Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (2022)

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 y-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

\large{y = mx + b}

is in slope-intercept formwhere:

m is the slope, and b is the y-intercept

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (1)

Quick notes:

  • The slope m measures how steep the line is with respect to the horizontal. Given two points \left( {{x_1},{y_1}} \right) and \left( {{x_2},{y_2}} \right) found in the line, the slope is computed as
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (2)
  • The y-intercept b is the point where the line crosses the y-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 y-intercept.
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (3)

Let’s go over some examples of how to write the equation of a straight line in linear form y = mx + b.

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 - \,5 and a y-intercept of 3.

The needed information to write the equation of the line in the form y = mx + b are clearly given in the problem since

m = - \,5 (slope)

b = 3 (y-intercept)

Substituting in y = mx + b, we obtain

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (4)

By having a negative slope, the line is decreasing/falling fromleft to right, and passing through the y-axis at point \left( {0,3} \right).

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (5)

Example 2: Write the equation of the line in slope-intercept form with a slope of 7 and a y-intercept of - \,4.

The slope is given as m = 7 and the y-intercept as b = - \,4. Substituting into the slope-intercept formula y = mx + b, we have

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (6)

The slope is positivethus the line is increasing or rising from left to right, but passing through the y-axis at point \left( {0, - \,4} \right).

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (7)
(Video) How to Write Slope Intercept Form of Equation of a Line (Hindi)

Example 3: Write the equation of the line in slope-intercept with a slope of 9 and passing through the point \left( {0, - \,2} \right).

This problem is slightly different from the previous two examplesbecause the y-intercept b is not given to us upfront. So our next goal is to somehow figure out the value of b first.

However, if we examine the slope-intercept form, it shouldlead us to believe that we have enough information to solve for b.How?

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (8)

That means m = 9, and from the given point \left( {0, - \,2} \right) we have x = 0 and y = - \,2.Let’s substitute these known values into the slope-intercept formula and solve for the missing value of b.

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (9)

Now it is possible to write the slope-intercept form as

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (10)

Example 4: Find the slope-intercept form of the line with a slope of - \,3 and passing through the point \left( { - 1,\,15} \right).

Again, the value of y-intercept b is not directly provided to us. But we can utilize the given slope and a point to find it.

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (11)

Substitute the known values into the slope-intercept formula, and then solve for the unknown value of b.

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (12)

Back substitute the value of the slope and the solved value of the y-intercept into y = mx + b.

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (13)

Example 5: A line with the slope of - \,8 and passing through the point \left( { - \,4,\, - 1} \right).

The given slope is m = - \,8 and from the given point \left( { - \,4,\, - 1} \right), we have x = - \,4 and y = - \,1.Now, we are going to substitute the known values into the slope-intercept form of the lineto solve for b.

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (14)

Since m = - \,8 and b = - \,33,the slope-intercept form of the line becomes

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (15)

Example 6: Write the slope-intercept form of the line with a slope of{3 \over 5} and through the point \left( {5,\, - 2} \right).

We have a slope here that is notan integer, i.e. the denominator is other than positive or negative one, \pm 1. 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

(Video) Slope-intercept form | Algebra I | Khan Academy

  • Given slope:
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (16)
  • Given point:
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (17)

Plug thevalues into y = mx + b and solve for b.

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (18)

As you can see the common factors of 5 in the numerator and denominator nicely cancel each other out which greatly simplifies the process of solving for b.

Putting this togetherin the form y = mx + b

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (19)

Example 7: Slope of{{\, - 3} \over 2} and through the point \left( { - 1,\, - 1} \right).

The given slope is m = {{\, - 3} \over 2} and from the given point\left( { - 1,\, - 1} \right),the values of x and y can easily be identified.

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (20)

Now plug in the known values into the slope-intercept form y = mx + b to solve for b.

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (21)

Make sure that when you add or subtract fractions, you generate a common denominator.

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (22)

After getting the value of b,we can now write the slope-intercept form of the line.

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (23)

Example 8: Slope of - \,6 and through the point \left( {{1 \over 2},{1 \over 3}} \right).

The slope is given as m = - \,6 and from the point, we have x = {1 \over 2} and y = {1 \over 3}.

Substitute the known values into y = mx + b. Then solve the missing value of b.

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (24)

Therefore, the slope-intercept form of the line is

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (25)

Example 9: Slope of{{\,7} \over 3} andthrough the point\left( {{{ - \,2} \over 5},{5 \over 2}} \right).

Identifying the known values

(Video) Slope-Intercept Form y=mx+b

  • Given slope:
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (26)
  • Given point:
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (27)

The setup to find b becomes

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (28)

That makes the slope-intercept form of the line as

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (29)

Example 10: A line passing through the given two points \left( {4,\,5} \right) and \left( {0,\,3} \right).

In this problem, we are not provided with both the slope m and y-intercept b.However, weshould realize that the slope is easily calculated when two points are known using the Slope Formula.

Slope Formula

The slope, m, of a line passing through two arbitrary points \left( {{x_1},{y_1}} \right) and \left( {{x_2},{y_2}} \right) is calculated as follows…

If we let \left( {4,\,5} \right) be the first point, then \left( {0,\,3} \right) 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 of a Straight Line (y = mx + b) | ChiliMath (31)

Based on the labeling above, now we know that

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (32)

Next, write the slope formula, plug in the known valuesand simplify.

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (33)

Great! We found the slope to be m = {{\,1} \over 2}\,.The only missing piece of the puzzle is to determine the y-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 y-intercept regardless of which point is selected for the calculation.

Finding the y-intercept

  • Using the first point \left( {4,\,5} \right).
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (34)
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (35)
  • Using the second point \left( {0,\,3} \right).
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (36)
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (37)

Indeed, the y-intercepts come out the same in both calculations.We can now write the linear equation in slope-intercept form.

