Question

How do you solve the system of equations $-2x+2y=32$ and $2x+3y=18$ ?

Answer 1

Hint: In order to find a solution to this problem, we will use a substitution method. The solution to the system of equations can be found using substitution method or Gaussian elimination method. Since Substitution is the easiest one we will find out by substitution method.

Complete step-by-step solution:
We have system of equations as:
$-2x+2y=32\to \left( 1 \right)$ and
 $2x+3y=18\to \left( 2 \right)$
Since we have a system of equations, we will use a substitution method.
First we will use on equation $\left( 1 \right)$,
Therefore, we will start by isolating $x$ for $-2x+2y=32$.
Now we will subtract $2y$ from both sides, we get:
$\Rightarrow -2x+2y-2y=32-2y$
On simplifying, we get:
$\Rightarrow -2x=32-2y$
Now on dividing both sides by $2$:
$\Rightarrow \dfrac{-2x}{2}=\dfrac{32}{2}-\dfrac{2y}{2}$
On simplifying, we get:
$\Rightarrow -x=16-y$
Now by adding minus$\left( - \right)$ sign on both sides, we get:
$\Rightarrow x=-16+y$
As we can see that we have a value of $x$ that is $x=-16+y$. Therefore, now we can find our solution by substituting the value of $x$ in equation $\left( 2 \right)$. That is we will find the value of $y$.
Therefore, we get:
$\left[ 2\left( -16+y \right)+3y=18 \right]$
$2\left( -16+y \right)+3y=18$
Now on simplifying left hand side, we get:
$\Rightarrow 2\left( -16+y \right)+3y$
On expanding the above expression, we get:
$\Rightarrow -32+2y+3y$
$\Rightarrow -32+5y=18$
With this now we will isolate $y$.
So on adding both sides by 32,
$\Rightarrow -32+5y+32=18+32$
On simplifying:
$\Rightarrow 5y=18+32$
$\Rightarrow 5y=50$
On simplifying, we get:
$\Rightarrow y=10$
Now, substitute $y=10$ in $x=-16+y$, we get the value of $x$.
$x=-16+10$
On simplifying,
$x=-6$
Therefore, the solution to the system of equation is:
$x=-6$ and $y=10$

Note: To find whether the value of $x$ and $y$ is correct, we will substitute it in the given equation and equate it.
$-2x+2y=32\to \left( 1 \right)$ and
 $2x+3y=18\to \left( 2 \right)$
On substituting $x=-6$ and $y=10$ in the left-hand side in equation $\left( 1 \right)$, we get:
$\Rightarrow -2\left( -6 \right)+2\left( 10 \right)$
On expanding we get:
\[\Rightarrow 12+10\]
On simplifying we get:
$\Rightarrow 32=32$
Also, On substituting $x=-6$ and $y=10$ in the left-hand side in equation $\left( 2 \right)$, we get:
$\Rightarrow 2\left( -6 \right)+3\left( 10 \right)$
On expanding we get:
$\Rightarrow -12+30$
On simplifying we get:
$\Rightarrow 18=18$
Since the left-hand side equals to the right-hand side in both equations $\left( 1 \right)$ and $\left( 2 \right)$, we can conclude that the answer is correct.