To determine which statements are true, we can solve the system of equations:
Equation A: 3x + y = 23
Equation B: 4x - y = 12
To find the solution, we can use the method of substitution or elimination. Let's use the elimination method:
Multiplying Equation A by 4:
4(3x + y) = 4(23)
12x + 4y = 92
Now we can subtract Equation B from this equation:
(12x + 4y) - (4x - y) = 92 - 12
This simplifies to:
12x + 4y - 4x + y = 80
8x + 5y = 80
Now we have a new equation:
8x + 5y = 80
Equation B: 4x - y = 12
Solving this system of equations, we can use the substitution method or elimination method. Let's use the elimination method again:
Multiplying Equation B by 5:
5(4x - y) = 5(12)
20x - 5y = 60
Now we can subtract Equation A from this equation:
(20x - 5y) - (8x + 5y) = 60 - 80
This simplifies to:
20x - 5y - 8x - 5y = -20
12x - 10y = -20
6x - 5y = -10
Now we have a new equation:
6x - 5y = -10
Equation B: 4x - y = 12
From Equation B, we can isolate y:
y = 4x - 12
Substituting this into the new equation:
6x - 5(4x - 12) = -10
Simplifying this equation leads to:
6x - 20x + 60 = -10
-14x + 60 = -10
-14x = -70
x = 5
Substituting x = 5 into Equation B:
4(5) - y = 12
20 - y = 12
-y = 12 - 20
-y = -8
y = 8
The solution to the system of equations is (5, 8). Let's now check which statements are true:
(2, -4) is a solution to Equation B:
Plugging in x = 2 and y = -4 into Equation B:
4(2) - (-4) = 12
8 + 4 = 12
12 = 12
This statement is false.
The y-intercept of Equation B is (0, 12):
To find the y-intercept, we set x = 0 in Equation B:
4(0) - y = 12
0 - y = 12
-y = 12
y = -12
The y-intercept of Equation B is (0, -12), not (0, 12). So, this statement is false.
The slope of Equation A is m = -3:
Equation A is in the form y = mx + b, where m is the slope. Comparing Equation A to this form:
3x + y = 23
y = -3x + 23
The slope of Equation A is indeed -3. So, this statement is true.
The solution to the system of equations is (5, 8):
We have already determined that (5, 8) is the solution to the system of equations. So, this statement is true.
(2, -4) is a solution to neither Equation A or B:
We have already determined that (2, -4) is not a solution to Equation B. Let's also check if it is a solution to Equation A:
3(2) + (-4) = 6 - 4 = 2
2 ≠ 23
Therefore, (2, -4) is not a solution to Equation A as well. So, this statement is true.
To summarize, the true statements are:
- The slope of Equation A is m = -3
- The solution to the system of equations is (5, 8)
- (2, -4) is a solution to neither Equation A or B.
Equation A: 3x+y=233x+y=23
Equation B: 4x−y=124x−y=12 Use the system of equations to determine which of the following statements are true. Select all that apply. (3 points) Responses (2,−4)(2,−4) is a solution to Equation B open paren 2 comma negative 4 close paren is a solution to Equation B The y-intercept of Equation B is (0,12)(0,12)The y-intercept of Equation B is open paren 0 comma 12 close paren The slope of Equation A is m = −3−3The slope of Equation A is m = negative 3 The solution to the system of equations is (5,8)(5,8)The solution to the system of equations is open paren 5 comma 8 close paren (2,−4)(2,−4) is a solution to neither Equation open paren 2 comma negative 4 close paren is a solution to neither Equation
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