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Chapter 5: Problem 10

Use back-substitution to solve the system of linear equations. $$\left\\{\begin{aligned} 4 x-2 y+z &=8 \\\\-y+z &=4 \\ z &=2 \end{aligned}\right.$$

Short Answer

Expert verified
The solution to the system of equations is \(x = 3\), \(y = -2\), \(z = 2\).

Step by step solution

01

Solve for Z

The third equation of the system is \(z = 2\), which directly gives us the value of \(z\).
02

Substitute Z into the second equation

Now, take the value of \(z\) and substitute it into the second equation: \(-y + z = 4\). This gives \(-y + 2 = 4\), which when simplified provides the value of \(y\).
03

Substitute Y and Z into the first equation

Substitute the value of \(y\) and \(z\) into the first equation: \(4x - 2y + z = 8\). Solve the equation to find the value of \(x\).

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Key Concepts

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

Back-Substitution
Back-substitution is like solving a puzzle by starting from the last piece and working your way back to the first one. This method is commonly used when solving a system of equations that has been arranged in a staircase or triangular form. In this exercise, back-substitution begins from the equation that has only the variable \(z\), making it easy to find its value right away.

**Steps in Back-Substitution:**
  • Solve for the simplest variable: Begin with the equation that makes it easiest to find a variable. In the given system, it's the third equation, \(z = 2\), which directly provides the value of \(z\).
  • Use the found values in other equations: Next, use the known value of \(z\) in the second equation to solve for \(y\). This results in \(-y + 2 = 4\), simplifying to find \(y\).
  • Continue substituting upward: Finally, substitute the values of \(y\) and \(z\) into the first equation to solve for \(x\). This is how back-substitution works upward from the last equation to find all the variables.
Each step builds on the previous one, making it simple to solve the entire system! That鈥檚 the beauty of back-substitution.
System of Equations
A system of equations is a collection of two or more equations with the same set of unknowns. Solving a system involves finding values for each variable that satisfy all the equations simultaneously.

In this exercise, we have a system of three linear equations:
  • \(4x - 2y + z = 8\)
  • \(-y + z = 4\)
  • \(z = 2\)
A system of equations can tell us a lot depending on the situation:
  • **Unique Solution:** One set of values for the variables that makes all equations true, as seen here using back-substitution.
  • **No Solution:** The equations are contradictory and cannot be satisfied by any set of values.
  • **Infinite Solutions:** There are endless solutions, often when the equations are multiples of each other.
Recognizing the type of system helps decide the appropriate method of solving it.
Solving Equations
Solving equations involves finding the value of unknown variables that make the equation true, a fundamental aspect of mathematics. In this problem, equations are solved sequentially using a special technique.

**Process of Solving Equations in the System:**
  • Isolate variables: The method starts by isolating the easiest variable first, here \(z = 2\).
  • Substitute back: Using the value of \(z\), substitute it into the equations above it to solve for other variables. This systematic approach reduces complexity progressively.
  • Solve linearly: Once substitutions are made, it usually simplifies to basic arithmetic, like finding integer values that satisfy the equalities. For instance, finding \(y\) and \(x\) by substitution.
With this layered way, solving equations in systems becomes manageable and logical, breaking down complex problems into smaller, tractable parts.

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Most popular questions from this chapter

Optimal Revenue An accounting firm charges $$\$ 2500$$ for an audit and $$\$ 350$$ for a tax return. Research and available resources have indicated the following constraints. \- The firm has 900 hours of staff time available each week. \- The firm has 155 hours of review time available each week. \- Each audit requires 75 hours of staff time and 10 hours of review time. \- Each tax return requires \(12.5\) hours of staff time and \(2.5\) hours of review time. What numbers of audits and tax returns will bring in an optimal revenue?

Objective function: $$ z=4 x+5 y $$ Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+y & \geq 8 \\ 3 x+5 y & \geq 30 \end{aligned} $$

MAKE A DECISION: DIET SUPPLEMENT A dietitian designs a special diet supplement using two different foods. Each ounce of food \(\mathrm{X}\) contains 12 units of calcium, 10 units of iron, and 20 units of vitamin \(\mathrm{B}\). Each ounce of food \(\mathrm{Y}\) contains 15 units of calcium, 20 units of iron, and 12 units of vitamin B. The minimum daily requirements for the diet are 300 units of calcium, 280 units of iron, and 300 units of vitamin \(\mathrm{B}\). (a) Find a system of inequalities describing the different amounts of food \(\mathrm{X}\) and food \(\mathrm{Y}\) that the dietitian can use in the diet. (b) Sketch the graph of the system. (c) A nutritionist normally gives a patient 10 ounces of food \(\mathrm{X}\) and 12 ounces of food \(\mathrm{Y}\) per day. Supplies of food \(\mathrm{Y}\) are running low. What other combinations of foods \(\mathrm{X}\) and \(\mathrm{Y}\) can be given to the patient to meet the minimum daily requirements?

Graphical Reasoning Two concentric circles have radii \(x\) and \(y\), where \(y>x .\) The area between the circles must be at least 10 square units. (a) Find a system of inequalities describing the constraints on the circles. (b) Use a graphing utility to graph the system of inequalities in part (a). Graph the line \(y=x\) in the same viewing window. (c) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in the context of the problem.

Optimal Profit A company makes two models of doghouses. The times (in hours) required for assembling, painting, and packaging are shown in the table. $$ \begin{array}{|l|c|c|} \hline \text { Process } & \text { Model A } & \text { Model B } \\ \hline \text { Assembling } & 2.5 & 3 \\ \hline \text { Painting } & 2 & 1 \\ \hline \text { Packaging } & 0.75 & 1.25 \\ \hline \end{array} $$ The total times available for assembling, painting, and packaging are 4000 hours, 2500 hours, and 1500 hours, respectively. The profits per unit are $$\$ 60$$ for model \(\mathrm{A}\) and $$\$ 75$$ for model \(\mathrm{B}\). What is the optimal production level for each model? What is the optimal profit?
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