Question

How do you solve the system of equations $4x-2y=-30\text{ and }-2x-10y=-18$ ?

Answer 1

Hint: In this question, we have to find the value of x and y from a system of equations. Therefore, we will use the elimination method to get the solution. As we know, in the elimination method, we will eliminate one variable by either subtracting or adding the equations to get an equation in one variable. Thus, we see that in both the equations, the coefficient of x is not the same; therefore we will multiply 2 on both sides in the second equation, to get 4x. Then, we will subtract both the equations to get an equation in one variable. After the necessary calculations, we get the value of y. Thus, we will substitute the value of y in any one of the equations and solve for x, which gives the required result for the solution.

Complete step by step solution:
According to the problem, we have to find the value of x and y from the system of equations.
Thus, we will use the elimination method to get the solution.
The equations given to us is $4x-2y=-30\text{ and }-2x-10y=-18$ .
Let us say that $4x-2y=-30$ -------- (1) and $-2x-10y=-18$ --------- (2)
Now, we see that in both equation (1) and (2), the coefficient of x is not the same, thus we will make them same by multiplying 2 on both sides in the equation (2), we get
$\text{2(-2}x-10y)=2(-18)$
Now, we will apply the distributive property $a(b+c)=ab+ac$ on the left-hand side in the above equation, we get
$\text{2(-2}x)+2(-10y)=2(-18)$
$-4x-20y=-36$ ----------- (3)
Thus, we see that the coefficient of x is the same but with different signs in both equation (1) and (3), thus we will add both the equations, that is
$\begin{align}
  & \text{ 4}x-2y\text{ }=-30 \\
 & \underset{(+)}{\mathop{{}}}\,\left( -4x-20y=-36 \right) \\
\end{align}$
Thus, opening the brackets of the second equation, we get
$\begin{align}
  & \text{ 4}x-\text{ }2y\text{ }=-30 \\
 & \underline{\underset{(+)}{\mathop{-}}\,4x\underset{(+)}{\mathop{-}}\,20y=\underset{(+)}{\mathop{-36}}\,} \\
\end{align}$
As we know, the same terms with opposite signs cancel out each other, thus we get
$\begin{align}
 & \text{ 0 }-22y=-66 \\
\end{align}$
Now, we will divide 22 on both sides in the above equation, we get
$\begin{align}
  & \text{ 0 }-\dfrac{22}{22}y=-\dfrac{66}{22} \\
\end{align}$
On further simplification, we get
$\begin{align}
 & \text{ 0 }-y=-3 \\
\end{align}$
Now, we will multiply (-1) on both sides in the above equation, we get
$\begin{align}
 & \text{ 0 }-y.(-1)=-3.(-1) \\
\end{align}$
Therefore, we get
$y=3$
Now, we will substitute the above value of y in equation (1), we get
$4x-2.(3)=-30$
On further simplification, we get
$4x-6=-30$
Now, we will add 6 on both sides in the above equation, we get
$4x-6+6=-30+6$
As we know, the same terms with opposite signs cancel out each other, thus we get
$4x=-24$
Now, we will divide 4 on both sides in the above equation, we get
$\dfrac{4}{4}x=\dfrac{-24}{4}$
On further solving, we get
$x=-6$

Therefore, for the system of equations $4x-2y=-30\text{ and }-2x-10y=-18$ , the value of x and y is -6 and 3 respectively.

Note: While solving this problem, do step-by-step calculations properly to avoid confusion and mathematical mistakes. You can also make the coefficient of y equal in both the equations, that is multiply 5 on both sides in the first equation and then solve, to get an accurate answer. One of the alternative methods to get the solution is either you can use the substitution method or the cross-multiplication method.