Question

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. $$ \begin{aligned}&-x+y=7\\\&2 x-2 y=-18\end{aligned} $$

Short answer

The linear system has no solutions. As they represent parallel lines, they do not intersect at any point.

Step-by-step solution

Arrange Equations

First, we arrange the equations in a format where x's, y's and constants align vertically. The equations would look like this:\n-1*x + 1*y = 7\n2*x - 2*y = -18

Multiply for coefficient match

In the linear combination method, it's necessary to match the coefficients of one of the variable. Here, we see the coefficients of y in both equations are '1' and '-2'. Multiply the first equation by 2 to match the coefficient '2' of y in the second equation. It results in the following system:\n-2*x + 2*y = 14\n2*x - 2*y = -18

Add Equations for Elimination

Now it's possible to add the equations to get rid of its y variable. Summing up the equations results in a new equation: 0*x = -4. The left-hand side simplifies to '0'. Hence further simplification gives that: 0=-4.

Analyze Result for Solution Space

Since the statement 0=-4 is not true, there is no solution to the system. This means that the lines represented by the two equations are parallel and never intersect.