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Chapter 6: Problem 64

Solve each system. Write solutions in terms of \(z\) if necessary. $$\begin{aligned} 3 x+6 y-3 z &=12 \\ -x-2 y+z &=16 \\ x+y-2 z &=20 \end{aligned}$$

Short Answer

Expert verified
The solutions in terms of \(z\) are: \(x = 36 + 3z\), \(y = -16 - z\), \(z = z\).

Step by step solution

01

Write the System of Equations

We start by clearly identifying the given system of equations: 1. \(3x + 6y - 3z = 12\) 2. \(-x - 2y + z = 16\) 3. \(x + y - 2z = 20\)
02

Simplify the Equations

Notice that the first equation \(3x + 6y - 3z = 12\) can be simplified by dividing every term by 3, resulting in: \(x + 2y - z = 4\).
03

Express System in Terms of x and y

With the simplified first equation, we now have: 1. \(x + 2y - z = 4\) 2. \(-x - 2y + z = 16\) 3. \(x + y - 2z = 20\). Add equations 1 and 2 to eliminate \(z\): \((x + 2y - z) + (-x - 2y + z) = 4+16\), which simplifies to \(0 = 20\). This equation is true regardless of \(x\), \(y\), and \(z\), indicating dependence and an infinite number of solutions.
04

Solve for x and y in Terms of z

Substitute \(z = z\) into the simplified first equation: \(x + 2y - z = 4\) rewritten as \(x + 2y = z + 4\). Then rearrange equation 3 in terms of \(x\): \(x = 20 - y + 2z\).
05

Substitute and Solve for x, y in Terms of z

Using the rearranged equation 3, substitute \(x = 20 - y + 2z\) into \(x + 2y = z + 4\): \((20 - y + 2z) + 2y = z + 4\), which yields \(20 + y + 2z = z + 4\). Solve for \(y\): \(y = -16 - z\).
06

Express Final Solutions in Terms of z

Now substitute the expression for \(y\) back into \(x = 20 - y + 2z\): \(x = 20 + 16 + z + 2z\), simplifying this gives \(x = 36 + 3z\). Thus, the solutions are \(x = 36 + 3z\), \(y = -16 - z\), and \(z = z\).

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Key Concepts

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

Linear Equations
Linear equations form the foundation of solving more complex systems. In simple terms, a linear equation is an equation involving two variables that creates a straight line when graphed. Consider the equations given: each represents a line in a three-dimensional space, where variables such as \(x\), \(y\), and \(z\) depict the directions along each axis.
  • Example: The equation \(3x + 6y - 3z = 12\) is linear because it represents a plane in a three-dimensional space.
Linear equations maintain the principle of balance. That means if you add, subtract, multiply, or divide the same number from both sides of the equation, the equality remains true.
The trick when dealing with systems of linear equations is to use combinations of the equations to find solutions for the variables involved. In our exercise, we combined equations to simplify and see their relationships more clearly.
Algebraic Manipulation
Algebraic manipulation is crucial when working with systems of equations. It allows us to rearrange and simplify equations, which is essential to find solutions for unknown variables. Let's shed light on a few key steps in the algebraic process from our example:
  • Simplification: The first equation \(3x + 6y - 3z = 12\) was divided by 3, simplifying it to \(x + 2y - z = 4\). This step makes further calculations easier and more manageable.
  • Substitution: By expressing one variable in terms of another, for example, \(z = z\), you obtain expressions for \(x\) and \(y\) relative to \(z\).
  • Rearrangement: Rearrange the equations to isolate variables, helping to uncover relationships between them. Example: Rearranging \(x + 2y = z + 4\) aids in finding \(y\).
Remember, effective algebraic manipulation requires practice and creativity in arranging equations to simplify and solve them.
Infinite Solutions
Encountering a system of equations with infinite solutions can seem confusing at first. It means that there are endless combinations of values for the variables that satisfy all equations simultaneously. A consistent and dependent system like this one indicates equations that are not independent and hence overlap in some way in their graphical representation.
  • In the exercise, when we added equations 1 and 2, we obtained \(0 = 20\)鈥攁 true statement. This reflects that the equations are dependent and infinitely many solutions exist as there is no contradiction.
  • The solution involves writing remaining variables, such as \(x\) and \(y\), in terms of a variable such as \(z\). This creates forms like \(x = 36 + 3z\) and \(y = -16 - z\), where \(z\) remains arbitrary.
Understanding systems yielding infinite solutions highlights the interconnectedness within the system of equations, promoting a deeper comprehension of linear algebra concepts.

