Problem 80 - A Graphical Approach to Precalculus with Limits

Question

Use substitution to solve the nonlinear system of equations in three variables. Note that solutions are ondered triples. $$\begin{aligned} &x^{2}+y^{2}+z^{2}=4\\\ &x+y+z=2\\\ &x-y \quad=0 \end{aligned}$$

Step-by-Step Solution

Answer

The solutions are \((0,0,2)\) and \(\left(\frac{4}{3}, \frac{4}{3}, -\frac{2}{3}\right)\).

Step 1: Analyze System of Equations

We are given three equations: \[ x^2 + y^2 + z^2 = 4 \] \[ x + y + z = 2 \] \[ x - y = 0 \] The goal is to find the values of \(x\), \(y\), and \(z\) that satisfy all three equations.

Step 2: Use the Third Equation

The third equation, \( x - y = 0 \), implies that \( x = y \). We will substitute \( y \) with \( x \) in the remaining equations.

Step 3: Substitute into Second Equation

Substituting \( y = x \) into the second equation \( x + y + z = 2 \), we get:\[ x + x + z = 2 \] Thus, \[ 2x + z = 2 \].

Step 4: Solve for z in Terms of x

From \( 2x + z = 2 \), we can express \( z \) as follows:\[ z = 2 - 2x \].

Step 5: Substitute into First Equation

Substitute \( y = x \) and \( z = 2 - 2x \) into the first equation:\[ x^2 + x^2 + (2 - 2x)^2 = 4 \]. Simplify to obtain:\[ 2x^2 + 4 - 8x + 4x^2 = 4 \].

Step 6: Simplify and Solve Quadratic Equation

Simplify the equation:\[ 6x^2 - 8x = 0 \].Factor out common terms:\[ 2x(3x - 4) = 0 \].Solving gives \( x = 0 \) or \( 3x = 4 \), which means \( x = \frac{4}{3} \).

Step 7: Find y and z for Each x

For \( x = 0 \):- \( y = x = 0 \)- \( z = 2 - 2\times0 = 2 \).Ordered triple is \((0,0,2)\).For \( x = \frac{4}{3} \):- \( y = x = \frac{4}{3} \)- \( z = 2 - 2\times\frac{4}{3} = -\frac{2}{3} \).Ordered triple is \(\left(\frac{4}{3}, \frac{4}{3}, -\frac{2}{3}\right)\).

Key Concepts

Substitution Method

The substitution method is a powerful tool for solving systems of equations, especially useful when dealing with nonlinear systems. In this approach, you replace one variable in one equation with its equivalent expression from another equation. This reduces the number of variables and simplifies the system, making it easier to solve.
Let's break it down further:

• First, identify an equation where one variable is easily solvable.
• Once identified, solve for that variable in terms of others.
• Substitute this expression into the remaining equations.

In our example, we start with the simplest equation, which was the third equation, \( x - y = 0 \). Solving this gives us \( x = y \). By substituting \( y \) with \( x \) in the other equations, we can continue simplifying until the system becomes easier to solve.

Quadratic Equations

Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). They are fundamental in algebra because they often arise in various mathematical contexts, including when solving nonlinear systems.
In our exercise, after substituting and simplifying, we ended up with a quadratic: \( 6x^2 - 8x = 0 \). To solve quadratic equations, we can:

• Factor the equation if possible, or
• Use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

For our specific need, we factored directly, which resulted in \( 2x(3x - 4) = 0 \). This gave solutions \( x = 0 \) and \( x = \frac{4}{3} \).
Factoring is often faster when applicable, providing solutions quickly by setting each factor to zero and solving for the variables.

Ordered Triples

An ordered triple is a way to express the solution to a system of equations with three variables, typically written in the form \((x, y, z)\). The order is crucial as it indicates the exact values of each variable.
When solving systems with three variables, you may find more than one solution. These solutions are represented as multiple ordered triples.
In our problem, we ended up with two ordered triples, \((0, 0, 2)\) and \(\left(\frac{4}{3}, \frac{4}{3}, -\frac{2}{3}\right)\), each representing a distinct set of values that satisfy all equations in the system.
Ordered triples allow us to express complex relationships in a straightforward manner. They are essential when describing geometrical interpretations of solutions in three-dimensional space.