use elimination to solve the system of equations: 1) 3x-3y=-6 2)-5x+6y=12
Several ways to do this, as usual
notice that the first can be simplified
3x - 3y = -6
x - y = -2
x = y-2
now use substitution into the 2nd
-5x + 6y = 12
-5(y-2) + 6y = 12
-5y + 10 + 6y = 12
..
solve for y, then sub that into x = y-2 to get your x
works out easily
x-y=-2
-5x+6y=12
To solve the system of equations using the elimination method, we need to eliminate one of the variables by adding or subtracting the equations.
Let's start by multiplying the first equation by 2 to make the coefficients of y opposite:
1) 2(3x - 3y) = 2(-6)
6x - 6y = -12
Now we have:
6x - 6y = -12 (equation 1)
-5x + 6y = 12 (equation 2)
Adding the two equations together:
(6x - 6y) + (-5x + 6y) = -12 + 12
x - 0y = 0
Simplifying:
x = 0
Now, we substitute the value of x into one of the original equations to find y. Let's use equation 1:
3x - 3y = -6
3(0) - 3y = -6
-3y = -6
y = -6 / -3
y = 2
So, the solution to the system of equations is x = 0 and y = 2.
To solve the system of equations using elimination, follow these steps:
Step 1: Multiply each equation by a constant so that the coefficients of one of the variables in both equations are the same (but with opposite signs). In this case, we can multiply the first equation by 2 and the second equation by 3, which will make the coefficients of y in both equations opposite each other:
Equation 1: 2(3x - 3y) = 2*(-6) => 6x - 6y = -12
Equation 2: 3(-5x + 6y) = 3*12 => -15x + 18y = 36
Step 2: Add the two modified equations together to eliminate one variable (in this case, y). The goal is to obtain an equation with only one variable.
(6x - 6y) + (-15x + 18y) = -12 + 36
Simplify the equation:
6x - 6y - 15x + 18y = 24
Combine like terms:
-9x + 12y = 24
Step 3: Solve the resulting equation for one variable (either x or y). Let's solve for x:
-9x = -12y + 24
Dividing both sides by -9:
x = (12y - 24) / -9
Simplify:
x = -4y + 8/3
Step 4: Substitute the value of x obtained in step 3 into either of the original equations. Let's substitute it into Equation 1:
3(-4y + 8/3) - 3y = -6
Multiply through by 3 to eliminate fractions:
-12y + 8 - 9y = -18
Combine like terms:
-21y = -26
Divide both sides by -21:
y = -26 / -21
Simplify:
y = 26/21
Step 5: Substitute the value of y back into the expression for x obtained in step 3:
x = -4(26/21) + 8/3
Simplify:
x = -104/21 + 56/21
Combine like terms:
x = -48/21
Simplify further (if necessary):
x = -16/7
So the solution to the system of equations is x = -16/7 and y = 26/21.