[/latex]. Use this information together with the value of slope to solve for the [latex]y[/latex]-intercept [latex]b[/latex].
![Slope-Intercept Form (53) Slope-Intercept Form (53)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex13c.png)
- Now, put them together. Since [latex]m = – \,1[/latex] and [latex]b = 3[/latex], the slope-intercept form of the line is
![Slope-Intercept Form (54) Slope-Intercept Form (54)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2019/03/sif-ex13d.png)
- Using a graphing utility, show that the line passes through the two given points.
![Slope-Intercept Form (55) Slope-Intercept Form (55)](https://i0.wp.com/www.chilimath.com/wp-content/uploads/2017/03/ex13_g.gif)
You may also be interested in these related math lessons or tutorials:
Types of Slopes of a Line
Slope Formula of a Line
Point-Slope Form of a Line