Given the graphs of f(x) = 3x − 3 and g(x) = −x + 5, which statements are true?
Responses
A The point of intersection is the solution.The point of intersection is the solution.
B The system has two solutions.The system has two solutions.
C Any point on either line is a solution.Any point on either line is a solution.
D The system has infinitely many solutions.The system has infinitely many solutions.
E Point (2, 3) is the only solution.
A - The point of intersection is the solution.
To determine which statements are true, we can find the point of intersection between the two graphs, which corresponds to the solutions of the system.
1. Set the two functions equal to each other:
3x - 3 = -x + 5
2. Rearrange the equation:
3x + x = 5 + 3
4x = 8
3. Solve for x:
x = 8/4
x = 2
4. Substitute the value of x back into one of the original equations to find the corresponding y-value:
f(x) = 3(2) - 3
f(x) = 6 - 3
f(x) = 3
Therefore, the point of intersection between the two graphs is (2, 3).
Now, let's evaluate each statement:
A. The point of intersection is the solution. (True) - The point (2, 3) is the solution to the system of equations.
B. The system has two solutions. (False) - The system of equations only has one solution, which is (2, 3).
C. Any point on either line is a solution. (False) - Only the point of intersection (2, 3) is a solution to both equations. Points on each individual line are not solutions to the other equation.
D. The system has infinitely many solutions. (False) - The system of equations only has one solution, which is (2, 3).
E. Point (2, 3) is the only solution. (True) - Point (2, 3) is indeed the only solution to the system of equations.
Therefore, the true statements are A (The point of intersection is the solution) and E (Point (2, 3) is the only solution).
To determine which statements are true, we need to analyze the graphs of the given functions, f(x) = 3x - 3 and g(x) = -x + 5.
1. First, let's plot the graphs of the functions on a coordinate plane.
The graph of f(x) = 3x - 3 is a straight line with a slope of 3 and y-intercept of -3. It will pass through the point (0, -3) and will continue in a positive direction from there.
The graph of g(x) = -x + 5 is also a straight line, but with a slope of -1 and a y-intercept of 5. It will pass through the point (0, 5) and slope downward from there.
2. Now, let's look for the point(s) of intersection between the two graphs.
For these particular equations, we can find the point of intersection by setting f(x) equal to g(x) and solving for x.
3x - 3 = -x + 5
Adding x to both sides, we get:
4x - 3 = 5
Adding 3 to both sides, we have:
4x = 8
Dividing both sides by 4, we find:
x = 2
Now, substitute this value of x back into either of the equations to find the y-coordinate of the point of intersection:
f(2) = 3(2) - 3 = 6 - 3 = 3
Therefore, the point of intersection is (2, 3).
3. Analyzing the statements:
A. The point of intersection is the solution.
✓ True. The point (2, 3) is the solution to the system of equations.
B. The system has two solutions.
✖ False. The system only has one solution, which is the point (2, 3).
C. Any point on either line is a solution.
✖ False. Not every point on either line will satisfy both equations simultaneously. Only the point (2, 3) satisfies both f(x) and g(x).
D. The system has infinitely many solutions.
✖ False. The system has only one solution, not infinitely many.
E. Point (2, 3) is the only solution.
✓ True. The point (2, 3) is indeed the only solution to the system of equations.
Therefore, the true statements are A and E.