Question

How do you solve the system of equations -6x+5y=8 and 5x+3y=22?

Answer 1

Hint: In this question we will be using elimination method to solve it. For this method, we require the component of either the x-component or the y-component of both the equations to be similar so that when subtracted the similar terms will get cancelled and we can solve for others. But for this question, we multiply both the by 3 and 5 in the respective order so that their y-component gets similar and can be cancelled. We can then proceed to get the value of \[x\]. We will substitute the value of \[x\], in any of the equations and we get the value of \[y\] on solving.

Complete step by step solution:
According to the given question, we have been given a set of two equations having two variables and we have to solve for \[x\] and \[y\]. And we will be using the elimination method to solve this system of equations.
Let’s take the equations as,
-6x+5y=8-----(1)
5x+3y=22-----(2)
We have to make either the x-component same or the y-component same in both the equations by multiplying the equation with a factor that would do the same.
So, we will multiply the equation (1) by 3 and equation (2) by 5, we get,
\[\left. -6x+5y=8 \right\}\times 3\]
\[\Rightarrow -18x+15y=24\]----(4)
\[\Rightarrow 25x+15y=110\]----(5)
We will now subtract equation (5) - equation (4), we get,
      \[25x+15y=110\]
\[\underline{-(-18x+15y=24)}\]
       \[43x=86\]----(6)
We get the value of x as,
\[\Rightarrow x=2\]
Now, substituting the value of \[x=2\] in the equation (2), we have,
5x+3y=22
\[\Rightarrow 5(2)+3y=22\]
Solving this expression, we get,
\[\Rightarrow 10+3y=22\]
\[\Rightarrow 3y=22-10\]
\[\Rightarrow 3y=12\]
\[\Rightarrow y=4\]

Therefore, \[x=2\And y=4\].

Note: This method is also very helpful in solving questions having more than two equations. But since it involves multiplication of a term to equalize either the x-component or the y-component and at times this can increase the complexity of the question and can be time consuming. The above question can also be done using a substitution method. If we compare elimination method and substitution method, substitution method is faster.