Given Information:

The provided system of equations is \(\displaystyle{7}{\left({x}-{y}\right)}={3}-{5}{x}\) and \(\displaystyle{4}{\left({3}{x}-{y}\right)}=-{2}{x}\).

Formula used:

Solving a System of Equations by Using the Addition Method:

Step1: Write both equations in standard form: \(\displaystyle{A}{x}+{B}{y}={C}\)

Step2: Clear fractions or decimals.

Step3: Multiply one or both equations by nonzero constants to create

opposite coefficients for one of the variables.

Step4: Add the equations from step3 to eliminate one variable.

Step5: Solve for the remaining variable.

Step6: Substitute the known value found in step5 into one of the original equations to solve for the other variable.

Step7: Check the ordered pair in each equation and write the solution set.

Calculation:

Consider the provided system of equations, \(\displaystyle{7}{\left({x}-{y}\right)}={3}-{5}{x}\) and \(\displaystyle{4}{\left({3}{x}-{y}\right)}=-{2}{x}\)

Convert the equations into standard form \(\displaystyle{A}{x}+{B}{y}={C}\):

\(\displaystyle{7}{\left({x}-{y}\right)}={3}-{5}{x}\)

\(\displaystyle{7}{x}-{7}{y}={3}-{5}{x}\) ............(1)

\(\displaystyle{7}{x}-{2}{y}={3}\)

And,

\(\displaystyle{4}{\left({3}{x}-{y}\right)}=-{2}{x}\)

\(\displaystyle{12}{x}-{4}{y}=-{2}{x}\)............(2)

\(\displaystyle{14}{x}-{4}{y}=-{0}\)

Now, multiply by —2 in the equation (1):

\(-14x+4y=-6\)............(3)

Now, add equations (2) and (3):

\(\displaystyle{14}{x}—{4}{y}={0}\)

\(\displaystyle-{l}{4}{x}+{\left.{d}{y}\right.}=-{6}\)

\(\displaystyle\overline{{{0}=-{6}}}\)

The system of equations is reduced to a contradiction. Hence, the system is inconsistent.

Therefore, the solution for the system of equations \(\displaystyle{7}{\left({x}-{y}\right)}={3}-{5}{x}\) yand \(\displaystyle{4}{\left({3}{x}-{y}\right)}=-{2}{x}\) is {} and the system is inconsistent.