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)
Responses
The y-intercept of Equation B is (0,12)
The y-intercept of Equation B is open paren 0 comma 12 close paren
The solution to the system of equations is (5,8)
The solution to the system of equations is open paren 5 comma 8 close paren
(2,−4)
is a solution to Equation B
open paren 2 comma negative 4 close paren is a solution to Equation B
The slope of Equation A is m = −3
The slope of Equation A is m = negative 3
(2,−4)
is a solution to neither Equation
To determine which statements are true, we need to solve the system of equations A and B.
Equation A: 3x + y = 23
Equation B: 4x - y = 12
To solve the system of equations, we can use either substitution or elimination method. Let's use the elimination method.
Adding these two equations together will cancel out the y term:
(3x + y) + (4x - y) = 23 + 12
7x = 35
x = 5
Now we substitute this value of x back into either equation A or B to find the value of y. Let's use equation A:
3(5) + y = 23
15 + y = 23
y = 8
So the solution to the system of equations is (5, 8).
Now let's evaluate each statement:
1. The y-intercept of Equation B is (0,12).
To find the y-intercept of Equation B, we need to set x = 0 and solve for y in equation B.
4(0) - y = 12
-y = 12
y = -12
The y-intercept of Equation B is (0, -12), not (0, 12).
Therefore, this statement is false.
2. The solution to the system of equations is (5, 8).
We found earlier that the solution to the system of equations is indeed (5, 8).
Therefore, this statement is true.
3. (2,-4) is a solution to Equation B.
Let's substitute these values into equation B and see if it holds true:
4(2) - (-4) = 12
8 + 4 = 12
12 = 12
The equality holds true, so (2, -4) is a solution to Equation B.
Therefore, this statement is true.
4. The slope of Equation A is m = -3.
In equation A, rearrange it to slope-intercept form (y = mx + b) to determine the slope:
3x + y = 23
y = -3x + 23
The slope (m) is -3, so this statement is true.
Overall, the true statements are:
- The solution to the system of equations is (5, 8).
- (2, -4) is a solution to Equation B.
- The slope of Equation A is m = -3.