Problem 11 - Introductory Algebra for College Students

Question

Solve each system by the substitution method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}-3 x+y=-1 \\\x-2 y=4\end{array}\right.$$

Step-by-Step Solution

Answer

The solution set to the system of equations is {(x, y) | x = -2, y = -11/5}.

Step 1: Express one of the variables in terms of the other

Let's express 'x' in terms of 'y' using the second equation. That is, rearrange x - 2y = 4 to get x = 4 + 2y.

Step 2: Substitute x into the other equation

Substitute x from step 1 into the first equation, and solve for y. We get -3(4+2y) + y = -1. This simplifies to -12 - 6y + y = -1, which further simplifies to -5y - 12 = -1. Adding 12 to both sides gives -5y = 11. Dividing by -5 results with y = -11/5.

Step 3: Substitute y into x to get its value

Substitute y = -11/5 back into the equation for x in step 1 to get x = 4 + 2(-11/5). Simplifies to x = 4 - 22/5 = -2.

Step 4: Express the solution set in set notation

The solution set in set notation will be given as {(x, y) | x = -2, y = -11/5}. This is an ordered pair representing the solution to the system of equations.