91Ó°ÊÓ

Equation of line in slope intercept form is expressed below.

$y=mx+c$

Where m is the slope and c is the intercept of y-axis.

Consider the first equation $2x+3y=7$.

Rewrite the equation in form of slope-intercept form.

Subtract both sides by $2x$.

$\begin{array}{c}2x+3y−2x=7−2x\\ 3y=7−2x\end{array}$

Next, divide both sides by 3 and rearrange the terms.

$\begin{array}{c}\frac{3y}{3}=\frac{7−2x}{3}\\ y=\frac{7}{3}−\frac{2x}{3}\\ y=−\frac{2}{3}x+\frac{7}{3}\end{array}$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $-\frac{2}{3}$and intercept of y-axis c is $\frac{7}{3}$.

Now, consider the second equation $2x-3y=7$.

Rewrite the equation in form of slope-intercept form.

Subtract both sides by $2x$.

$\begin{array}{c}2x−3y−2x=7−2x\\ −3y=7−2x\end{array}$

Next, divide both sides by $-3$ and rearrange the terms.

$\begin{array}{c}\frac{−3y}{−3}=\frac{7−2x}{−3}\\ y=−\frac{7}{3}+\frac{2x}{3}\\ y=\frac{2}{3}x−\frac{7}{3}\end{array}$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $\frac{2}{3}$ and intercept of y-axis c is $-\frac{7}{3}$.

Plot the equations on the same plane and the point where both the equations intersect is the solution of the system of the equations.

The red line denotes the equation $2x+3y=7$ and blue line denotes the equation $2x-3y=7$.

Therefore, the point of intersection is $\left(\frac{7}{2},0\right)$.

The point of intersection is solution of system of equations if the point satisfies both the equation.

Substitute the point $\left(\frac{7}{2},0\right)$in the equation $2x+3y=7$.

Substitute x as $\frac{7}{2}$ and y as $0$and check whether right hand side is equal to left hand side of the equation.

$\begin{array}{c}2\left(\frac{7}{2}\right)+3\left(0\right)=7\\ 7+0=7\\ 7=7\end{array}$

Since, this is true so the point $\left(\frac{7}{2},0\right)$satisfy the equation $2x+3y=7$.

Substitute the point $\left(\frac{7}{2},0\right)$in the equation $2x-3y=7$.

Substitute x as $\frac{7}{2}$ and y as 0 and check whether right hand side is equal to left hand side of the equation.

$\begin{array}{c}2\left(\frac{7}{2}\right)−3\left(0\right)=7\\ 7−0=7\\ 7=7\end{array}$

Since, this is true so the point $\left(\frac{7}{2},0\right)$satisfy the equation $2x-3y=7$.

Hence, the solution of the system of equations is $\left(\frac{7}{2},0\right)$.