Question

How do you solve the system of equation $-8x+5y=-8$ and $-16x-y=-16$ by elimination?

Answer 1

Hint: We will be using the elimination method as asked in the question. Firstly, we will take one of the equations, say, $-8x+5y=-8$ and we will multiply it with a certain factor so that one, either x-component or y-component, becomes similar so that it can be cancelled. And so we will multiply by 2, so the x-components get the same and are cancelled and then we proceed to calculate to find the value of y from the resulting expression. Substituting the value of y back in any one of the equations give us the value of x.

Complete step by step solution:
According to the question given, we have been given two sets of equations having two variables namely \[x\] and \[y\].
Elimination method requires the set of equations have one, either x-component or y-component, become similar so that it can be cancelled.
So first, we will take an equation, say, $-8x+5y=-8$ and now to make its x-component or y-component similar to the equation $-16x-y=-16$ we will have to multiply the former equation with a factor. If we observe closely, by multiplying the equation $-8x+5y=-8$ by 2, the x-component becomes similar to $-16x-y=-16$.
We have,
\[\left\{ -8x+5y=-8 \right\}\times 2\]
\[\Rightarrow -16x+10y=-16\]-----(1)
 $-16x-y=-16$------(2)
We will now subtract equation (1) – (2), we get,
\[-16x+10y=-16\]
\[\underline{-(-16x-y=-16)}\]
\[0x+11y=0\]-----(3)
Solving equation (3), we get the value of \[y\] as,
\[\Rightarrow y=0\]
Putting the value of \[y=0\] in equation (1), we get,
\[-16x+10y=-16\]
\[\Rightarrow -16x+10(0)=-16\]
\[\Rightarrow -16x=-16\]
\[\Rightarrow x=1\]

Therefore, \[x=1,y=0\].

Note: If in a set of equation using elimination method, making one of the component (either x or y component) similar in both the equations, both the components get similar when a factor is multiplied to it then, in that case the given question has just one equation and the second one is just another way of writing the first equation and the question cannot be solved.