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.
• (2, -4) is a solution to Equation B
• (2, -4) is a solution to neither Equation
• The y-intercept of Equation B is (0, 12)
• The solution to the system of equations is (5, 8)
• the slope of equation A is m=-3
To determine the truth of each statement, we can substitute the values given into each equation and check if the equation holds true.
1. (2, -4) is a solution to Equation B:
- Substituting x = 2 and y = -4 into Equation B: 4(2) - (-4) = 12
- This simplifies to 8 + 4 = 12, which is true.
- Therefore, (2, -4) is a solution to Equation B.
2. (2, -4) is a solution to neither Equation:
- We have already shown that (2, -4) is a solution to Equation B.
- However, we need to check if it is a solution to Equation A: 3(2) + (-4) = 6 - 4 = 2, which is not equal to 23.
- Therefore, (2, -4) is not a solution to Equation A.
- The statement " (2, -4) is a solution to neither Equation" is false.
3. 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 the equation 4x - y = 12.
- Substituting x = 0 into the equation, we get 4(0) - y = 12, which simplifies to -y = 12, or y = -12.
- The y-interceptor is (0, -12), not (0, 12).
- Therefore, the statement "The y-intercept of Equation B is (0, 12)" is false.
4. The solution to the system of equations is (5, 8):
- To check if (5, 8) is a solution to the system of equations, we substitute x = 5 and y = 8 into both Equation A and Equation B.
- For Equation A: 3(5) + 8 = 15 + 8 = 23, which is true.
- For Equation B: 4(5) - 8 = 20 - 8 = 12, which is also true.
- Therefore, (5, 8) is a solution to the system of equations.
5. The slope of equation A is m = -3:
- The equation is given in the form y = mx + b, where m is the slope.
- By rearranging Equation A, we can isolate y: 3x + y = 23 -> y = -3x + 23.
- Comparing this equation to the standard form, we can see that the slope is -3.
- Therefore, the statement "the slope of equation A is m = -3" is true.
In summary, the statements that are true are:
- (2, -4) is a solution to Equation B.
- The solution to the system of equations is (5, 8).
- The slope of equation A is m = -3.