Solve the system of equations.
−6x+2y=−32
4x+3y=−9 (1 point)
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To solve the system of equations, we can use the method of substitution or elimination.
Let's use the method of substitution:
1) Solve the first equation for x in terms of y:
-6x + 2y = -32
-6x = -2y - 32
x = (2y + 32)/-6
x = (-y/3) - (16/3) (equation 2)
2) Substitute the expression for x in the second equation:
4x + 3y = -9
4((-y/3) - (16/3)) + 3y = -9
(-4y/3) - (64/3) + 3y = -9
Multiply through by 3 to remove the fraction:
-4y - 64 + 9y = -27
5y - 64 = -27
5y = 37
y = 37/5
3) Substitute the value of y back into equation 2 to solve for x:
x = (-y/3) - (16/3)
x = (-(37/5)/3) - (16/3)
x = (-37/5)*(1/3) - (16/3)
x = -37/15 - 16/3
x = (-37/15) - (16/3)
x = (-37/15) - (80/15)
x = -117/15
x = -39/5
Therefore, the solution to the system of equations is (x, y) = (-39/5, 37/5).
To solve the system of equations:
1. Begin by labeling the two equations as Equation 1 and Equation 2:
Equation 1: -6x + 2y = -32
Equation 2: 4x + 3y = -9
2. Choose a method to solve the system of equations. The two common methods are substitution and elimination. Let's use the elimination method.
3. Multiply Equation 1 by 2 to eliminate the y term:
2(-6x + 2y) = 2(-32)
-12x + 4y = -64
4. Now, we can add Equation 2 and the modified Equation 1 to eliminate the x term:
(4x + 3y) + (-12x + 4y) = -9 + (-64)
4x - 12x + 3y + 4y = -9 - 64
-8x + 7y = -73
5. Now, we have a new equation: -8x + 7y = -73.
6. Next, we'll solve Equation 2: 4x + 3y = -9.
7. Multiply Equation 2 by 2 to eliminate the x term:
2(4x + 3y) = 2(-9)
8x + 6y = -18
8. Now, we can add Equation 1 and the modified Equation 2 to eliminate the x term:
(-6x + 2y) + (8x + 6y) = -32 + (-18)
-6x + 8x + 2y + 6y = -32 - 18
2x + 8y = -50
9. Simplify the equation: 2x + 8y = -50.
10. Now, we have a new equation: 2x + 8y = -50.
11. We now have two new equations: -8x + 7y = -73 and 2x + 8y = -50.
12. We can now solve these two equations simultaneously. We'll use the elimination method again.
13. Multiply the second equation by 4 to eliminate the x term:
4(2x + 8y) = 4(-50)
8x + 32y = -200
14. Add the first equation (-8x + 7y = -73) and the modified second equation (8x + 32y = -200):
(-8x + 7y) + (8x + 32y) = -73 + (-200)
-8x + 8x + 7y + 32y = -73 - 200
39y = -273
15. Solve for y by dividing both sides of the equation by 39:
39y / 39 = -273 / 39
y = -7
16. Now, substitute the value of y (-7) back into one of the original equations (Equation 1 or Equation 2) to solve for x. Let's use Equation 1:
-6x + 2y = -32
-6x + 2(-7) = -32
-6x - 14 = -32
-6x = -18
x = 3
17. Therefore, the solution to the system of equations is x = 3 and y = -7.