Question

Suppose you subtract a multiple of an equation in a system from another equation in the system. Explain why the two systems (before and after this operation) have the same solutions.

Short answer

If we do any type of operation in the equation to solve it. We have to do it on both sides of the equation which will lead to not changing the values of the answer.

Step-by-step solution

Equation

Suppose a system of two equations be

If we do any type of operation in the equation to solve it. We have to do it on both sides of the equation which will lead to not changing the values of the answer.

Explanation with example

Suppose two linear equations,

$x+3y=2,3x+4y=1$

The solution of this set of equation is$y=1,x=-1$$y=1,x=-1$

Now we do the desired operation in the above equation. Which is we will subtract the 3 times of equation 1 from equation 2.

$$\begin{gathered} x+3y=2 \\ 3x-3x+4y-9y=1-6 \\ -5y=-5 \\ \Rightarrow y=1,x=-1 \end{gathered}$$

Hence, the solution of the equation remains the same before and after the operation.

Therefore, if we do any type of operation in the equation to solve it. We have to do it on both sides of the equation which will lead to not changing the values of the answer.

Also, doing the same operation will lead to giving an equivalent system of equations.