Slope intercept form: An introduction with techniques and examples
Math
The slope intercept form is a well-known method of algebra that is widely used to determine the linear equation of the straight line. The equation of the straight line is further used in different branches of algebra for various purposes.
The other form of the linear equation of the line can be determined by using the point-slope form or an x & y-intercept form. In this article, we will briefly describe the slope-intercept form along with methods and examples.
What is the slope-intercept form?
In algebra, the slope-intercept form is a well-known and frequent technique for finding the linear equation of the straight line with the help of a general equation. By name, it is clear that this technique must use the slope and y-intercept to determine the line’s equation.
The general (main) equation of the slope-intercept form is:
Y = m * X + B
- X & Y = the fixed points of the line.
- “M” = the slope of the line.
- “B” = the y-intercept of the line.
A question will increase in the mind that the equation of the slope-intercept form is also linear then why do we need to calculate it to get the linear equation of the line? The response to this question is that the equation of the slope-intercept form is general.
The slope and the y-intercept have to be substituted to the equation to get the linear equation of the straight line. You also have some sound knowledge to determine the slope and y-intercept of the line.
Techniques of the slope-intercept form
The following three are the well-known techniques for calculating the slope-intercept form to find the straight-line equation.
- Two points technique
- One point and slope technique
- Slope and y-intercept technique
Below are a few examples of the slope-intercept form solved by using the above techniques.
- By using two points technique
Example 1: For positive points
Evaluate the straight line equation by using two points technique (x1, y1) = (4, 6) and (x2, y2) = (28, 34).
Solution
Step I: First of all, take the given coordinate points of the line.
X1 = 4, x2 = 28, y1 = 6, y2 = 34
Step II: Now determine the slope of the line with the help of coordinate points. Or use a slope calculator to find it easily.
m = [y2 – y1] / [x2 – x1]
= 34 – 6 / 28 – 4
= 28 / 24
= 14/12
= 7/6 = 1.17
Step III: Now write the general equation of the slope-intercept form.
Y = m * X + B
Step IV: Put any pair of points and the calculated slope of the line in the above equation to find the y-intercept of the line.
Y = m * X + B
6 = 1.17 * 4 + B
6 = 4.68 + B
6 – 4.68 = B
1.32 = b
b = 1.32
Step V: Substitute the values of the slope and the y-intercept of the line in the general equation of the slope-intercept form.
Y = m * X + B
Y = 1.17 * X + 1.32
Y = 1.17X + 1.32
So, Y = 1.17X + 1.32 is the linear equation of the straight line.
Example 2: For negative points
Evaluate the straight line equation by using two points technique (x1, y1) = (-2, -7) and (x2, y2) = (-32, -27).
Solution
Step I: First of all, take the given coordinate points of the line.
X1 = -2, x2 = -7, y1 = -32, y2 = -27
Step II: Now determine the slope of the line with the help of coordinate points.
m = [y2 – y1] / [x2 – x1]
= [-32 – (-2)] / [-27 – (-7)]
= [-32 + 2] / [-27 + 7]
= -30/-20 = 30/20
= 7/6 = 3/2 = 1.5
Step III: Now write the general equation of the slope-intercept form.
Y = m * X + B
Step IV: Put any pair of points and the calculated slope of the line in the above equation to find the y-intercept of the line.
Y = m * X + B
-7 = 1.5 * (-2) + B
-7 = -3 + B
-7 + 3 = B
-4 = b
b = -4
Step V: Substitute the values of the slope and the y-intercept of the line in the general equation of the slope-intercept form.
Y = m * X + B
Y = 1.5 * X + (-4)
Y = 1.5X – 4
So, Y = 1.5X – 4 is the linear equation of the straight line.
To get rid of the lengthy calculation of finding the linear equation of the line, use a slope intercept form calculator.
- By using the one point and slope technique
Example
Evaluate the straight line equation by using the one point and slope technique, if the point is (x1, y1) = (-3, 11) and the slope of the line is 5
Solution
Step I: Take the given coordinate point and the slope of the line.
Point of the line = (x1, y1) = (-3, 11)
Slope of the line = m = 5
Step II: Now write the general equation of the slope-intercept form.
Y = m * X + B
Step III: Put any pair of points and the calculated slope of the line in the above equation to find the y-intercept of the line.
Y = m * X + B
11 = 5 * (-3) + B
11 = -15 + B
11 + 15 = B
26 = B
B = 26
Step IV: Substitute the values of the slope and the y-intercept of the line in the general equation of the slope-intercept form.
Y = m * X + B
Y = 5 * X + (26)
Y = 5X + 26
So, y = Y = 5X + 26 is the linear equation of the straight line.
- By using the slope and y-intercept technique
Evaluate the line equation by using the slope and y-intercept method, if the slope of the line is 6 and the y-intercept of the line is -6.
Solution
Step I: Write the given data values.
Slope of the line = m = 6
Y-intercept of the line = B = -6
Step II: Now write the general equation of the slope-intercept form.
Y = m * X + B
Step III: Substitute the values of the slope and the y-intercept of the line in the general equation of the slope-intercept form.
Y = m * X + B
y = 6 * X + (-6)
Y = 6X – 6
Y = 6(X – 1)
So, Y = 6(X – 1) is the linear equation of the straight line.
Conclusion
In this article, we have discussed all the basic techniques for solving the slope-intercept form to determine the line equation. Now you can solve any problem of line equation easily by learning the techniques of this post.
Slope intercept form: An introduction with techniques and examples
