Question

1. Solve the system of two linear equations.

-3x - 4y = -23
3x - 5y = 5

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve this system of equations, we can use the method of elimination.

Step 1: Multiply the second equation by 4 to get:
12x - 20y = 20

Step 2: Add this new equation to the first equation to eliminate x:
-3x - 4y + 12x - 20y = -23 + 20
9x - 24y = -3

Step 3: Divide the resulting equation by 3 to simplify:
3x - 8y = -1

Now, we have a new equation:
3x - 8y = -1

Now we have two equations with two variables:
-3x - 4y = -23
3x - 8y = -1

Step 4: Add the two equations to eliminate x:
-3x - 4y + 3x - 8y = -23 - 1
-12y = -24

Step 5: Divide by -12 to solve for y:
y = 2

Step 6: Substitute y = 2 into the first equation to solve for x:
-3x - 4(2) = -23
-3x - 8 = -23
-3x = -15
x = 5

Therefore, the solution to the system of equations is x = 5, y = 2.

Answer 2

3. Solve the system of two equations using elimination with multiplication.

4x - 6y = -48
-2x + 4y = 30

Answer 3

[AI-generated answer — Bot GPT 3.5]

To solve this system of equations using the elimination method, we will first multiply both equations by suitable constants to create opposite coefficients for either x or y.

Given equations:
4x - 6y = -48
-2x + 4y = 30

We can multiply the first equation by 2 and the second equation by 3 to make the coefficients of y equal and opposite:

2(4x - 6y) = 2(-48)
3(-2x + 4y) = 3(30)

This gives us:
8x - 12y = -96
-6x + 12y = 90

Now, add the two equations together to eliminate y:
8x - 12y + (-6x + 12y) = -96 + 90
2x = -6
x = -3

Substitute x = -3 into one of the original equations to solve for y:
4(-3) - 6y = -48
-12 - 6y = -48
-6y = -36
y = 6

Therefore, the solution to the system of equations is x = -3, y = 6.

Answer 4

4. Solve the system of two equations.

3x - 4y = 40
5x - 5y = 55

Answer 5

[AI-generated answer — Bot GPT 3.5]

To solve this system of equations, we can use the method of substitution.

Given equations:
3x - 4y = 40
5x - 5y = 55

Step 1: Rearrange the second equation in terms of x:
5x - 5y = 55
5x = 5y + 55
x = y + 11

Step 2: Substitute x from the second equation into the first equation:
3(y + 11) - 4y = 40
3y + 33 - 4y = 40
-y + 33 = 40
-y = 7
y = -7

Step 3: Substitute y = -7 back into x = y + 11 to solve for x:
x = -7 + 11
x = 4

Therefore, the solution to the system of equations is x = 4, y = -7.