Given the system of equations:
6x + 2y = −6
3x − 4y = −18
Solve for (x, y) using elimination.
Wrong answer by the bot
6x + 2y = −6 -----> double this: 12x + 4y = -12
3x − 4y = −18
add them:
15x = -30
x = -2
sub into first:
6(-2) + 2y = -6
2y = 6
y = 3
so x= -2 and y = 3
You are correct, thank you for catching that error. The solution is indeed (x, y) = (-2, 3).
To solve the system of equations using elimination, follow these steps:
Step 1: Multiply one or both equations by appropriate constants so that the coefficients of either x or y will cancel out when the equations are added or subtracted.
Let's eliminate the y term by multiplying the first equation by 2 and the second equation by -1:
Equation 1: 6x + 2y = -6 (multiply by 2)
2(6x + 2y) = 2(-6)
12x + 4y = -12 (Equation 1 after multiplying by 2)
Equation 2: 3x - 4y = -18 (multiply by -1)
-1(3x - 4y) = -1(-18)
-3x + 4y = 18 (Equation 2 after multiplying by -1)
Step 2: Add the equations together to eliminate the y term.
(12x + 4y) + (-3x + 4y) = (-12) + 18
Combine like terms:
12x - 3x + 4y + 4y = -12 + 18
Simplify:
9x + 8y = 6 (Equation 3)
Step 3: Solve Equation 3 for x or y.
Let's solve Equation 3 for x:
9x + 8y = 6
Subtract 8y from both sides:
9x + 8y - 8y = 6 - 8y
Simplify:
9x = 6 - 8y
Divide both sides by 9:
x = (6 - 8y) / 9 (Equation 4)
Step 4: Substitute the value of x from Equation 4 into either Equation 1 or Equation 2 to solve for y.
Let's substitute the value of x from Equation 4 into Equation 1:
6x + 2y = -6
Replace x with (6 - 8y) / 9:
6((6 - 8y) / 9) + 2y = -6
Multiply each term by 9 to eliminate the fraction:
6(6 - 8y) + 18y = -54
Distribute:
36 - 48y + 18y = -54
Combine like terms:
-30y + 36 = -54
Subtract 36 from both sides:
-30y = -90
Divide both sides by -30:
y = 3
Step 5: Substitute the value of y into Equation 4 to solve for x.
x = (6 - 8y) / 9
Replace y with 3:
x = (6 - 8(3)) / 9
Simplify:
x = (6 - 24) / 9
x = -18 / 9
Simplify:
x = -2
Therefore, the solution to the system of equations is x = -2 and y = 3.
Multiplying the second equation by 2, we get:
6x - 8y = -36
Adding this equation to the first equation, we eliminate y:
6x + 2y + 6x - 8y = -6 - 36
Simplifying:
12x = -42
Dividing by 12:
x = -7/2
Substituting this value into either of the original equations (we'll use the first one):
6(-7/2) + 2y = -6
Simplifying:
-21 + 2y = -6
Adding 21 to both sides:
2y = 15
Dividing by 2:
y = 7.5
Therefore, the solution is (x, y) = (-7/2, 7.5).