In the following equations: Equation A: 3x + y = 23 Equation B: 4x - y =12
Use the system of equations to determine which of the following statement are true. Select all that apply.
A, (2, -4) is a solution to Equation B
B. The solution to the system of equation is (5, 8)
C. The y=intercept of Equation B is (0, 12)
D. (2, -4) is a solution to neither
E. The slope of Equation A is m= -3
To determine which statements are true, we need to solve the system of equations.
First, let's solve the system of equations by using the method of substitution:
We have Equation A: 3x + y = 23
And Equation B: 4x - y = 12
From Equation A, we can rewrite it as y = 23 - 3x
Substituting y in Equation B with 23 - 3x, we get:
4x - (23 - 3x) = 12
4x - 23 + 3x = 12
7x - 23 = 12
7x = 35
x = 5
Now, we can substitute x = 5 into Equation A to find y:
3(5) + y = 23
15 + y = 23
y = 8
So the solution to the system of equations is (5, 8).
Now, let's analyze each statement:
A) (2, -4) is a solution to Equation B.
To check if this is true, substitute x = 2 and y = -4 into Equation B:
4(2) - (-4) = 12
8 + 4 = 12
12 = 12
This is true, so statement A is correct.
B) The solution to the system of equations is (5, 8).
We have already found that the solution to the system of equations is indeed (5, 8). So statement B is correct.
C) The y-intercept of Equation B is (0, 12).
To find the y-intercept of Equation B, set x = 0:
4(0) - y = 12
-y = 12
y = -12
So the y-intercept of Equation B is (0, -12), not (0, 12). Therefore, statement C is incorrect.
D) (2, -4) is a solution to neither.
We have already determined that (2, -4) is a solution to Equation B. So statement D is incorrect.
E) The slope of Equation A is m = -3.
To find the slope of Equation A, rearrange it into slope-intercept form (y = mx + b):
3x + y = 23
y = -3x + 23
The slope of Equation A is -3, so statement E is correct.
In conclusion, the correct statements are:
A) (2, -4) is a solution to Equation B.
B) The solution to the system of equations is (5, 8).
E) The slope of Equation A is m = -3.