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Magazine Chapter 8: Problem 57
Solve each system. $$\begin{aligned} x y &=z^{2} \\ x+y+z &=7 \\ x^{2}+y^{2}+z^{2} &=133 \end{aligned}$$
Short Answer
Expert verified
The solutions are \(x=2, y=3, z=2\) and \(x=3, y=2, z=2\).
Step by step solution
01
Analyze the given system of equations
The system consists of three equations: 1. \(xy = z^2\)2. \(x + y + z = 7\)3. \(x^2 + y^2 + z^2 = 133\). The goal is to find the values of x, y, and z that satisfy all three equations simultaneously.
02
Express one variable in terms of others using Equation 2
From Equation 2, express y in terms of x and z:\(y = 7 - x - z\)
03
Substitute y in Equations 1 and 3
Substitute \(y = 7 - x - z\) into Equations 1 and 3.For Equation 1: \(x(7 - x - z) = z^2\)For Equation 3: \(x^2 + (7 - x - z)^2 + z^2 = 133\)
04
Simplify and solve the equations
Substitute and simplify Equation 1: \(x(7 - x - z) = z^2\) leads to \(7x - x^2 - xz = z^2\).Substitute and simplify Equation 3: \((7 - x - z)^2 = 133 - x^2 - z^2\). Upon expanding and simplifying, solve the resulting quadratic equations.
05
Solve the resulting system for x and z
Using solved simplified equations from Step 4, solve for values of x and z. Check solutions by substituting back into original equations to confirm they satisfy all three equations.
06
Find corresponding y values
For each pair of x and z solutions, find corresponding y value using \(y = 7 - x - z\). Ensure these values satisfy all original equations.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
substitution method
The substitution method is a powerful tool for solving systems of equations. Here's how it works: you start by solving one of the equations for one variable and then substitute that expression into the other equations. In our exercise, we started by isolating y from Equation 2:
\( y = 7 - x - z \).
This replaces y in Equations 1 and 3, turning them into equations with just x and z. Once substituted, these equations become easier to manage. Simplifying them helps to find solutions step-by-step.
\( y = 7 - x - z \).
This replaces y in Equations 1 and 3, turning them into equations with just x and z. Once substituted, these equations become easier to manage. Simplifying them helps to find solutions step-by-step.
- First, isolate one variable.
- Substitute it into the other equations.
- Simplify and solve.
quadratic equations
Quadratic equations are polynomial equations of degree 2, often appearing as \( ax^2 + bx + c = 0 \). They are a core part of solving nonlinear systems. In our exercise, after substitution, we encountered quadratic equations such as
\( x(7 - x - z) = z^2 \), which simplifies to
\( x^2 + xz - 7x + z^2 = 0 \).
To solve these quadratics, use methods like factoring, completing the square, or the quadratic formula
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Key steps include:
\( x(7 - x - z) = z^2 \), which simplifies to
\( x^2 + xz - 7x + z^2 = 0 \).
To solve these quadratics, use methods like factoring, completing the square, or the quadratic formula
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Key steps include:
- Identifying coefficients a, b, and c.
- Applying the quadratic formula or other methods.
- Checking all potential solutions in the original equations.
polynomial equations
Polynomial equations involve variables raised to powers, with more than one term. They play a critical role in solving nonlinear systems like the given set.
For our exercise, simplifying the equations led to polynomial forms, e.g., \( (7 - x - z)^2 = 133 - x^2 - z^2 \).
To solve polynomial equations:
For our exercise, simplifying the equations led to polynomial forms, e.g., \( (7 - x - z)^2 = 133 - x^2 - z^2 \).
To solve polynomial equations:
- Combine like terms.
- Simplify the expressions.
- Use algebraic identities such as
\( (a + b)^2 = a^2 + 2ab + b^2 \) to expand and simplify.
nonlinear systems
Nonlinear systems consist of at least one equation that is not a straight line. These include terms like \( x^2 \), \( y^2 \), and products of variables such as \( xy = z^2 \). Solving such a system involves:
- Recognizing the type of equations involved.
- Using methods like substitution or elimination.
- Simplifying step-by-step to find possible solutions.