(Video) Deriving Slope Y intercept form of Straight Line y=mx+c

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (38)

Below is the graph of the line passing through the given two points.

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (39)

Example 11: A line passing through the given two points \left( { - \,7,\,4} \right) and \left( { - \,2,\,19} \right).

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 of a Straight Line (y = mx + b) | ChiliMath (40)
  • Step 2: Substitute the known values into the slope formula, and simplifyif necessary.
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (41)
  • Step 3: Pick any of the two given points. Suppose we pick the point \left( { - \,7,\,4} \right).That means x = - \,7 and y = 4.Using the calculated value of slope in step 2, we can now find the y-intercept b.
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (42)
  • Step 4: Putting them together in y = mx + b form, since m = 3 and b = 25,we have the slope-intercept form of the line as
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (43)
  • Step 5:Using a graphing utility, show that the solved linear equation in slope-intercept form passes through the two points.
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (44)

Example 12: A line passing through the given two points \left( { - \,6,\, - \,3} \right) and \left( { - \,7,\, - 1} \right).

  • Find the slope
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (45)
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (46)
  • Pick any of the two given points. Suppose, we chose the second point which is
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (47)

Substitute known values in the slope-intercept form y = mx + b to solve for b.

Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (48)
  • Putting them together. Since m = - \,2 and b = - \,15,the slope-intercept form of the line is
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (49)
  • This is the graph of the line showing that it passes both ofthe two points.
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (50)

Example 13: A line passing through the given two points \left( {5,\, - \,2} \right) and \left( { - \,2,\,5} \right).

  • Determine the slope from the given two points
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (51)
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (52)
  • Pick any of the two given points. Let’s say we chose the first one, \left( {5,\, - \,2} \right). That means x = 5, and y = - \,2. Use this information together with the value of slope to solve for the y-intercept b.
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (53)
  • Now, put them together. Since m = - \,1 and b = 3, the slope-intercept form of the line is
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (54)
  • Using a graphing utility, show that the line passes through the two given points.
Slope-Intercept Form of a Straight Line (y = mx + b) | ChiliMath (55)
(Video) Slope Intercept Form | Equation of Straight line | General Form of Straight line Equation

You might also be interested in:

Types of Slopes of a Line
Slope Formula of a Line
Point-Slope Form of a Line

FAQs

How do you find the slope-intercept form of Y MX B? ›

Graphing linear equations using y = mx + b (Slope - Intercept) - YouTube

What is the slope of a line Y MX B? ›

In the equation y = mx + b, m is the slope of the line and b is the intercept. X and y represent the distance of the line from x-axis and y-axis, respectively. The value of b is equal to y when x = 0, and m shows how steep the line is. The slope of the line is also called the gradient.

What is the slope-intercept form of a straight line? ›

The graph of the linear equation y = mx + c is a line with m as slope, m and c as the y-intercept. This form of the linear equation is called the slope-intercept form, and the values of m and c are real numbers. The slope, m, represents the steepness of a line.

What is the y-intercept of the straight line Y MX? ›

The y-intercept 'c'

The y -intercept is the point of intersection between the straight line y = mx + c , and the y -axis. The y -intercept is the value of y when x = 0 .

How do you solve slope intercept? ›

How to Find the Slope-Intercept Form of a Line
  1. Solve for the slope (m). Remember, the slope is equal to the change in y divided by the change in x (rise over run).
  2. Solve for the y-value of the y-intercept (b). ...
  3. Replace m and b with their values in the y=mx+b formula.

How do you find the slope-intercept form example? ›

Examples. y = 5x + 3 is an example of the Slope Intercept Form and represents the equation of a line with a slope of 5 and and a y-intercept of 3. y = −2x + 6 represents the equation of a line with a slope of −2 and and a y-intercept of 6.

What is the y-intercept in the equation y MX B? ›

In the equation of a straight line (when the equation is written as "y = mx + b"), the slope is the number "m" that is multiplied on the x, and "b" is the y-intercept (that is, the point where the line crosses the vertical y-axis).

How do you find y MX B with slope and a point? ›

These are the two methods to finding the equation of a line when given a point and the slope: Substitution method = plug in the slope and the (x, y) point values into y = mx + b, then solve for b. Use the m given in the problem, and the b that was just solved for, to create the equation y = mx + b.

What is straight line formula? ›

The equation of a straight line is y=mx+c y = m x + c m is the gradient and c is the height at which the line crosses the y -axis, also known as the y -intercept.

How do you draw a straight line with a slope? ›

Drawing a Line with a Given Slope - YouTube

What's the slope of a vertical line? ›

Vertical lines are said to have "undefined slope," as their slope appears to be some infinitely large, undefined value.

How find the slope of an equation? ›

How to Find the Slope from an Equation - YouTube

What slope is perpendicular? ›

Vertical lines and horizontal lines are perpendicular to each other. The slope of the perpendicular line in this case would be the slope of a horizontal line which would be 0. The slope of the parallel line is undefined and the slope of the perpendicular line is 0.

Videos

1. Slope Intercept Form of equation | Straight Line | MATHS | Class XI | JEE
(Unacademy - Accelerate)
2. Slope Intercept Form - Understand In 10 Minutes
(TabletClass Math)
3. Slope Intercept Form Y=mx+b
(The Organic Chemistry Tutor)
4. Slope Intercept Form (GMAT/GRE/CAT/Bank PO/SSC CGL) | Don't Memorise
(Don't Memorise)
5. Graphing linear equations using y = mx + b (Slope - Intercept)
(tecmath)
6. What is Slope Intercept Form of a Line , y = mx + b , Intermediate Algebra , Lesson 65
(Learn Math Tutorials)

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