Solve the following system of linear equations by addition. Indicate whether the given system of linear equations has one solution, has no solution, or has an infinite number of solutions. If the system has one solution, find the solution.
{−2x+2y=−4 6x+3y=−15
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To solve the system of linear equations by addition, we need to eliminate one variable by adding the two equations together.
First, let's multiply the first equation by 3 and the second equation by 2 to make the coefficients of y equal:
-2x + 2y = -4 (equation 1)
6x + 3y = -15 (equation 2)
Multiply equation 1 by 3:
(3)(-2x + 2y) = (3)(-4)
-6x + 6y = -12 (equation 3)
Multiply equation 2 by 2:
(2)(6x + 3y) = (2)(-15)
12x + 6y = -30 (equation 4)
Now, add equation 3 and equation 4 together:
(-6x + 6y) + (12x + 6y) = (-12) + (-30)
6x + 12y = -42 (equation 5)
So, the system of equations becomes:
6x + 12y = -42 (equation 5)
6x + 3y = -15 (equation 2)
We can see that both equations have the same left-hand side (6x), but different right-hand sides. This means that the system has no solution.