A linear equation is usually called an equation of the line, when it is used to represent a line on cartesian coordinates, this concept is widely used in mathematics and physics. It also plays a wide role in the field of engineering.
In engineering, it allows the engineers to find out the expected load on the bridge (that is under construction) at any point. Further in this article, we will study the basic definition, formula, and some different cases for calculating the slope-intercept form of a linear equation with the assistance of a bunch of examples.
Slope intercept form:
The slope-intercept form is a method used to determine the equation of a line. In this method, the y-intercept and slope are used to evaluate the equation of a line. If we don’t have these values, we will calculate them first.
General Equation:
Generally, we use the following formula to represent the slope-intercept form.
Y = mx + c
- “x” and “y” are the representative points, these are variables
- “c” is the y-intercept
- “m” is the slope of the line.
This form can be obtained using different ways and cases.
General Cases:
The following three cases are the ways by which we can find the slope-intercept form:
- Using two points (two ordered pairs).
- Using x, y, and the slope “m”.
- Using the slope “m” and “c”.
Example section:
In this section, we’re going to elaborate on all three cases with the assistance of mathematical problems.
Case 1: Using two points.
Example 1:
Find out the linear equation of a line if it has points (7, 4) and (9, -6). Represent it in the slope-intercept form.
Solution:
Step 1: Extract the given data:
x1 = 7, y1 = 4, x2 = 9, y2 = -6
Step 2: Compute the slope “m”.
Slope = m = Rise / run
Slope = m = (y2 – y1) / (x2 – x1)
Slope = m = (-6 – (4)) / (9 – (7))
Slope = m = (-6 – 4) / (9 – 7)
Slope = m = -10/2
Slope = m = -5
Step 3: Find out “c” using 1st point (ordered pair).
y = mx + c
4 = -5 (7) + c
4 = -35 + c
4 + 35 = c
c = 39
Step 4: Place all values in the general formula of the slope-intercept form.
y = mx + c
y = -5x + 39
Example 2:
Calculate using the slope-intercept form formula, x1 = 12, x2 = 9, y1 = 2, y2 = 14
Solution:
Step 1: Extract the given data:
x1 = 12, x2 = 9, y1 = 2, y2 = 14
Step 2: Compute the slope “m”.
Slope = m = Rise / run
Slope = m = (y2 – y1) / (x2 – x1)
Slope = m = (14 – 2) / (9 – 12)
Slope = m = (12) / (-3)
Slope = m = -4
Step 3: Find out “c” using 1st point (ordered pair).
y = mx + c
2 = -4 (12) + c
2 = -48 + c
2 + 48 = c
c = 50
Step 4: Place all values in the general formula of the slope-intercept form.
y = mx + c
m = -4, c = 50
y = -4x + 50
A slope intercept form calculator can be used to get step-by-step solutions of the problems.
Case 2: Using x, y, and the slope “m”.
Example 1:
Evaluate y = mx + c if m = 4, x = 2, and y = – 3.
Solution:
Step 1: Extract the given data.
y = -3, x = 2, and m = 4
Step 2: Calculate “c” using the given data
y = -3, x = 2, and m = 4
-3 = 4 (2) + c
-3 = 8 + c
-11 = c
c = -11
Step 3: Represent it in “y = mx + c” form
m = 4, c = -11
y = mx + c
y = 4x – 11
Example 2:
Evaluate the slope-intercept form of the line if it has m = 3, x = 9, and y = – 12.
Solution:
Step 1: Extract the given data.
y = -12, x = 9, and m = 3
Step 2: Calculate “c” using the given data
y = -12, x = 9, and m = 3
-12 = 3 (9) + c
-12 = 27 + c
-39 = c
c = -39
Step 3: Represent it in “y = mx + c” form
m = 3, c = -39
y = mx + c
y = 3x – 39
Case 3: Using the slope “m” and “c”
Example 1:
If c (y-intercept) of a line is 9, and its slope m = -1 then calculate its slope-intercept form.
Solution:
Step 1: Extract the given data.
c = 9, m = -1
Step 2: Put the values in the general formula.
y = mx + c
m = -1, c = 9
y = -x + 9
Summary:
In this article, we have studied the basic definition of the slope-intercept form, its uses, and applications. Moreover, we have read the general formula and the cases by which we can evaluate an equation’s y = mx + c form.
In the example section, we have discussed the methods to find the slope-intercept form, now you can solve all the problems related to this topic.
