To solve the system of equations x - 3y = 32 and x + y = -2,
we can use the method of substitution or elimination.
Method 1: Substitution
Step 1: Solve one equation for one variable in terms of the other variable.
From the second equation, we have x = -2 - y.
Step 2: Substitute the expression for x into the other equation.
In the first equation, replace x with -2 - y: (-2 - y) - 3y = 32.
Step 3: Simplify and solve for y.
-2 - y - 3y = 32
-2 - 4y = 32
-4y = 34
y = -34/4
y = -17/2
Step 4: Substitute the value of y back into one of the original equations to solve for x.
Using the second equation: x + (-17/2) = -2
x - 17/2 = -2
x = -2 + 17/2
x = -4/2 + 17/2
x = 13/2
So the solution to the system of equations is x = 13/2 and y = -17/2.
Method 2: Elimination
Step 1: Multiply the second equation by 3 to make the coefficients of y in both equations equal.
Original equations: x - 3y = 32 and x + y = -2.
After multiplying the second equation by 3: 3x + 3y = -6.
Step 2: Add the two equations together to eliminate the y terms.
(x - 3y) + (3x + 3y) = 32 + (-6)
4x = 26
x = 26/4
x = 13/2
Step 3: Substitute the value of x into one of the original equations and solve for y.
Using the second equation: (13/2) + y = -2
y = -2 - 13/2
y = -4/2 - 13/2
y = -17/2
So the solution to the system of equations is x = 13/2 and y = -17/2.
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