Question

How do you solve the system of equations $2x-2y=0$ and $4x+9y=0$ ?

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 the substitution method or Gaussian elimination method. Since Substitution is the easiest one we will find out by substitution method and get the required answer.

Complete step-by-step solution:
We have system of equations as:
$2x-2y=0\to \left( 1 \right)$ and
$4x+9y=0\to \left( 1 \right)$
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 $2x-2y=0$.
Now we will add $2y$ on both sides, we get:
$\Rightarrow 2x-2y+2y=0+2y$
On simplifying, we get:
$\Rightarrow 2x=2y$
Now on dividing both sides by $2$:
$\Rightarrow \dfrac{2x}{2}=\dfrac{2y}{2}$
On simplifying, we get:
$\Rightarrow x=y$
As we can see that we have a value of $x$ that is $x=y$. 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:
$\left[ 4\left( y \right)+9y=0 \right]$
$4\left( y \right)+9y=0$
Now on simplifying left hand side, we get:
$\Rightarrow 4y+9y$
On simplifying, we get:
$\Rightarrow 13y$
With this now we will isolate $y$.
So on dividing both sides by $13$,
$\Rightarrow \dfrac{13y}{13}=\dfrac{0}{13}$
On simplifying:
$\Rightarrow y=0$
Now, substitute $y=0$ in $x=y$, we get the value of $x$.
$x=0$
Therefore, the solution to the system of equation is:
$x=0$ and $y=0$

Note: To find whether the value of $x$ and $y$ is correct, we will substitute it in the given equation and equate it.
$2x-2y=0\to \left( 1 \right)$ and
$4x+9y=0\to \left( 1 \right)$
On substituting $x=0$ and $y=0$ in the left-hand side in equation $\left( 1 \right)$, we get:
$\Rightarrow 2\left( 0 \right)-2\left( 0 \right)$
On expanding we get:
\[\Rightarrow 0-0\]
On simplifying we get:
$\Rightarrow 0=0$
Also, On substituting $x=0$ and $y=0$ in the left-hand side in equation $\left( 2 \right)$, we get:
$\Rightarrow 4\left( 0 \right)+9\left( 0 \right)$
On expanding we get:
$\Rightarrow 0+0$
On simplifying we get:
$\Rightarrow 0=0$
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.