Question

How do you solve the system \[2y=4x-16\] and \[5y=4x+8\] ?

Answer 1

Hint: We can solve this question using basic linear equation concepts. We can make them a layered problem. First we will rearrange the terms as variables containing terms on one side and constants on the other side. we use algebra to manipulate the expressions to eliminate one unknown and solve for the other, Then we take that solved variable and use it to find the second one. Then we can find values of both the variables.

Complete step by step solution:
Given equations are
\[2y=4x-16\]
\[5y=4x+8\]
By rearranging the terms as variable containing terms and constants.
We will get
\[2y-4x=-16\]
\[5y-4x=8\]
Now we will solve a layered problem.
Here we can see that we already have equal coefficients so there is no need for any changes on equations.
If we don’t have any terms with equal coefficients then we have to make any one term to be equal by multiplying or dividing. If both of them are not having opposite signs we have to subtract otherwise we have to add.
Here we have the same direction so we have to subtract them.
After subtracting both the equations we get
\[\begin{align}
  & 2y-4x=-16 \\
 & \underline{5y-4x=8} \\
 & -3y=-24 \\
\end{align}\]
Now we have to divide the equation with 3 on both sides.
\[\Rightarrow -\dfrac{3y}{3}=-\dfrac{24}{3}\]
We can multiply the equation with – as it is both sides of the equation. We will get
\[\Rightarrow \dfrac{3y}{3}=\dfrac{24}{3}\]
By simplifying, we will get
\[\Rightarrow y=8\]
Now we will substitute this y value in the first equation given to get the x value.
Our first equation is
\[2y=4x-16\]
Now we have to substitute y value in it. We will get
\[\Rightarrow 2\left( 8 \right)=4x-16\]
\[\Rightarrow 16=4x-16\]
Now we have to add 16 on both sides we will get
\[\Rightarrow 16+16=4x-16+16\]
\[\Rightarrow 32=4x\]
Now we can rewrite the equation as
\[\Rightarrow 4x=32\]
Now we divide the equation with 4 on both sides of the equation.
\[\Rightarrow \dfrac{4x}{4}=\dfrac{32}{4}\]
We will get
\[\Rightarrow x=8\]
So the x and y values by solving the equations given is
\[x=8,y=8\]

Note: We can check the solution by back substituting the values in both the equations and check if they are satisfied or not. Don’t check using only one equation we have checked both the equations then only we can say the solutions are correct.