Question

try it!

1. how many solutions does the system of equations \\(\

$$\begin{cases} y = x \\\\ y = x^2 \\end{cases}$$

\\) have? explain. use the tool to support your answer.

Explanation:

Equate the two functions

To find the intersection points, set the equations equal to each other.
Using the Graphing Quadratic Functions knowledge point

$$ x^2 = x $$

Solve the quadratic equation

Rearrange the equation to solve for \(x\).

$$ x^2 - x = 0 $$

$$ x(x - 1) = 0 $$

$$ x = 0 \quad \text{or} \quad x = 1 $$

Determine the corresponding y-values

Substitute the \(x\)-values back into either original equation to find the \(y\)-coordinates.
For \(x = 0\):

$$ y = 0 $$

For \(x = 1\):

$$ y = 1 $$

The intersection points are \((0, 0)\) and \((1, 1)\).

Interpret the graphical representation

The linear function \(y = x\) and the quadratic function \(y = x^2\) intersect at exactly two distinct points on the coordinate plane. Therefore, the system has two real solutions.

By setting the two equations equal to each other, we get \(x^2 = x\), which simplifies to \(x(x - 1) = 0\). This yields two \(x\)-values: \(x = 0\) and \(x = 1\). Substituting these back into the equations gives the intersection points \((0, 0)\) and \((1, 1)\). Graphically, the line \(y = x\) intersects the parabola \(y = x^2\) at exactly these two points, confirming there are 2 solutions.
</answer>

Answer:

The system of equations has 2 solutions.

Explanation:
By setting the two equations equal to each other, we get \(x^2 = x\), which simplifies to \(x(x - 1) = 0\). This yields two \(x\)-values: \(x = 0\) and \(x = 1\). Substituting these back into the equations gives the intersection points \((0, 0)\) and \((1, 1)\). Graphically, the line \(y = x\) intersects the parabola \(y = x^2\) at exactly these two points, confirming there are 2 solutions.