Questions: Solve the following system by graphing. [ leftbeginarrayl y=-x^2+4 x+1 y=4 x+1 endarrayright. ] Use the graphing tool to graph the system. Click to enlarge graph

Solve the following system by graphing.
[
leftbeginarrayl
y=-x^2+4 x+1
y=4 x+1
endarrayright.
]

Use the graphing tool to graph the system.
Click to enlarge graph

Transcript text: Solve the following system by graphing. \[ \left\{\begin{array}{l} y=-x^{2}+4 x+1 \\ y=4 x+1 \end{array}\right. \] Use the graphing tool to graph the system. Click to enlarge graph

Solution

Solution Steps

Step 1: Identify the Equations

The system of equations given is:

$ y = -x^2 + 4x + 1 $
$ y = 4x + 1 $

Step 2: Set the Equations Equal to Each Other

To find the points of intersection, set the equations equal to each other: \[ -x^2 + 4x + 1 = 4x + 1 \]

Step 3: Simplify the Equation

Subtract $4x + 1$ from both sides: \[ -x^2 + 4x + 1 - 4x - 1 = 0 \] \[ -x^2 = 0 \]

Step 4: Solve for $x$

Solving the equation: \[ x^2 = 0 \implies x = 0 \]

Step 5: Solve for $y$

Substitute $x = 0$ into one of the original equations, for example, $y = 4x + 1$: \[ y = 4(0) + 1 = 1 \]

Final Answer

The solution to the system is the point of intersection: $(0, 1)$.

{"axisType": 3, "coordSystem": {"xmin": -5, "xmax": 5, "ymin": -5, "ymax": 5}, "commands": ["y = -x**2 + 4_x + 1", "y = 4_x + 1"], "latex_expressions": ["$y = -x^2 + 4x + 1$", "$y = 4x + 1$"]}

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