Question

How do you solve the system of equations $-8x-8y=3$ and $-7x-8y=16$ using elimination?

Answer 1

Hint: For solving the given system of two linear equations by the method of elimination, which is given as $-8x-8y=3$ and $-7x-8y=16$, we have to remove one variable to get an equation in one variable. For this, we have to make the coefficients of one of the variables same in both the equations. From the two equations given, $-8x-8y=3$ and $-7x-8y=16$ we can observe that the coefficient of $y$ is already the same and is equal to $-8$ in both. So we just need to subtract both the equations to obtain the equation in $x$, on solving which, we will get the value of $x$. Finally substituting this value in any of the two equations, we will obtain the value of $y$.

Complete step by step solution:
The given system of equations is
$\begin{align}
  & \Rightarrow -8x-8y=3.........\left( i \right) \\
 & \Rightarrow -7x-8y=16........\left( ii \right) \\
\end{align}$
In the above question, we are directed to solve these using the method of elimination. We know that in this method we remove one of the two variables to get an equation in the single variable. For this, we make the coefficient of one of the variables same in both the equations. But from the above two equations, we can see that the coefficient of $y$ is already same and is equal to $-8$ in both the equations. So we subtract the equation (i) from (ii) to get
$\begin{align}
  & \Rightarrow -7x-8y-\left( -8x-8y \right)=16-3 \\
 & \Rightarrow -7x-8y+8x+8y=13 \\
 & \Rightarrow x=13 \\
\end{align}$
Now, we substitute this in (i) to get
$\begin{align}
  & \Rightarrow -8\left( 13 \right)-8y=3 \\
 & \Rightarrow -104-8y=3 \\
\end{align}$
Adding $104$ both the sides, we get
$\begin{align}
  & \Rightarrow -104-8y+104=3+104 \\
 & \Rightarrow -8y=107 \\
\end{align}$
Finally, dividing both sides by $-8$ we get
$\begin{align}
  & \Rightarrow \dfrac{-8y}{-8}=\dfrac{107}{-8} \\
 & \Rightarrow y=-\dfrac{107}{8} \\
\end{align}$

Hence, we have solved the given system of equations by elimination and obtained the solution as $x=13$ and $y=-\dfrac{107}{8}$.

Note: After obtaining the value of the second variable by substituting the value of the eliminated variable in one of the equations, we must check the solution by substituting the values of both the variables in the other equation as well. This is done to check for the calculation mistakes which may occur while solving.