How to Find the Slope and Y Intercept

y=mx+b

y = mx + b is the slope intercept form of writing the equation of a straight line. In the equation 'y = mx + b', 'b' is the point, where the line intersects the 'y axis' and 'm' denotes the slope of the line. The slope or gradient of a line describes how steep a line is. It can have either a positive or a negative value. When a standard form of a linear equation is of the form Ax + By = C, where 'x' and 'y' and 'C' are variables and 'A', 'B' are constants, the slope-intercept form is the most preferred way of expressing a straight line due to its simplicity, as it is very easy to find the slope and the 'y intercept' from the given equation.

1.	Meaning of y = mx + b
2.	How to Find y = mx + b?
3.	Writing an Equation in the Slope Intercept Form
4.	Solved Examples on y mx b
5.	Practice Questions on y mx b
6.	FAQs on y mx b

Meaning of y = mx + b

y = mx + b is the slope-intercept form of a staight line. In the equation y = mx + b for a straight line, m is called the slope of the line and b is the y-intercept of a line. y = mx+b, where

y ⇒ how far up or down is the line,

x ⇒ how far along is the line,

b ⇒ the value of y when x = 0 and

m ⇒ how steep the line is.

This is determined by m = (difference in y coordinates)/ (difference in x coordinates). Note that difference in y coordinates is indicated as rise or fall and difference in x coordinates is indicated as run.

y mx b, the equation of a straight line in the slope-intercept form.

How To Find y = mx + b?

y = mx + b is the formula used to find the equation of a straight line, when we know the slope(m) and the y-intercept (b) of the line. To determine m, we apply a formula based on the calculations. Let's derive this formula using the equation for the slope of a line. Let us consider a line whose slope is 'm' and whose y-intercept is 'b'. Let (x,y) be any other random point on the line whose coordinates are not known. We obtain the graph as follows.

y-intercept of a line

We know that the equation for the slope of a line in the slope-intercept form is y = mx+b

Rewriting this, we get m = (y-b) / x

Thus the formula to find m = change in y/ change in x

finding slope of a line

Let us derive the formula to find the value of the slope if two points \((x_{1},y_{1})\) and \((x_{2},y_{2})\) on the straight line are known. Then we have \(y_{1} = mx_{1} + b\) and \(y_{2} = mx_{2} + b\)

We know that, slope = change in y/ change in x

Substituting the values of y₁ and y₂, we get \[\begin{align}\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}&= \dfrac{(mx_{2}+b) - (mx_{1}+b)}{x_{2}-x_{1}}\\\\&=\dfrac{mx_{2}-mx_{1}}{x_{2}-x_{1}}\\\\&= \dfrac{m(x_{2}-x_{1})}{x_{2}-x_{1}}\\\\ &=m\end{align}\]

Thus we find that the slope (m) is calculated as (change in y)/ (change in x)

m = (difference in y coordinates)/ (difference in x coordinates)

To find the y-intercept or 'b', substitute the value of 'x' as 0 in the equation of a straight line, which is of the form Ax + By + C = 0. Consider an equation of a straight line : 3x + 5y = 10. To find the y-intercept, substitute the value of 'x' as 0 in the equation and solve for 'y'. On substituting 'x = 0' in the equation 3x + 5y =10, we get, 3(0) + 5y = 10
⇒5y = 10 and thus y = 10/5 ⇒ y = 2 or 'b' = 2.

Writing an Equation in The Slope Intercept Form

If the slope 'm' and y-intercept 'b' are given, then the equation of the straight line can be written in the form of 'y = mx +b'. For example, if the slope(m) for a line is 2 and the y-intercept 'b' is -1, then the equation of the straight line is written as y = 2x - 1. The slope value can be positive or negative. As we discussed in the earlier sections, in y = mx + b, 'm' represents the slope of the equation. To find the slope of a line, given its equation, we have to rearrange its terms to the slope-intercept form y = mx + b. Here, 'm' gives the slope and 'b' gives the y-intercept of the equation.

Let us consider the equation 2x + 3y = 6. We are required to find the slope and the y-intercept from the equation which is of the form Ax + By = C

We rewrite the standard form of the equation of the line to the slope-intercept form y = mx + b.

2x + 3y = 6
3y = 2x + 6
y = (-2/3) x + 2

Comparing the final equation with y = mx + b, we obtain the slope of the equation is m = -2/3 and the y-intercept of the equation is, b = 2 or (0,2).

Important Notes:

The equation of the slope-intercept form of a line whose slope is 'm' and whose y-intercept is 'b' or (0,b) is y = mx + b.
The equation of a horizontal line passing through (a,b) is of the form y = b.
The equation of a vertical line passing through (a,b) is of the form x = a.
m is calculated using the formula rise over run or (change in y)/ (change in x)

Topics Related to y = mx + b

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Example 1: Find the equation of the line whose graph contains the points (1,3) and (3,7)

Solution:
The required equation of the line is y = mx + b
Using the formula for slope, m = change in y / change in x = \(\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
= (7-3)/ (3-1) = 4/2 ⇒ m = 2
To find the y-intercept b, we consider any one of the coordinates.
Let us use(1,3) and m = 2 and substitute the values in the equation \(y_{1} = mx_{1} + b\)
3 = 2(1) + b ⇒ b = 3 - 2 = 1
Applying, m =2 and b = 1 in the equation of the line(y = mx + b), we get y = 2x + 1 Thus the equation of the straight line is y = 2x + 1
Example 2: Find the slope-intercept form of a line with slope -2 and which passes through the point (-1.4).

Solution:
We know that the slope-intercept form of a line is y = mx + b.
It is given that slope (m) = -2 and the coordinates through which the line is passing through is (-1,4). Substituting the given values in the slope-intercept form equation we get, 4 = (-2) (-1) + b.
4 = 2 + b b = 4 - 2 = 2.
The slope intercept form of the line is y = - 2 x + 2.

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FAQs on y mx b

What is y = mx + b?

y = mx + b is a representation of equation of a straight line. It is called as the slope intercept form. 'm' is referred to as the slope of the line, and 'b' refers to the 'y -intercept' of the line.

How to Find the Slope of a Line?

For two coordinates, (x₁,y₁) and (x₂, y₂), the slope of a line is the ratio of difference between the difference between the y coordinates and the difference between x coordinates, also known as the rise over the run. The formula to find the slope of a line is m = (y₂-y₁)/(x₂-x₁)

What is Slope-Intercept Form?

The equation of a straight line which is of the form y = mx + b, is called the slope intercept form. Here 'm' is the slope of the line and 'b' is the point at which the line intercepts the y - axis. An example for slope intercept form equation is y = 3x + 5

What is a Line With a Negative Slope?

A line for which the slope in negative is said to move from left to right in a graph. The slope of a line is found by the ration of difference in y-coordinates to the difference in x-coordinates. If this value is negative for a line, then the line has a negative slope.

What Does the Slope of a Line Mean?

The direction of a line is described by its slope. The slope can be positive or negative, based on its direction. A negative slope moves downward from left to right and a line with positive slope moves in the upward direction from right to left.

How to Find the Slope and Y Intercept

Source: https://www.cuemath.com/geometry/y-mx-b/

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