Question

How do you graph using slope and intercept of $3x + 2y = 21$?

Answer 1

In this question we have to plot a graph using slope and intercept of a given straight line. Firstly, we will convert the given equation into a slope intercept form of a straight line. It can be done by first subtracting $3x$ from both sides of the given equation. Then dividing each term by 2 and rearranging the obtained equation. We then compare the given equation of a line with the standard slope intercept form of a line and find the slope and intercept. We substitute different values of x and obtain the values of y. Then we plot the points $(x,y)$ in the x-y plane and we will have a required graph of the given equation.

Complete step by step answer:
Given the equation of a straight line $3x + 2y = 21$ …… (1)
We are asked to draw the graph using the slope and intercept of the given line.
So firstly we will try to find out the slope of a line given in the equation (1).
To find this, we need to convert our given equation into slope intercept form of a straight line.
The general equation of a straight line in slope intercept form is given by,
$y = mx + c$ …… (2)
where $m$ is the slope or gradient of a line and $c$ is the intercept of a line.
Now we convert the given equation of a line into slop intercept form by rearranging the terms.
Consider the equation of a line given in the equation (1).
Subtracting $3x$ from both sides of the equation (1), we get,
$\Rightarrow 3x + 2y - 3x = 21 - 3x$
Combining the like terms we get,
$\Rightarrow 3x - 3x + 2y = 21 - 3x$
$\Rightarrow 0 + 2y = 21 - 3x$
$\Rightarrow 2y = 21 - 3x$
Now dividing throughout by 2 we get,
$\Rightarrow \dfrac{{2y}}{2} = \dfrac{{21 - 3x}}{2}$
$\Rightarrow y = \dfrac{{21}}{2} - \dfrac{3}{2}x$
Rearranging the above equation we get,
$\Rightarrow y = - \dfrac{3}{2}x + \dfrac{{21}}{2}$ …… (3)
Comparing with the standard slope intercept form given in the equation (2), we get,
$m = - \dfrac{3}{2}$ and $c = \dfrac{{21}}{2}$.
Now to draw a graph of a linear equation, we first assume some values for the variable x and substitute in the above equation and obtain the values of the other variable y.
Then plotting these values of x and y on the x-y plane, we get the graph of the given equation.
We first let different values of x.
Substituting $x = 0$ in the equation (3), we have,
$y = - \dfrac{3}{2}(0) + \dfrac{{21}}{2}$
$\Rightarrow y = \dfrac{{21}}{2}$
$\Rightarrow y = 10.5$
Substituting $x = 1$ in the equation (3), we have,
$y = - \dfrac{3}{2}(1) + \dfrac{{21}}{2}$
$\Rightarrow y = \dfrac{{ - 3 + 21}}{2}$
$\Rightarrow y = \dfrac{{18}}{2}$
$\Rightarrow y = 9$
Substituting $x = 2$ in the equation (3), we have,
$y = - \dfrac{3}{2}(2) + \dfrac{{21}}{2}$
$\Rightarrow y = \dfrac{{ - 6 + 21}}{2}$
$\Rightarrow y = \dfrac{{15}}{2}$
$\Rightarrow y = 7.5$
Substituting $x = 3$ in the equation (3), we have,
$y = - \dfrac{3}{2}(3) + \dfrac{{21}}{2}$
$\Rightarrow y = \dfrac{{ - 9 + 21}}{2}$
$\Rightarrow y = \dfrac{{12}}{2}$
$\Rightarrow y = 6$

x	0	1	2	3
y	10.5	9	7.5	6

Note: Graph of a linear equation is always a straight line. Remember the general form of an equation of a straight line given by $y = mx + c$, where m is the slope of the line and c is the intercept. If while calculating the points, if someone has made a mistake then all the points obtained after calculations will not come on a straight line. So, we need to calculate carefully while doing calculations for points and also while plotting in x-y plane.