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Chapter 6: Problem 41

\(\begin{aligned} 2 x-y+2 z &=6 \\ -x+y+z &=0 \\ -x-3 z &=-6 \end{aligned}\)

Short Answer

Expert verified
The solution is \(x = 6\), \(y = 6\), \(z = 0\).

Step by step solution

01

Solve for x from equation 3

Start by considering the third equation \(-x - 3z = -6\). To solve for \(x\), rearrange the terms to get \(x = 3z + 6\). We will use this expression for \(x\) in the other equations.
02

Substitute x in equations 1 and 2

Substitute \(x = 3z + 6\) into the first equation: \(2x - y + 2z = 6\), which becomes \(2(3z + 6) - y + 2z = 6\) or \(6z + 12 - y + 2z = 6\). Simplify it to \(8z - y = -6\) or \(y = 8z + 6\).Next, substitute \(x = 3z + 6\) into the second equation: \(-x + y + z = 0\), which becomes \(-(3z + 6) + y + z = 0\) or \(-3z - 6 + y + z = 0\). Simplify it to \(y - 2z = 6\) or \(y = 2z + 6\).
03

Equate expressions for y

We have two expressions for \(y\): \(y = 8z + 6\) and \(y = 2z + 6\). Equate these two expressions: \(8z + 6 = 2z + 6\). Solve for \(z\) by subtracting \(2z\) from both sides to get \(6z = 0\), giving \(z = 0\).
04

Substitute back to find y and x

Use \(z = 0\) in \(y = 2z + 6\) to find \(y\). Substituting \(z = 0\) gives \(y = 6\).Next, use \(z = 0\) in \(x = 3z + 6\) to find \(x\). Substituting \(z = 0\) gives \(x = 6\).
05

Verify the solution

Verify the solution by substituting \(x = 6\), \(y = 6\), and \(z = 0\) back into the original equations.1. For \(2x - y + 2z = 6\): Substitute to get \(2(6) - 6 + 2(0) = 12 - 6 = 6\), which matches.2. For \(-x + y + z = 0\): Substitute to get \(-6 + 6 + 0 = 0\), which matches.3. For \(-x - 3z = -6\): Substitute to get \(-6 - 3(0) = -6\), which matches.All equations are satisfied, confirming the solution is correct.

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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 that create a straight line when graphed on a coordinate plane. They consist of variables and constants with linear relationships. In this problem, we have three linear equations:
  • \(-x - 3z = -6\)
  • \(-x + y + z = 0\)
  • \(2x - y + 2z = 6\)
These equations combine variables \(x\), \(y\), and \(z\) with constant coefficients. The goal is to find values that satisfy each equation simultaneously. Linear equations are foundational in algebra, forming the basis for modeling and solving real-world problems.
Substitution Method
The substitution method is a technique used to solve systems of equations, particularly effective for systems like linear ones. Here鈥檚 how it is applied in our problem:Start by solving one of the equations for a specific variable. For instance, from the third equation, solve for \(x\):
  • Equation: \(-x - 3z = -6\)
  • Rearrange to find \(x\): \(x = 3z + 6\)
Next, substitute this expression for \(x\) into the remaining equations:
  • Substitute \(x = 3z + 6\) into the first equation: \(2(3z + 6) - y + 2z = 6\)
  • This simplifies to: \(8z - y = -6\) or \(y = 8z + 6\)
  • Also, substitute \(x = 3z + 6\) into the second equation: \(-3z - 6 + y + z = 0\)
  • Simplifying it yields: \(y = 2z + 6\)
By equating these two expressions for \(y\), we can solve for \(z\). Using substitution helps simplify the problem by reducing the number of variables step-by-step.
Verification of Solutions
Verifying solutions is the final step in solving systems of equations. It confirms that the solutions satisfy all given equations.For this problem, once we found \(x = 6\), \(y = 6\), and \(z = 0\), we need to substitute these values back into the original equations to verify:
  • Substitute into \(2x - y + 2z = 6\):
    \(2(6) - 6 + 2(0) = 12 - 6 = 6\)
    This equation is satisfied.
  • Check \(-x + y + z = 0\):
    \(-6 + 6 + 0 = 0\)
    This equation checks out as well.
  • Confirm with \(-x - 3z = -6\):
    \(-6 - 3(0) = -6\)
    Matched exactly.
If all equations are true with the found values, it verifies the correctness of the solution. This step is essential to ensure there are no errors in the calculations and that the solutions are accurate and reliable.

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Most popular questions from this chapter

Use a graphing calculator to solve each system. Express solutions with approximations to the nearest thousandth. $$\begin{aligned}&\pi x+e y=3\\\&e x+\pi y=4\end{aligned}$$

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?

Explain how one can determine whether a system is inconsistent or has dependent equations when using the substitution or elimination method.

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(c(A+B)=c A+c B,\) for any real number \(c\)

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities.$$\begin{aligned} &x+y \geq 2\\\ &x+y \leq 6 \end{aligned}$$
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