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Chapter 5: Problem 19

Solve the system of equations. If a system does not have one unique solution, determine the number of solutions to the system. $$ \begin{array}{l} 2 x+5 z=2 \\ 3 y-7 z=9 \\ -5 x+9 y=22 \end{array} $$

Short Answer

Expert verified
The system has a unique solution: \((x, y, z) = (1, 3, 0)\).

Step by step solution

01

Analyze the given system of equations

The given system is: \(2x + 5z = 2\) \(3y - 7z = 9\) \(-5x + 9y = 22\).Notice that there are three equations with three variables: \(x\), \(y\), and \(z\).
02

Isolate one variable

From the first equation, solve for \(x\):\(2x = 2 - 5z\) \(x = 1 - \frac{5z}{2}\).
03

Substitute \(x\) into the third equation

Substitute \(x = 1 - \frac{5z}{2}\) into \(-5x + 9y = 22\):\(-5(1 - \frac{5z}{2}) + 9y = 22\) \(-5 + \frac{25z}{2} + 9y = 22\) \(9y = 22 + 5 - \frac{25z}{2}\) \(9y = 27 - \frac{25z}{2}\).
04

Solve for \(y\)

Isolate \(y\) from the equation obtained in Step 3:\(9y = 27 - \frac{25z}{2}\) \(y = 3 - \frac{25z}{18}\).
05

Check for consistency with the second equation

Substitute \(y = 3 - \frac{25z}{18}\) into the second original equation: \(3(3 - \frac{25z}{18}) - 7z = 9\) \(9 - \frac{75z}{18} - 7z = 9\) \(9 - \frac{75z + 126z}{18} = 9\) \(9 - \frac{201z}{18} = 9\) \(9 - \frac{67z}{6} = 9\).
06

Analyze the result for consistency

Simplify the equation in Step 5: \(\frac{67z}{6} = 0\) \(z = 0\).We find that \(z = 0\) must hold for the system to be consistent.
07

Solve the system using \(z = 0\)

Substitute \(z = 0\) back into the original equations:For \(2x + 5z = 2\): \(2x = 2\) \(x = 1\).For \(3y - 7z = 9\): \(3y = 9\) \(y = 3\).
08

Conclusion

With \(z = 0\), we found that \(x = 1\) and \(y = 3\). So, the solution to the system is \((x, y, z) = (1, 3, 0)\). This indicates a unique solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
A system of linear equations involves multiple equations where each equation represents a line in a multi-dimensional space. In this problem, we have three linear equations: \(2x + 5z = 2\), \(3y - 7z = 9\) and \(-5x + 9y = 22\). Each equation involves the variables \(x\), \(y\), and \(z\). We're looking to find values for these variables that satisfy all three equations simultaneously. Linear equations are foundational in algebra and are key to solving more complex problems in higher mathematics.
Variable Isolation
Isolating a variable means solving one of the equations for a single variable. This helps simplify the problem and allows you to substitute that expression into other equations. For instance, in this solution, we isolate \(x\) from the first equation:
  • \(2x = 2 - 5z\)
  • \(x = 1 - \frac{5z}{2}\)
By expressing \(x\) in terms of \(z\), we can substitute this back into the other equations, making it easier to solve for the remaining variables. By isolating variables step by step, you reduce complexity and systematically approach the solution.
Consistency in Systems
A consistent system of linear equations has at least one set of values satisfying all equations. To test for consistency, we substitute isolated variables back into the original equations and simplify. In our problem, substituting \(y = 3 - \frac{25z}{18}\) into the second equation helps us check if there are common solutions:
  • \(3(3 - \frac{25z}{18}) - 7z = 9\)
  • Simplifying gives \(\frac{67z}{6} = 0\), which means \(z = 0\).
This consistency shows that when \(z = 0\), there are specific solutions for \(x\) and \(y\) that satisfy all equations, which leads us to the unique solution.
Unique Solution in Algebra
When solving systems of linear equations, finding a unique solution means there is exactly one set of values for the variables that satisfy all equations. In our solved system, after finding \(z = 0\), we substitute it back into the equations:
  • For \(2x + 5z = 2\): \(2x = 2\) which simplifies to \(x = 1\).
  • For \(3y - 7z = 9\): \(3y = 9\) which simplifies to \(y = 3\).
Hence, the unique solution for the system is \((x, y, z) = (1, 3, 0)\). This means, these values are the only ones that can satisfy all the given equations simultaneously. The existence of a unique solution indicates that the equations intersect at exactly one point in three-dimensional space.

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Most popular questions from this chapter

Use a graphing utility to approximate the solution(s) to the system of equations. Round the coordinates to 3 decimal places. $$ \begin{array}{l} x^{2}+y^{2}=32 \\ y=0.8 x^{2}-9.2 \end{array} $$

Solve the system of equations by using the addition method. (See Examples \(3-4)\) $$ \begin{array}{l} 3 x-7 y=1 \\ 6 x+5 y=-17 \end{array} $$

A sporting goods store sells two types of exercise bikes. The deluxe model costs the store $$\$ 540$$ from the manufacturer and the standard model costs the store $$\$ 420$$ from the manufacturer. The profit that the store makes on the deluxe model is $$\$ 180$$ and the profit on the standard model is $$\$ 120$$. The monthly demand for exercise bikes is at most \(30 .\) Furthermore, the store manager does not want to spend more than $$\$ 14,040$$ on inventory for exercise bikes. a. Determine the number of deluxe models and the number of standard models that the store should have in its inventory each month to maximize profit. (Assume that all exercise bikes in inventory are sold.) b. What is the maximum profit? c. If the profit on the deluxe bikes were $$\$ 150$$ and the profit on the standard bikes remained the same, how many of each should the store have to maximize profit?

Describe the solution set to the system of inequalities. \(x \geq 0, y \geq 0, x \leq 1, y \leq 1\)

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