Question

How do you solve the system of equations $2x+5y=-1$ and $3x+2y=4$ ?

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.

Complete step by step answer:
We have system of equations as:
$2x+5y=-1\to \left( 1 \right)$ and
$3x+2y=4\to \left( 2 \right)$
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 $2x+5y=-1$.
Now we will subtract $5y$ from both sides, we get:
$\Rightarrow 2x+5y-5y=-1-5y$
On simplifying, we get:
$\Rightarrow 2x=-1-5y$
Now on dividing both sides by $2$:
$\Rightarrow \dfrac{2x}{2}=-\dfrac{1}{2}-\dfrac{5y}{2}$
On simplifying, we get:
$\Rightarrow x=-\dfrac{1}{2}-\dfrac{5y}{2}$
As we can see that we have value of $x$ that is $x=-\dfrac{1}{2}-\dfrac{5y}{2}$. Therefore, now we can find our solution by substituting value of $x$ in equation $\left( 2 \right)$. That is we will find value of $y$.
Therefore, we get:
$\left[ 3\left( -\dfrac{1}{2}-\dfrac{5y}{2} \right)+2y=4 \right]$
$3\left( -\dfrac{1}{2}-\dfrac{5y}{2} \right)+2y=4$
Now on simplifying left hand side, we get:
$\Rightarrow 3\left( -\dfrac{1}{2}-\dfrac{5y}{2} \right)+2y$
$\Rightarrow -\dfrac{3}{2}-\dfrac{15y}{2}+2y$
On expanding the above expression, we get:
$\Rightarrow \dfrac{-3-11y}{2}$
So we get:
$\Rightarrow \dfrac{-3-11y}{2}=4$
With this now we will isolate $y$.
So multiply both the sides by 2,
$\Rightarrow 2\times \dfrac{-3-11y}{2}=4\times 2$
On simplifying:
$\Rightarrow -3-11y=8$
Now, on adding $3$ on both sides,
$\Rightarrow -11y=8+3$
On simplifying:
$\Rightarrow -11y=11$
Now by dividing both sides by $11$ and simplifying, we get:
$\Rightarrow -\dfrac{11y}{11}=\dfrac{11}{11}$
$\Rightarrow y=-1$
Now, substitute $y=-1$ in $x=-\dfrac{1}{2}-\dfrac{5y}{2}$, we get the value of $x$.
$x=-\dfrac{1}{2}-\dfrac{5\left( -1 \right)}{2}$
On simplifying,
$x=\dfrac{-1+5}{2}$
$x=2$

Therefore, the solution to the system of equation is:
$x=2$ and $y=-1$.

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