How many solutions does the system have? y=−2x+4
y=x^2−2x+4
Enter your answer in the box.
The correct Answer and Explanation is:
To determine how many solutions the system of equations has, we need to solve the system:
- First equation: ( y = -2x + 4 )
- Second equation: ( y = x^2 – 2x + 4 )
Since both equations are set equal to ( y ), we can set them equal to each other to find the points where they intersect (i.e., the values of ( x ) where the two equations have the same ( y )-value):
[
-2x + 4 = x^2 – 2x + 4
]
Now, simplify the equation by subtracting ( (-2x + 4) ) from both sides:
[
0 = x^2
]
This simplifies to:
[
x^2 = 0
]
Now, solve for ( x ):
[
x = 0
]
So, ( x = 0 ) is the only solution for ( x ). To find the corresponding value of ( y ), substitute ( x = 0 ) into either equation. We’ll use the first equation:
[
y = -2(0) + 4 = 4
]
Thus, the solution is ( (0, 4) ).
How many solutions?
Since we found only one value of ( x ) that satisfies both equations, the system has one solution, which is ( (0, 4) ).
Explanation:
This system consists of a linear equation ( y = -2x + 4 ), which represents a straight line, and a quadratic equation ( y = x^2 – 2x + 4 ), which represents a parabola. The number of solutions to this system corresponds to the number of intersection points between the line and the parabola.
After solving, we found that the equations intersect at exactly one point, ( (0, 4) ), which means the system has one solution. The quadratic equation forms a parabola that touches the line at one point, which is known as a “tangent” intersection. Therefore, the system has exactly one solution.