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.
2x² = 3y + 23
y = 2x - 5

Verified step by step guidance

1. Identify the system of equations: \$2x^2 = 3y + 23\( and \)y = 2x - 5$.
2. Substitute the \(y\) value from the second equation into the first equation to verify the points: replace \(y\) with \$2x - 5\( in \)2x^2 = 3y + 23\( to get \)2x^2 = 3(2x - 5) + 23$.
3. Check the point \((0, -1)\) by substituting \(x = 0\) and \(y = -1\) into both equations: verify if \$2(0)^2 = 3(-1) + 23\( and if \)-1 = 2(0) - 5$ hold true.
4. Check the point \((3, 6)\) by substituting \(x = 3\) and \(y = 6\) into both equations: verify if \$2(3)^2 = 3(6) + 23\( and if \)6 = 2(3) - 5$ hold true.
5. If both points satisfy both equations, then they are solutions to the system, confirming the points of intersection on the graph.

Key Concepts

System of Nonlinear Equations

A system of nonlinear equations consists of two or more equations involving variables where at least one equation is nonlinear, such as quadratic. Solutions are points where the graphs of the equations intersect, satisfying all equations simultaneously.

Substitution Method

The substitution method involves replacing one variable with an expression from another equation to verify if a point satisfies both equations. This method is useful for checking if given points are solutions to the system by direct substitution.

Graphical Interpretation of Solutions

The points of intersection on the graph represent solutions to the system of equations. Each intersection point's coordinates satisfy both equations, providing a visual confirmation of the solutions.