Solve the system of linear equations given below using substitution or elimination. 3x+2y=12 −x+y=11 (1 point) (, )
To solve the system of equations using substitution, we can solve one equation for one variable and substitute it into the other equation.
Let's solve the second equation for x:
-x + y = 11
x = y + 11
Now substitute this value of x into the first equation:
3x + 2y = 12
3(y + 11) + 2y = 12
3y + 33 + 2y = 12
5y + 33 = 12
5y = -21
y = -21/5
Now substitute this value of y back into x = y + 11:
x = (-21/5) + 11
x = (-21 + 55)/5
x = 34/5
Therefore, the solution to the system of equations is (34/5, -21/5).
To solve the system of linear equations using substitution or elimination, we'll start by using elimination.
First, let's multiply the second equation by 3 to make the coefficients of x the same:
-3x + 3y = 33
Now, let's add this equation to the first equation:
(3x + 2y) + (-3x + 3y) = 12 + 33
Simplifying, we have:
5y = 45
Dividing both sides by 5:
y = 9
Now substitute the value of y into either of the original equations. Let's use the second equation:
-x + 9 = 11
Adding x to both sides:
9 + x = 11
Subtracting 9 from both sides:
x = 2
Therefore, the solution to the system of linear equations is x = 2 and y = 9.
The coordinates of the solution are (2, 9).
To solve the system of linear equations using substitution or elimination, we'll go through each method step-by-step.
Method 1: Substitution:
Step 1: Solve one equation for one variable.
Let's solve the second equation for y:
-x + y = 11
y = x + 11
Step 2: Substitute the expression for y back into the other equation:
3x + 2y = 12
Replace y with x + 11:
3x + 2(x + 11) = 12
Step 3: Simplify and solve for x:
3x + 2x + 22 = 12
5x + 22 = 12
5x = 12 - 22
5x = -10
x = -10/5
x = -2
Step 4: Substitute the value of x back into one of the original equations to solve for y:
-x + y = 11
Substitute x = -2:
-(-2) + y = 11
2 + y = 11
y = 11 - 2
y = 9
Therefore, the solution to the system of linear equations is x = -2 and y = 9.
Method 2: Elimination:
Step 1: Multiply the second equation by 3 to make the coefficients of x in both equations the same:
3(-x + y) = 3(11)
-3x + 3y = 33
Step 2: Add the equations together to eliminate the variable x:
(3x + 2y) + (-3x + 3y) = 12 + 33
3x - 3x + 2y + 3y = 45
5y = 45
y = 45/5
y = 9
Step 3: Substitute the value of y back into one of the original equation to solve for x:
3x + 2y = 12
3x + 2(9) = 12
3x + 18 = 12
3x = 12 - 18
3x = -6
x = -6/3
x = -2
So, the solution to the system of linear equations is x = -2 and y = 9.