Solve the system of linear equations given below using substitution or elimination. 3x+2y=12 −x+y=11
Let's use the method of substitution.
We have the second equation: -x + y = 11.
Solving this equation for x, we get: x = y - 11.
Now we can substitute this expression for x into the first equation.
We have: 3x + 2y = 12.
Substituting x = y - 11, we get: 3(y - 11) + 2y = 12.
Simplifying this equation gives us: 3y - 33 + 2y = 12.
Combining like terms: 5y - 33 = 12.
Adding 33 to both sides of the equation: 5y = 45.
Dividing both sides by 5, we get: y = 9.
Now that we have the value of y, we can substitute it back into one of the original equations to solve for x.
Using the equation -x + y = 11, we have: -x + 9 = 11.
Subtracting 9 from both sides of the equation: -x = 2.
Dividing both sides by -1, we get: x = -2.
Therefore, the solution to the system of equations is x = -2 and y = 9.
To solve the system of linear equations using substitution or elimination, let's begin with the substitution method.
Step 1: Solve one of the equations for one variable.
Let's solve the second equation for x:
-x + y = 11
x = y + 11
Step 2: Substitute the expression for x into the other equation.
Now, substitute the value of x (which is y + 11) into the first equation:
3x + 2y = 12
3(y + 11) + 2y = 12
3y + 33 + 2y = 12
5y + 33 = 12
5y = 12 - 33
5y = -21
Step 3: Solve for the remaining variable.
Divide both sides of the equation by 5:
5y/5 = -21/5
y = -21/5
Step 4: Substitute the value of y back into the equation to find x.
Now we substitute the value of y into the equation we found in Step 1:
x = y + 11
x = -21/5 + 11
x = -21/5 + 55/5
x = 34/5
Therefore, the solution to the system of linear equations is:
x = 34/5
y = -21/5
To solve the system of linear equations using substitution or elimination, we will need to choose a method that best suits the system. In this case, let's use the substitution method.
Step 1: Solve one equation for one variable.
Let's solve the second equation, −x+y=11, for x:
x = y - 11
Step 2: Substitute the expression obtained in step 1 into the other equation.
Substitute x = y - 11 into the first equation: 3x + 2y = 12
3(y - 11) + 2y = 12
Step 3: Simplify and solve for y.
3y - 33 + 2y = 12
5y - 33 = 12
5y = 45
y = 9
Step 4: Substitute the found value of y into one of the original equations and solve for x.
Using the second equation, −x + y = 11:
-x + 9 = 11
-x = 2
x = -2
Therefore, the solution to the system of linear equations 3x+2y=12 and −x+y=11 is x = -2 and y = 9.