solve by addition method: please show work.
-3x + y = 3
2x -3y =7
To solve this system of equations using the addition method, we need to eliminate one variable by adding or subtracting the equations. Let's start by analyzing the equations:
Equation 1: -3x + y = 3 (Equation 1)
Equation 2: 2x - 3y = 7 (Equation 2)
Step 1: Multiply equation 1 by 2 and equation 2 by -3 to make the coefficients of x match up when we add the equations.
2 * (-3x + y) = 2 * 3
-3 * (2x - 3y) = -3 * 7
This gives us:
-6x + 2y = 6 (Equation 3)
-6x + 9y = -21 (Equation 4)
Step 2: Now we can add equation 3 and equation 4 to eliminate the x variable:
(-6x + 2y) + (-6x + 9y) = 6 + (-21)
Simplifying the equation gives us:
-12x + 11y = -15 (Equation 5)
Step 3: Now we have a new equation with only y, so we can solve for y:
-12x + 11y = -15
Move the -12x to the other side of the equation:
11y = -12x - 15
Divide both sides of the equation by 11:
y = (-12/11)x - (15/11) (Equation 6)
Step 4: Finally, substitute equation 6 back into either equation 1 or equation 2 to solve for x. Let's use equation 1:
-3x + y = 3
Substitute y = (-12/11)x - (15/11):
-3x + (-12/11)x - (15/11) = 3
Now, solve for x:
(-33/11)x + (-12/11)x = 3 + (15/11)
(-45/11)x = (33 + 15)/11
(-45/11)x = 48/11
x = (48/11) * (-11/45)
x = -16/15
Step 5: We have found the value of x (-16/15). Now substitute this value back into equation 1 or equation 2 to find y. Let's use equation 1:
-3x + y = 3
Substitute x = -16/15:
-3(-16/15) + y = 3
(48/15) + y = 3
(16/5) + y = 3
y = 3 - (16/5)
y = (15/5) - (16/5)
y = -1/5
Therefore, the solution to the system of equations is x = -16/15 and y = -1/5.