Question

Equation A: 3x + y = 23

Equation B: 4x - y = 12

Use the system of equations to determine which of the following statements are true. Select all that apply.

(3 points)

The y-intercept of Equation B is (0, 12)

The slope of Equation A is m = - 3

(2, - 4) is a solution to Equation B

(2, - 4) is a solution to neither Equation

The solution to the system of equations is (5, 8)

Answer 1

To determine which of the statements are true, we can solve the system of equations:

Equation A: 3x + y = 23
Equation B: 4x - y = 12

First, let's solve for y in Equation A:
3x + y = 23
y = 23 - 3x

Now, let's substitute y in Equation B with this expression:
4x - (23 - 3x) = 12
4x - 23 + 3x = 12
7x - 23 = 12
7x = 35
x = 5

Substituting the value of x back into y in Equation A:
y = 23 - 3(5)
y = 23 - 15
y = 8

So, the solution to the system of equations is (5, 8).

Now, let's evaluate each statement:

The y-intercept of Equation B is (0, 12):
To find the y-intercept of Equation B, we set x = 0 in Equation B:
4(0) - y = 12
-y = 12
y = -12

The statement is false.

The slope of Equation A is m = - 3:
The slope of Equation A is the coefficient of x, which is 3.

The statement is false.

(2, - 4) is a solution to Equation B:
Let's substitute x = 2 and y = -4 into Equation B:
4(2) - (-4) = 12
8 + 4 = 12
12 = 12

The statement is true.

(2, - 4) is a solution to neither Equation:
We have already shown that (2, -4) is a solution to Equation B.

The statement is false.

The solution to the system of equations is (5, 8):
We have already determined that the solution to the system of equations is indeed (5, 8).

The statement is true.

Therefore, the true statements are:
- (2, -4) is a solution to Equation B
- The solution to the system of equations is (5, 8)