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Chapter 4: Problem 8

Solve each system of equations. $$\begin{aligned}x+y-z &=4 \\\y &=6 \\\y+z &=13\end{aligned}$$

Short Answer

Expert verified
The solution is x = 5, y = 6, and z = 7.

Step by step solution

01

- Substitute y in the second equation

The second equation gives the value of y as 6. Substitute y = 6 into the first and third equations.
02

- Substitute y in the first equation

Substitute y = 6 into the first equation: \[ x + 6 - z = 4\] This simplifies to: \[ x - z = -2\]
03

- Substitute y in the third equation

Substitute y = 6 into the third equation: \[ 6 + z = 13\] This simplifies to: \[ z = 7\]
04

- Solve for x using the simplified first equation

The simplified first equation is: \[ x - z = -2\] Substitute z = 7 into this equation: \[ x - 7 = -2\] Solve for x: \[ x = 5\]
05

- Verify the solution

Verify that the obtained values of x, y, and z satisfy the original equations. Original equations: \[ x + y - z = 4\]\[ y = 6\]\[ y + z = 13\]Substituting x = 5, y = 6, and z = 7, we get: \[ 5 + 6 - 7 = 4\]\[ 6 = 6\]\[ 6 + 7 = 13\] Therefore, the solution is verified.

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Key Concepts

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

linear equations
A linear equation is an equation that makes a straight line when it is graphed. These equations are in the form of \(Ax + By + Cz = D\) where x, y, and z represent variables, and A, B, C, and D are constants.
  • They only involve powers of 1 for the variables.
  • No variables are multiplied together.
  • No variables are raised to a higher power.
In our exercise, the system of equations: \( \begin {aligned} x + y - z = 4 \ y = 6 \ y + z = 13 \end {aligned} \), is an example of linear equations.
Understanding how to identify and solve linear equations is crucial because they appear frequently in various fields like physics, engineering, and economics.
substitution method
The substitution method is a way to solve systems of linear equations. The main idea is to solve one equation for one of the variables and then substitute that expression into the other equations.
Let's walk through the substitution method using our exercise.
  1. Begin with the second equation: \( y = 6 \).
  2. Substitute \( y = 6 \) into the first equation: \ x + 6 - z = 4 \. This simplifies to: \ x - z = -2 \.
  3. Now, substitute \( y = 6 \) into the third equation: \ 6 + z = 13 \. This simplifies to: \ z = 7 \.
  4. With \ z = 7 \ known, substitute it back into the simplified first equation: \ x - 7 = -2 \. Solving this gives us: \ x = 5 \.
Therefore, the solution obtained is \ x = 5 \, \ y = 6 \, and \ z = 7 \.
algebraic verification
Algebraic verification is a step where we check the solution of our system of equations to ensure it is correct. Let's verify the obtained values \ x = 5 \, \ y = 6 \, and \ z = 7 \.
1. Substitute these values back into the original equations:
  • First Equation: \ x + y - z = 4 \Rightarrow \( 5 + 6 - 7 = 4 \) which holds true.
  • Second Equation: \ y = 6 \Rightarrow \( 6 = 6 \) which is true.
  • Third Equation: \ y + z = 13 \Rightarrow \( 6 + 7 = 13 \) which also holds true.
Since all the original equations are satisfied with \ x = 5 \, \ y = 6 \, and \ z = 7 \, our solution is verified as correct.

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Most popular questions from this chapter

Solve each system using the Gauss-Jordan elimination method. $$ \begin{array}{r} x+y=6 \\ -x+y=8 \end{array} $$

Write a step-by-step procedure for solving any system of two linear equations in two variables by the Gauss-Jordan elimination method. Have a classmate evaluate your procedure by using it to solve a system.

Determine the row operation that was used to convert each given augmented matrix into the equivalent augmented matrix that follows it. $$ \left[\begin{array}{rr|r} 1 & -1 & -1 \\ 0 & 1 & 3 \end{array}\right],\left[\begin{array}{ll|l} 1 & 0 & 2 \\ 0 & 1 & 3 \end{array}\right] $$

Solve each problem using a system of linear equations and the Gauss-Jordan elimination method. Photo size. The length of a rectangular photo is 2 inches greater than the width. The perimeter is 20 inches. Find the length and width.

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