Geometry Y=Mx+B – Coordinate Geometry: Slope

y = mx + bis 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.

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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 + bis 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 anddifference in x coordinates is indicated as run.

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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 determinem, we apply a formula based on the calculations. Let's derive this formula using the equation forthe 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.

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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

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Let us derive the formula to find the value of the slope iftwo 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<egin{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}}\\ &=mend{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 anequation 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 thusy = 10/5⇒y = 2 or 'b' = 2.

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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 = 63y = 2x + 6y = (-2/3) x + 2

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

Important Notes:

The equation of the slope-intercept formof a line whose slope is 'm' and whose y-intercept is 'b' or (0,b) isy = mx + b.The equation of a horizontal line passing through (a,b) is of the form y = b.The equation of a vertical linepassing 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)

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Topics Related to y = mx + b

Check out some interesting articles related to y = mx + b.

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 + bUsing 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 = 2To 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 = 1Applying, 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).

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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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