Verify that the points of intersection specified on the graph of each nonlinear system are solutions of the system by substituting directly into both equations.
y = 3x²
x² + y² = 10

Verified step by step guidance

1. Step 1: Identify the system of equations and the points to verify. The system is given by \(y = 3x^2\) and \(x^2 + y^2 = 10\). The points to check are \((-3, 2)\) and \((3, 2)\).
2. Step 2: Substitute the point \((-3, 2)\) into the first equation \(y = 3x^2\). Calculate \(3(-3)^2\) and check if it equals \(y = 2\).
3. Step 3: Substitute the point \((-3, 2)\) into the second equation \(x^2 + y^2 = 10\). Calculate \((-3)^2 + 2^2\) and check if it equals 10.
4. Step 4: Repeat the substitution process for the point \((3, 2)\). First, substitute into \(y = 3x^2\) by calculating \$3(3)^2\( and check if it equals \)y = 2$.
5. Step 5: Substitute \((3, 2)\) into \(x^2 + y^2 = 10\) by calculating \((3)^2 + 2^2\) and check if it equals 10. If both points satisfy both equations, they are solutions to the system.

Key Concepts

Nonlinear Systems of Equations

A nonlinear system consists of two or more equations involving variables raised to powers other than one or multiplied together. Solutions are points where the graphs of the equations intersect. Verifying solutions involves substituting the points into each equation to check if they satisfy both.

Substitution Method

The substitution method involves replacing one variable with an equivalent expression from another equation. This allows checking if a given point satisfies both equations by direct substitution, confirming whether the point lies on both curves.

Graph Interpretation of Intersection Points

Intersection points on a graph represent solutions common to both equations in a system. Identifying these points visually helps in understanding where the equations meet, and these points can be tested algebraically to verify their validity as solutions.