Math Problem Statement

Solve the nonlinear system of equations for real solutions. left brace Start 2 By 1 Matrix 1st Row 1st Column y equals x squared minus 8 2nd Row 1st Column y equals x squared minus 8 x EndMatrix

Solution

We are solving the system of nonlinear equations:

y = x^2 - 8, \\ y = x^2 - 8x. \end{cases}$$ ### Step 1: Set the two equations equal to each other Since both equations are equal to $$y$$, we can set them equal to each other: $$x^2 - 8 = x^2 - 8x.$$ ### Step 2: Simplify the equation Cancel out $$x^2$$ from both sides: $$-8 = -8x.$$ Divide through by $$-8$$: $$x = 1.$$ ### Step 3: Solve for $$y$$ Substitute $$x = 1$$ into either equation to find $$y$$. Using $$y = x^2 - 8$$: $$y = 1^2 - 8 = -7.$$ ### Step 4: Verify the solution Substitute $$x = 1$$ and $$y = -7$$ into both equations: 1. $$y = x^2 - 8 \quad \implies -7 = 1^2 - 8 \quad \text{(True)}$$, 2. $$y = x^2 - 8x \quad \implies -7 = 1^2 - 8(1) \quad \text{(True)}$$. Thus, the solution is: $$\boxed{(x, y) = (1, -7)}$$ Would you like to explore the reasoning further or solve a similar problem? Here are some related questions: 1. What happens when the system includes more complicated nonlinear terms? 2. How do solutions change if constants in the equations are altered? 3. Can such systems have more than one solution? Why or why not? 4. How can graphing these equations help visualize solutions? 5. What are other methods to solve nonlinear systems of equations? **Tip:** When solving nonlinear systems, always check for extraneous solutions by substituting back into the original equations.

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Math Problem Analysis

Mathematical Concepts

Nonlinear Systems of Equations
Algebra
Quadratic Equations

Formulas

Setting equations equal: If y = f(x) and y = g(x), then f(x) = g(x)
Basic quadratic form: ax^2 + bx + c = 0

Theorems

Properties of Equality
Simplification Techniques for Equations

Suitable Grade Level

Grades 9-11

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