Problem 65 - A Graphical Approach to College Algebra

Question

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} x^{2}+y^{2} &=10 \\ 2 x^{2}-y^{2} &=17 \end{aligned}$$

Step-by-Step Solution

Answer

The real solutions are \((3, 1)\), \((3, -1)\), \((-3, 1)\), and \((-3, -1)\).

Step 1: Identify the System

We are given two equations: 1. \( x^2 + y^2 = 10 \) 2. \( 2x^2 - y^2 = 17 \). We need to solve these equations simultaneously to find all real solutions for \(x\) and \(y\).

Step 2: Express One Equation in Terms of Another

From the first equation, \( x^2 + y^2 = 10 \), isolate \( y^2 \) to obtain:\[ y^2 = 10 - x^2. \]

Step 3: Substitute to Solve for One Variable

Substitute \( y^2 = 10 - x^2 \) from Step 2 into the second equation: \[ 2x^2 - (10 - x^2) = 17. \]Simplify the equation to:\[ 2x^2 - 10 + x^2 = 17. \]Combine terms to get:\[ 3x^2 - 10 = 17. \]

Step 4: Solve for \(x^2\)

Add 10 to both sides to give:\[ 3x^2 = 27 \]Divide both sides by 3:\[ x^2 = 9. \]

Step 5: Solve for \(x\)

Take the square root of both sides to find \(x\):\[ x = 3 \quad \text{or} \quad x = -3. \]

Step 6: Substitute Back to Find \(y\)

Substitute \( x = 3 \) and \( x = -3 \) back into the equation \( y^2 = 10 - x^2 \).For \( x = 3 \): \[ y^2 = 10 - 9 \quad \Rightarrow \quad y^2 = 1 \quad \Rightarrow \quad y = 1 \quad \text{or} \quad y = -1. \]For \( x = -3 \): The same operation gives:\[ y^2 = 10 - 9 \quad \Rightarrow \quad y^2 = 1 \quad \Rightarrow \quad y = 1 \quad \text{or} \quad y = -1. \]

Step 7: List All Solutions

We combine the \( x \) and \( y \) values to give the solutions for the system: 1. \( (3, 1) \) 2. \( (3, -1) \) 3. \( (-3, 1) \) 4. \( (-3, -1) \).

Key Concepts

Analytical Solution

An analytical solution refers to a way of solving mathematical problems using algebraic expressions and known formulas, rather than numerical or graphical means. In solving nonlinear systems of equations, like the example provided, the main goal is to find exact values for the variables.

The steps employed involved:

• Identifying the equations as a system, noting that each equation contains the same set of variables.
• Using algebraic methods to manipulate the equations, such as substitution or elimination, to express one variable in terms of another.
• Simplifying to solve for each variable individually, ensuring all possible solutions are accounted for.

It's important to follow each step carefully and verify each solution by substituting back into the original equations. This extensive process allows us to obtain precise and dependable answers.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, and they appear frequently in algebra. In the context of the given exercise, the equations involve terms like \( x^2 \) and \( y^2 \), which are characteristic of quadratics.

A basic quadratic equation looks like \( ax^2 + bx + c = 0 \). Here, 'a', 'b', and 'c' are constants. Solving these equations typically involves:

• Factoring the polynomial expression, if possible.
• Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find solutions when the quadratic cannot be factored easily.
• Completing the square, another method less often used but effective in certain conditions.

In our exercise, we specifically dealt with \( x^2 = 9 \), which simplifies to \( x = \pm 3 \). This is a typical scenario for solving quadratics where only the squared terms are involved, leading to straightforward solutions.

Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables, often requiring a solution for all variables involved at the same time. These can be linear or nonlinear based on the degree of the equations.

When dealing with nonlinear systems, like in our exercise, where one equation is quadratic, we often use methods such as:

• Substitution, as shown in the steps, where you solve one equation for a variable and substitute into another.
• Elimination, which involves adding or subtracting equations to eliminate a variable.
• Graphical methods, though less precise, involve plotting each equation to find points of intersection, giving a visual solution.

The goal with simultaneous equations is always consistency in variables and satisfying both original equations. The resultant solutions must make sense in the context of all given equations, validating their correctness.