2x-3y=-2
-2y+3x=12
~Explanation too, please!
I really need help!
Please and thanks!
Rewrite your equations as:
6x -9y = -6
6x -4y = 24
Note that I multipled each of your equations by a constant to get the same coefficients for x.
Then subtract the lower equation from the upper.
-5 y = -30
y = 6
2x - 18 = -2
2x = 16
x = 8
Thank you so much. :P
To solve the system of equations:
2x - 3y = -2 ........(equation 1)
-2y + 3x = 12 ........(equation 2)
We can use the method of substitution or elimination to find the solution.
Method 1: Substitution
Step 1: Solve one equation for one variable (in terms of the other variable).
Let's solve equation 1 for x:
2x - 3y = -2
2x = 3y - 2
x = (3y - 2) / 2
Step 2: Substitute the expression for x in the other equation.
Substitute (3y - 2) / 2 for x in equation 2:
-2y + 3((3y - 2) / 2) = 12
-2y + (9y - 6) / 2 = 12
Multiply through by 2 to eliminate the denominator:
-4y + 9y - 6 = 24
Combine like terms:
5y - 6 = 24
Add 6 to both sides:
5y = 30
Divide by 5:
y = 6
Step 3: Substitute the value of y back into one of the original equations to solve for x.
Substitute y = 6 into equation 1:
2x - 3(6) = -2
2x - 18 = -2
Add 18 to both sides:
2x = 16
Divide by 2:
x = 8
Therefore, the solution to the system of equations is x = 8, y = 6.
Method 2: Elimination
Step 1: Multiply one or both equations by a constant such that the coefficients of either x or y in one equation will be equal to (or multiples of) the coefficients in the other equation.
Let's multiply equation 1 by 2 and equation 2 by 3 to eliminate the x-terms:
(2)(2x - 3y) = (2)(-2)
(3)(-2y + 3x) = (3)(12)
Which becomes:
4x - 6y = -4 ........(equation 3)
-6y + 9x = 36 ........(equation 4)
Step 2: Add equation 3 and equation 4 to eliminate the y-terms.
(4x - 6y) + (-6y + 9x) = -4 + 36
4x - 6y - 6y + 9x = 32
Combine like terms:
13x - 12y = 32 ........(equation 5)
Step 3: Solve equation 5 for x or y.
Let's solve equation 5 for x:
13x = 12y + 32
x = (12y + 32) / 13
Step 4: Substitute the expression for x in either equation 1 or equation 2 to solve for y.
Substitute (12y + 32) / 13 for x in equation 1:
2((12y + 32) / 13) - 3y = -2
Multiply through by 13 to eliminate the denominator:
24y + 64 - 39y = -26
Combine like terms:
-15y + 64 = -26
Subtract 64 from both sides:
-15y = -90
Divide by -15:
y = 6
Step 5: Substitute the value of y back into equation 1 or equation 2 to solve for x.
Substitute y = 6 into equation 1:
2x - 3(6) = -2
2x - 18 = -2
Add 18 to both sides:
2x = 16
Divide by 2:
x = 8
Therefore, the solution to the system of equations is x = 8, y = 6.
Both methods yield the same solution.