Question

How do you solve the system $-3x+7y=-16$ and $-9x+5y=16$

Answer 1

Hint: To solve the system of equations we will use the elimination method. For this first we will equate the coefficient of either x or y in the given equations. Then add or subtract both equations and obtain a value of x or y. Then substitute the obtained value in the given equation to get the value of another variable.

Complete step-by-step solution:
We have been given the system of equations: $-3x+7y=-16$ and $-9x+5y=16$
We have to solve the given equations.
The given equations are linear equations in two variables.
Now, to solve the equations we will use elimination method, for these first let us equate the coefficient of x in both the equations.
$-3x+7y=-16.............(i)$
$-9x+5y=16............(ii)$
So, by multiplying by 3 in equation (i) we will get
$\begin{align}
  & \Rightarrow -3x\times 3+3\times 7y=-16\times 3 \\
 & \Rightarrow -9x+21y=-48..........(iii) \\
\end{align}$
Now, subtracting equation (ii) from equation (iii) we will get
$\begin{align}
  & -9x+21y=-48 \\
 & \underline{-9x+5y=16} \\
 & 16y=-64 \\
\end{align}$
Now, simplifying the obtained equation we will get
\[\begin{align}
  & \Rightarrow 16y=-64 \\
 & \Rightarrow y=\dfrac{-64}{16} \\
 & \Rightarrow y=-4 \\
\end{align}\]
Now, we have got the value of y, so we can substitute it in equation (i). Then we will get
$\begin{align}
  & \Rightarrow -3x+7y=-16 \\
 & \Rightarrow -3x+3\times \left( -4 \right)=-16 \\
 & \Rightarrow -3x-12=-16 \\
 & \Rightarrow -3x=-16+12 \\
 & \Rightarrow -3x=-4 \\
 & \Rightarrow x=\dfrac{-4}{-3} \\
 & \Rightarrow x=\dfrac{4}{3} \\
\end{align}$
So, on solving the given system of equations we get the values $x=\dfrac{4}{3}$ and $y=-4$.

Note: Alternatively one can use the substitution method also to solve the equations. In the substitution method we get the value of one variable in terms of another variable and substitute it into the given equation. By simplifying the equation we will get the values.