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Chapter 5: Problem 71

Use substitution to solve the system for the set of ordered triples \((x, y, \lambda)\) that satisfy the system. $$ \begin{array}{l} 2=2 \lambda x \\ 6=2 \lambda y \\ x^{2}+y^{2}=10 \end{array} $$

Short Answer

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Step by step solution

01

Solve for \(\lambda\) in terms of \(x\)

From the first equation, solve for \(\lambda\): \(2 = 2\lambda x\). Divide both sides by \(2x\) to get \(\lambda = \frac{1}{x}\).

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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. It involves solving one of the equations for one variable and then substituting that expression into the other equations. This reduces the number of variables and makes the system easier to solve.
In the provided exercise, you start by solving the first equation for \(\backslash lambda\) in terms of \(x\). Once you have \(\lambda = \frac{1}{x}\), you can substitute it into the remaining equations.
By doing this, the equations transform and you can simplify or solve further to find the values of all variables.
ordered triples
Ordered triples are sets of three numbers that represent coordinates in a three-dimensional space. In this exercise, the ordered triples \((x, y, \backslash lambda)\) represent the solution to the system of equations.
Finding an ordered triple means you are determining the exact values of \(x\), \(y\), and \(\lambda\) that satisfy all equations simultaneously.
This can involve multiple steps of substitution and solving to ensure that each variable fits perfectly within the constraints given by the algebraic equations.
algebraic equations
Algebraic equations are mathematical statements where two expressions are set equal to each other. In this exercise, you have three such equations: \(2 = 2 \lambda x\), \(6 = 2 \lambda y\), and \(x^2 + y^2 = 10\).
These equations link the variables in specific ways. For instance, the first equation \(2 = 2 \lambda x\) can be rearranged to solve for \(\backslash lambda\) as \(\lambda = \frac{1}{x}\).
Similarly, you can manipulate the second and third equations to find expressions or numerical solutions for the other variables.
system of equations
A system of equations consists of multiple equations that share common variables. Solving a system of equations means finding values of the variables that satisfy all the equations simultaneously.
In this exercise, the system is made up of three equations. The goal is to find the values of \((x, y, \backslash lambda)\) that work in all of them.
Using methods like substitution helps simplify and solve such systems. It often involves combining equations to reduce the number of variables step by step until all values are determined.

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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. $$ \begin{array}{l} x^{2}+y^{2}=40 \\ y=-x^{2}+8.5 \end{array} $$

Solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples \(5-6\) ) $$ \begin{array}{r} 3 x+y=6 \\ x+\frac{1}{3} y=2 \end{array} $$

Solve the system using any method. $$ \begin{array}{l} 3 x-10 y=1900 \\ 5 y+800=x \end{array} $$

To protect soil from erosion, some farmers plant winter cover crops such as winter wheat and rye. In addition to conserving soil, cover crops often increase crop yields in the row crops that follow in spring and summer. Suppose that a farmer has 800 acres of land and plans to plant winter wheat and rye. The input cost for 1 acre for each crop is given in the table along with the cost for machinery and labor. The profit for 1 acre of each crop is given in the last column. $$ \begin{array}{|l|c|c|c|} \hline & \begin{array}{c} \text { Input Cost } \\ \text { per Acre } \end{array} & \begin{array}{c} \text { Labor/Machinery } \\ \text { Cost per Acre } \end{array} & \begin{array}{c} \text { Profit } \\ \text { per Acre } \end{array} \\ \hline \text { Wheat } & \$ 90 & \$ 50 & \$ 42 \\ \hline \text { Rye } & \$ 120 & \$ 40 & \$ 35 \\ \hline \end{array} $$ Suppose the farmer has budgeted a maximum of $$\$ 90,000$$ for input costs and a maximum of $$\$ 36,000$$ for labor and machinery. a. Determine the number of acres of each crop that the farmer should plant to maximize profit. (Assume that all crops will be sold.) b. What is the maximum profit? c. If the profit per acre for wheat were $$\$ 40$$ and the profit per acre for rye were $$\$ 45$$, how many acres of each crop should be planted to maximize profit?

A system of equations is given in which each equation is written in slope- intercept form. Determine the number of solutions. If the system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. $$ \begin{array}{l} y=\frac{1}{2} x+3 \\ y=2 x+\frac{1}{3} \end{array} $$
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