Question

How do you solve the system of equations $-2x+4y=10$ and $3x-6y=-15$ ?

Answer 1

Hint: In order to find a solution to this problem, we will use 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.

Complete step by step answer:
We have system of equations as:
$-2x+4y=10\to \left( 1 \right)$ and
$3x-6y=-15\to \left( 2 \right)$
Since we have system of equation, we will use substitution method.
First we will use on equation $\left( 1 \right)$,
Therefore, we will start by isolating$x$ for $-2x+4y=10$.
Now we will subtract $4y$ from both sides, we get:
$\Rightarrow -2x+4y-4y=10-4y$
On simplifying, we get:
$\Rightarrow -2x=10-4y$
Now on dividing both sides by $2$:
$\Rightarrow \dfrac{-2x}{2}=\dfrac{10}{2}-\dfrac{4y}{2}$
On simplifying, we get:
$\Rightarrow -x=5-2y$
Now by adding minus$\left( - \right)$ sign on both sides, we get:
$\Rightarrow x=-5+2y$
As we can see that we have value of $x$ that is $x=-5+2y$. Therefore, now we can find our solution by substituting value of $x$ in equation $\left( 2 \right)$.
Therefore, we get:
$\left[ 3\left( -5+2y \right)-6y=-15 \right]$
$3\left( -5+2y \right)-6y=-15$
Now on simplifying left hand side, we get:
$\Rightarrow 3\left( -5+2y \right)-6y$
On expanding the above expression, we get:
$\Rightarrow -15+6y-6y$
Now we will use a similar element property that is, $6y-6y=0$.
So we get:
$\Rightarrow -15$
With this now, we will write left hand side and right hand side simultaneously. So, we get:
$\Rightarrow -15=-15$
That is,
$\left[ -15=-15 \right]$
We can conclude that both equations represent the same line so there are infinite solutions.

Note: To find whether the value of $x$ is correct, we will substitute it in the given equations and equate it.
$-2x+4y=10\to \left( 1 \right)$ and
$3x-6y=-15\to \left( 2 \right)$
On substituting $x=-5+2y$ in the left-hand side in equation $\left( 1 \right)$ we get:
$\Rightarrow -2\left( -5+2y \right)+4y$
On expanding we get:
$\Rightarrow 10-4y+4y$
On simplifying we get:
$\Rightarrow 10=10$
Also, On substituting $x=-5+2y$ in the left-hand side in equation $\left( 2 \right)$ we get:
$\Rightarrow 3\left( -5+2y \right)-6y$
On expanding we get:
$\Rightarrow -15+6y-6y$
On simplifying we get:
$\Rightarrow -15=-15$