How many solutions does the system of equations have?
y=−2x+4
y=2x
A. One solution
B. Infinitely many solutions
C. No solution
The Correct Answer and Explanation is:
To determine the number of solutions for the system of equations given by:
- (y = -2x + 4)
- (y = 2x)
we can analyze the equations graphically or algebraically.
Graphical Interpretation
- First Equation: The first equation (y = -2x + 4) is in slope-intercept form ((y = mx + b)), where the slope ((m)) is (-2) and the y-intercept ((b)) is (4). This means the line descends steeply from left to right, crossing the y-axis at (y = 4).
- Second Equation: The second equation (y = 2x) also follows the slope-intercept form, with a slope of (2) and a y-intercept of (0). This line ascends from left to right, crossing the origin (0,0).
Algebraic Method
To find the intersection point (solution) of these two lines, we set the equations equal to each other:
[
-2x + 4 = 2x
]
Next, we solve for (x):
[
4 = 2x + 2x
]
[
4 = 4x
]
[
x = 1
]
Now that we have (x = 1), we can substitute this value back into either original equation to find (y). Using the second equation:
[
y = 2(1) = 2
]
Thus, the point of intersection is ((1, 2)). This means there is exactly one point where these two lines meet.
Conclusion
Since the two lines have different slopes (the first has a slope of (-2) while the second has a slope of (2)), they are not parallel and will intersect at exactly one point. Therefore, the system of equations has one solution.
The correct answer is A. One solution.