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Chapter 8: Problem 14

Solve each system. $$ \left\\{\begin{aligned} x+3 y+5 z &=20 \\ y-4 z &=-16 \\ 3 x-2 y+9 z &=36 \end{aligned}\right. $$

Short Answer

Expert verified
The solution to the system of equations is \( x = 44 \), \( y = 0 \), and \( z = 4 \)

Step by step solution

01

Express y from Equation 2

First, isolate y from Equation 2 to make it easier to substitute y into other equations. This can be done by rearranging equation 2 as \( y = 4z -16 \).
02

Substitute y into Equation 1 and 3

Substitute \( y = 4z - 16 \) into Equation 1 and 3, which gives us:From equation 1, we get: \( x+3(4z -16)+5z = 20 \), simplify to get \( x = -7z + 68 \).From equation 3, we get: \( 3x - 2(4z - 16) + 9z = 36 \), substitute \( x = -7z + 68 \) in to get \( 3(-7z + 68) - 8z + 32 = 36 \), solve to get \( z = 4 \).
03

Backward Substitution

Now substitute \( z = 4 \) into the isolations of \( x \) and \( y \) from step 2, to find the values of \( x \) and \( y \). Therefore, \( x = -7*4 + 68 = 44 \) and \( y = 4*4 - 16 = 0 \).
04

Conclusion

So, the solution to the system of equations is \( x = 44 \), \( y = 0 \), and \( z = 4 \)

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Key Concepts

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

Linear Equations
Linear equations are mathematical expressions involving variables raised to the power of one. They form a straight line when graphed. In a system of linear equations, we deal with multiple such equations that work together to determine a common solution for all included variables.
In our specific system of equations:
  • Equation 1: \( x + 3y + 5z = 20 \)
  • Equation 2: \( y - 4z = -16 \)
  • Equation 3: \( 3x - 2y + 9z = 36 \)
Each equation contains three variables: \( x \), \( y \), and \( z \). The key is to find the values of these variables that satisfy all equations simultaneously.
Substitution Method
The substitution method is a powerful technique for solving systems of equations. It involves solving one of the equations for one variable and then substituting this expression into the other equations.
This method is useful because it reduces the number of variables, making the system easier to solve. Here, we solved Equation 2 for \( y \):
  • \( y = 4z - 16 \).
Then, this expression for \( y \) was substituted into Equations 1 and 3, allowing us to express \( x \) in terms of \( z \) and eventually solving for \( z \).
Solution of Equations
Finding the solution of the system involves determining the specific values for the variables that satisfy each linear equation.
Substituting \( y = 4z - 16 \) into Equation 1 gives:
  • \( x + 3(4z - 16) + 5z = 20 \)
  • Simplifying gives: \( x = -7z + 68 \)
Substituting into Equation 3 gives:
  • \( 3x - 2(4z - 16) + 9z = 36 \)
  • With x substituted, solving reveals \( z = 4 \)
This process progressively narrows down the values of \( x \), \( y \), and \( z \).
Backward Substitution
Backward substitution is the final step where you plug the known value of one variable back into the equations set aside from previous steps to find the other unknowns.
After finding \( z = 4 \), substitute back:
  • \( x = -7 * 4 + 68 \) gives \( x = 44 \)
  • \( y = 4 * 4 - 16 \) gives \( y = 0 \)
In conclusion, using this systematic approach ensures we find precise solutions, confirming \( x = 44 \), \( y = 0 \), and \( z = 4 \). Each value checks out against the original system of equations, proving the solution's validity.

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Most popular questions from this chapter

Exercises 120鈥�122 will help you prepare for the material covered in the next section. a. Graph the solution set of the system: $$\left\\{\begin{array}{r} {x+y \geq 6} \\ {x \leq 8} \\ {y \geq 5} \end{array}\right.$$ b. List the points that form the corners of the graphed region in part (a). c. Evaluate \(3 x+2 y\) at each of the points obtained in part (b).

Suppose that \(\sin \alpha=\frac{3}{5}\) and \(\cos \beta=-\frac{12}{13}\) for quadrant II angles \(\alpha\) and \(\beta .\) Find the exact value of each of the following: a. \(\cos \alpha\) b. \(\sin \beta\) c. \(\cos (\alpha+\beta)\) d. \(\sin (\alpha+\beta)\) (Section \(6.2, \text { Example } 5)\)

determine whether each statement makes sense or does not make sense, and explain your reasoning. Because \(x+5\) is linear and \(x^{2}-3 x+2\) is quadratic, I set up the following partial fraction decomposition: $$\frac{7 x^{2}+9 x+3}{(x+5)\left(x^{2}-3 x+2\right)}=\frac{A}{x+5}+\frac{B x+C}{x^{2}-3 x+2}$$

Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. On June \(24,1948,\) the former Soviet Union blocked all land and water routes through East Germany to Berlin. A gigantic airlift was organized using American and British planes to bring food, clothing, and other supplies to the more than 2 million people in West Berlin. The cargo capacity was \(30,000\) cubic feet for an American plane and \(20,000\) cubic feet for a British plane. To break the Soviet blockade, the Western Allies had to maximize cargo capacity but were subject to the following restrictions: \(\cdot\) No more than 44 planes could be used. \(\cdot\) The larger American planes required 16 personnel per flight, double that of the requirement for the British planes. The total number of personnel available could not exceed 512 . \(\cdot\) The cost of an American flight was 9000 and the cost of a British flight was 5000 . Total weekly costs could not exceed $300,000 Find the number of American and British planes that were used to maximize cargo capacity.

What is a system of linear equations? Provide an example with your description.
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