Solve the system of equations
-6x + 2y =-32
4x + 3y = -9
what a load of rubbish.
-6x + 2y = -32
4x + 3y = -9
-18x + 6y = -96
8x + 6y = -18
26x = 78
x = 3
so y = -7
Apologies for the mistake in the previous response. You're correct. Let's solve the system of equations correctly:
Starting with the given equations:
-6x + 2y = -32
4x + 3y = -9
We can eliminate the y variable by multiplying the first equation by 3 and the second equation by 2:
-18x + 6y = -96
8x + 6y = -18
Next, we can add the two equations to eliminate the y variable:
(-18x + 6y) + (8x + 6y) = -96 + (-18)
-18x + 8x + 6y + 6y = -96 - 18
-10x + 12y = -114
Simplifying further, we get:
2x + 12y = -114
Let's multiply the second equation by 2 to match the coefficients of x:
2(4x + 3y) = 2(-9)
8x + 6y = -18
Now, we can subtract this equation from the previous equation to eliminate x:
(2x + 12y) - (8x + 6y) = -114 - (-18)
2x + 12y - 8x - 6y = -114 + 18
-6x + 6y = -96
Further simplification yields:
-x + y = -16
Now, we have the system of equations:
-6x + 6y = -96
-x + y = -16
We can solve the second equation for x by subtracting y from both sides:
-x = -16 - y
x = 16 + y
Substituting this value of x into the first equation:
-6(16 + y) + 6y = -96
-96 - 6y + 6y = -96
-96 = -96
The equation -96 = -96 is always true, which means that the system of equations is dependent and has infinitely many solutions. This implies that for any value of y, we can find a corresponding value of x that satisfies both equations.
However, if we assume y = -7 as you mentioned, we can solve for x:
x = 16 + y
x = 16 + (-7)
x = 9
Therefore, the solution to the system of equations is x = 9 and y = -7.
To solve the system of equations, we can use either the substitution method or the elimination method. Let's solve it using the elimination method:
Step 1: Multiply the first equation by 2 and the second equation by 3 to make the coefficients of x in both equations equal:
-6x + 2y = -32 (multiply by 2)
4x + 3y = -9 (multiply by 3)
This gives us:
-12x + 4y = -64
12x + 9y = -27
Step 2: Now, add the two equations together to eliminate x:
(-12x + 4y) + (12x + 9y) = -64 + (-27)
This simplifies to:
13y = -91
Step 3: Divide both sides of the equation by 13 to solve for y:
y = -91/13
Step 4: Substitute the value of y back into one of the original equations to solve for x. Let's use the first equation:
-6x + 2(-91/13) = -32
Multiply both sides by 13:
-78x - 182 = -416
Add 182 to both sides:
-78x = -234
Divide both sides by -78:
x = 3
Step 5: The solution to the system of equations is x = 3 and y = -91/13.
To solve the system of equations, we can use the method of elimination or substitution.
Let's use the method of elimination:
First, we need to multiply the first equation by 2 to match the coefficients of y:
-6x + 2y = -32
2(-6x + 2y) = 2(-32)
-12x + 4y = -64
Now, we can add this modified equation to the second equation to eliminate the x variable:
-12x + 4y = -64
+ 4x + 3y = -9
_________________
-8x + 7y = -73
Now we can solve for y by multiplying the second equation by 8 and the first equation by 7 to eliminate the x variable again:
8(4x + 3y) = 8(-9)
32x + 24y = -72
7(-12x + 4y) = 7(-64)
-84x + 28y = -448
Now we can add these two equations to eliminate the x variable:
32x + 24y = -72
+ (-84x + 28y) = -448
_______________________
-52x + 52y = -520
Next, we can multiply the initial second equation by 6 to match the x coefficients:
6(4x + 3y) = 6(-9)
24x + 18y = -54
And now we can add this equation to the second equation to eliminate the y variable:
-52x + 52y = -520
+ (24x + 18y) = -54
_______________________
-28x - 30 = -574
To solve for x, we can divide both sides of the equation by -28:
(-28x - 30)/(-28) = (-574)/(-28)
x + (15/14) = 41/2
x = 41/2 - 15/14
x = (287 - 15)/14
x = 272/14
x = 136/7
Now that we have the value of x, we can substitute it into one of the original equations to solve for y. Let's use the first equation:
-6x + 2y = -32
-6(136/7) + 2y = -32
-816/7 + 2y = -32
2y = -32 + 816/7
2y = (-32 * 7 + 816)/7
2y = (224 - 32 + 816)/7
2y = 1008/7
y = (1008/7)/2
y = (1008/7)/(14/7)
y = 1008/14
y = 72/1
y = 72
Therefore, the solution to the system of equations is x = 136/7 and y = 72.