 To calculate: The solution for the system of equations 7\left(x-y\r xcl3411 2021-11-15 Answered
To calculate: The solution for the system of equations $$\displaystyle{7}{\left({x}-{y}\right)}={3}-{5}{x}$$ and $$\displaystyle{4}{\left({3}{x}-{y}\right)}=-{2}{x}$$, if the system does not have one unique solution, state whether the system is inconsistent, or whether the equations are dependent

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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.