Question

How do you solve the system $4x+6y=-12$ and $2x+3y=9$ by graphing?

Answer 1

Hint: To solve the given equations by graphing first we have to find the coordinates of x and y. First we have to find two solutions for each of the equations. Then by drawing two lines by using the coordinates we will analyze that if lines are intersecting or not.

Complete step-by-step solution:
We have been given the system $4x+6y=-12$ and $2x+3y=9$.
We have to solve the given system by graphing.
We have two equations in two variables.
$4x+6y=-12.........(i)$
$2x+3y=9..........(ii)$
To solve the given system by graphing we need to find the values of x and y.
Let us solve the equation (i), by putting the value of x equal to zero we will get
$\Rightarrow 4\times 0+6y=-12$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow 6y=-12 \\
 & \Rightarrow y=\dfrac{-12}{6} \\
 & \Rightarrow y=-2 \\
\end{align}$
So we get the point $\left( 0,-2 \right)$.
Now, if we put y equal to zero in equation (i) then we will get
$\Rightarrow 4x+6\times 0=-12$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow 4x=-12 \\
 & \Rightarrow x=\dfrac{-12}{4} \\
 & \Rightarrow x=-3 \\
\end{align}$
So we get the point $\left( -3,0 \right)$.
Let us draw a line on the graph by using these points we will get

Now, let us solve the equation (ii), by putting the value of x equal to zero we will get
$\Rightarrow 2\times 0+3y=9$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow 3y=9 \\
 & \Rightarrow y=\dfrac{9}{3} \\
 & \Rightarrow y=3 \\
\end{align}$
So we get the point $\left( 0,3 \right)$.
Now, if we put y equal to zero in equation (i) then we will get
$\Rightarrow 2x+3\times 0=9$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow 2x=9 \\
 & \Rightarrow x=\dfrac{9}{2} \\
 & \Rightarrow x=4.5 \\
\end{align}$
So we get the point $\left( 4.5,0 \right)$.
Let us draw a line on the graph by using these points we will get

Now, we can see that the lines are parallel to each other. As the lines are parallel and have no common point so the equations have no solution.

Note: The points to be noted are that if lines are parallel in graph equations has no solution. When the lines intersect each other at some point then the point becomes the solution of the equations. We can also verify the answer by putting the values in the given equations.