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Most popular questions from this chapter

To analyze population dynamics of the northern spotted owl, mathematical ecologists divided the female owl population into three categories: juvenile (up to 1 year old), subadult (1 to 2 years old), and adult (over 2 years old). They concluded that the change in the makeup of the northern spotted owl population in successive years could be described by the following matrix equation. $$\left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=\left[\begin{array}{rrr} 0 & 0 & 0.33 \\ 0.18 & 0 & 0 \\ 0 & 0.71 & 0.94 \end{array}\right]\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right]$$ The numbers in the column matrices give the numbers of females in the three age groups after \(n\) years and \(n+1\) years. Multiplying the matrices yields the following. $$\begin{aligned} &j_{n+1}=0.33 a_{n}\\\ &s_{n+1}=0.18 j_{n}\\\ &a_{n+1}=0.71 s_{n}+0.94 a_{n} \end{aligned}$$ (Source: Lamberson, R. H., R. McKelvey, B. R. Noon, and C. Voss, "A Dynamic Analysis of Northern Spotted Owl Viability in a Fragmented Forest Landscape," Conservation Biology, Vol. \(6, \text { No. } 4 .)\) (a) Suppose there are currently 3000 female northern spotted owls: 690 juveniles, 210 subadults, and 2100 adults. Use the preceding matrix equation to determine the total number of female owls for each of the next 5 years. (b) Using advanced techniques from linear algebra, we can show that, in the long run, $$ \left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=0.98359\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right] $$ What can we conclude about the long-term fate of the northern spotted owl? (c) In this model, the main impediment to the survival of the northern spotted owl is the number 0.18 in the second row of the \(3 \times 3\) matrix. This number is low for two reasons: The first year of life is precarious for most animals living in the wild, and juvenile owls must eventually leave the nest and establish their own territory. If much of the forest near their original home has been cleared, then they are vulnerable to predators while searching for a new home. Suppose that, due to better forest management, the number 0.18 can be increased to \(0.3 .\) Rework part (a) under this new assumption.

Solve each nonlinear system of equations analytically. $$\begin{array}{r}x^{2}+y^{2}=5 \\\\-3 x+4 y=2\end{array}$$

Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{array}{r}3 x-y=4 \\\x+y=0\end{array}$$

The relationship between a professional basketball player's height \(h\) in inches and weight \(w\) in pounds was modeled by using two samples of players. The resulting equations were $$\begin{aligned}&w=7.46 h-374\\\&w=7.93 h-405\end{aligned}$$ and Assume that \(65 \leq h \leq 85\) (a) Use each equation to predict the weight to the nearest pound of a professional basketball player who is 6 feet 11 inches. (b) Determine graphically the height at which the two models give the same weight. (c) For each model, what change in weight is associated with a 1 -inch increase in height?

In one study, a group of conditioned athletes was exercised to exhaustion. Let \(x\) represent an athlete's heart rate 5 seconds after stopping exercise and \(y\) the rate after 10 seconds. It was found that the maximum heart rate \(H\) for these athletes satisfied the two equations $$\begin{aligned}&H=0.491 x+0.468 y+11.2\\\&H=-0.981 x+1.872 y+26.4\end{aligned}$$ and If an athlete had maximum heart rate \(H=180\), determine \(x\) and \(y\) graphically. Interpret your answer. (Source: Thomas, V., Science and Sport, Faber and Faber.)
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