Question

Select all the systems of equations with the same solution

Answer 1

System of Equations

Given the system of equations:

`\begin{gathered} Ax+By=C \\ Dx+Ey=F \end{gathered}`

One of the principles of the reduction method is that we can multiply one equation by a non-zero number and add it to the other equation and the solutions will be the same, i.e., the system is not affected.

Multiply the second equation by -1:

`\begin{gathered} Ax+By=C \\ -Dx-Ey=-F \end{gathered}`

Add it to the first equation and replace it:

`(A-D)x+(B-E)y=C-F`

Joining the equations again, we get the system:

`\begin{gathered} (A-D)x+(B-E)y=C-F \\ Dx+Ey=F \end{gathered}`

This system has the same solutions as the original and the first choice is exactly like this system, thus this choice is correct.

Multiply the first equation by 2:

`2Ax+2By=2C`

Add it to the second equation and replace the first equation::

`\begin{gathered} \\ \begin{equation*} (2A+D)x+(2B+E)y=2C+F \end{equation*} \\ Dx+Ey=F \end{gathered}`

This system of equations has the same solutions, but it's not the system shown in the second choice, thus it's incorrect.

Multiplying the second equation by -3:

`\begin{gathered} Ax+By=C \\ -3Dx-3Ey=-3F \end{gathered}`

This is the system shown in the third choice, thus it's correct

Multiplying the second equation by -5 and adding to the first, we get:

`\begin{gathered} (A-5D)x+(B-5E)y=C-5F \\ Dx+Ey=F \end{gathered}`

This is the system shown in the fourth choice, thus it's correct.

The fifth choice is incorrect since there is no way to get the sums of the coefficients as shown.