Solve the system of linear equations given below using substitution or elimination.

Question

3x+2y=12
−x+y=11

Answer 1

We can solve this system of linear equations using 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 + y = 11
Rearranging the equation gives:
y = x + 11

Step 2: Substitute the value of y in terms of x into the other equation.
Plugging y = x + 11 into the first equation, we have:
3x + 2(x + 11) = 12
Simplifying, we get:
3x + 2x + 22 = 12
Combining like terms, we obtain:
5x + 22 = 12
Subtracting 22 from both sides gives:
5x = -10
Dividing both sides by 5, we have:
x = -2

Step 3: Substitute the value of x back into one of the original equations to solve for the other variable.
Using the second equation:
-y = -2 + 11
Simplifying:
-y = 9
Dividing both sides by -1, we find:
y = -9

Therefore, the solution to the system of equations is x = -2 and y = -9.

Method 2: Elimination
Step 1: Multiply the second equation by 2 to make the coefficients of x match.
The second equation -x + y = 11 becomes:
-2x + 2y = 22

Step 2: Add the two equations together to eliminate the y-term.
(3x + 2y) + (-2x + 2y) = 12 + 22
Simplifying gives:
x = 34

Step 3: Substitute the value of x back into one of the original equations to solve for the other variable.
Using the first equation, we have:
3(34) + 2y = 12
Simplifying:
102 + 2y = 12
Subtracting 102 from both sides yields:
2y = -90
Dividing both sides by 2 provides:
y = -45

Thus, the solution to the system of equations is x = 34 and y = -45.

Both methods give the same solution: x = -2, y = -9.