Solve each system of equations by graphing.
$\begin{array}{l}x+2y=6\\ 2x+y=9\end{array}$

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 $x+2y=6$.

Rewrite the equation in form of slope-intercept form.

Subtract both sides by x.

$\begin{array}{l}x+2y-x=6-x\\ 2y=6-x\end{array}$

Next, divide both sides by 2 and rearrange the terms.

$\begin{array}{l}\frac{2y}{2}=\frac{6-x}{2}\\ y=\frac{6}{2}-\frac{x}{2}\\ y=-\frac{1}{2}x+3\end{array}$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $-\frac{1}{2}$ and intercept of y-axis c is $3$.

Now, consider the second equation $2x+y=9$.

Rewrite the equation in form of slope-intercept form.

Subtract both sides by $2x$.

$\begin{array}{l}2x+y-2x=9-2x\\ y=-2x+9\end{array}$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is$-2$ and intercept of y-axis c is$9$.

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 $x+2y=6$ and blue line denotes the equation $2x+y=9$.

Therefore, the point of intersection is $\left(4,1\right)$.

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

Substitute the point $\left(4,1\right)$in the equation $x+2y=6$.

Substitute x as 4 and y as$1$and check whether right hand side is equal to left hand side of the equation.

$\begin{array}{l}4+2\left(1\right)=6\\ 4+2=6\\ 6=6\end{array}$

Since, this is true so the point $\left(4,1\right)$satisfy the equation$2x+y=9$.

Substitute the point $\left(4,1\right)$in the equation $2x+y=9$.

Substitute x as 4 and y as $1$and check whether right hand side is equal to left hand side of the equation

$\begin{array}{l}2\left(4\right)+1=9\\ 8+1=9\\ 9=9\end{array}$

Since, this is true so the point $\left(4,1\right)$satisfy the equation$2x+y=9$.

Hence, the solution of the system of equations is $\left(4,1\right)$.