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)
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)