Chapter 4: Problem 74

Solve each problem using a system of linear equations and the Gauss-Jordanelimination method. Jay Leno's garage. Jay Leno's collection of cars and motorcycles totals \(187 .\) When he checks the air in the tires he has 588tires to check. How many cars and how many motorcycles does he own? Assumethat the cars all have four tires and the motorcycles have two.

### Short Answer

Expert verified

Jay Leno owns 107 cars and 80 motorcycles.

## Step by step solution

01

## - Define Variables

Let the number of cars be denoted by \(C\) and the number of motorcycles be denoted by \(M\).

02

## - Set Up Equations

We know from the problem that the total number of vehicles is 187 and the total number of tires is 588. This translates into two equations: 1. \(C + M = 187\) 2. \(4C + 2M = 588\)

03

## - Simplify the Second Equation

The second equation can be simplified by dividing everything by 2: \(2C + M = 294\)

04

## - Form the Augmented Matrix

Translate the equations into an augmented matrix: \[\begin{pmatrix} 1 & 1 & | & 187 \ 2 & 1 & | & 294 \end{pmatrix}\]

05

## - Eliminate Using Row Operations

Using row operations to solve, subtract Row 1 from Row 2 to eliminate M: \[\begin{pmatrix} 1 & 1 & | & 187 \ 0 & -1 & | & -80 \end{pmatrix}\]

06

## - Solve for M

Solve the second equation for M: \[-M = -80 \implies M = 80\]

07

## - Substitute M Back Into the First Equation

Substitute \(M = 80\) into the first equation: \(C + 80 = 187 \implies C = 107\)

08

## - Interpret the Result

The solution to the system of equations is \(C = 107\) and \(M = 80\). Jay Leno owns 107 cars and 80 motorcycles.

## Key Concepts

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

###### system of linear equations

A system of linear equations is simply a set of two or more linear equations involving the same set of variables. Each equation in the system represents a line (when dealing with two variables) or a plane (for three variables) in a graphical sense. The solution to the system is the point or points where these lines or planes intersect. In our problem, we created a system using the variables C (cars) and M (motorcycles). Our system was derived from the total number of vehicles and the total number of tires:

- 1. \(C + M = 187\)2. \(4C + 2M = 588\)

Solving these equations simultaneously helps us determine the exact number of cars and motorcycles Jay Leno has in his collection.

###### augmented matrix

An augmented matrix is a compact representation of a system of linear equations. It combines the coefficients of the variables and the constants from the equations into a single matrix. This makes it easier to apply row operations and manipulate the system for solutions. For our problem, the system of equations:

- \(C + M = 187\)
- \(2C + M = 294\)

can be converted to an augmented matrix as follows: \[\begin{pmatrix}1 & 1 & | & 187 \ 2 & 1 & | & 294 \end{pmatrix}\]. The vertical bar separates the coefficients of the variables from the constants, which we need to solve the system. This matrix will be manipulated using row operations to find the values of \C\ and \M\.

###### row operations

Row operations are techniques used to simplify matrices and solve systems of linear equations. These include:

- 1. Swapping two rows
- 2. Multiplying a row by a nonzero constant
- 3. Adding or subtracting a multiple of one row to another row

In our Gauss-Jordan elimination method, we performed a series of row operations to transform the augmented matrix into reduced row-echelon form. The steps included:

- 1. Subtracting Row 1 from Row 2 to eliminate M: \[\begin{pmatrix}1 & 1 & | & 187 \ 0 & -1 & | & -80 \end{pmatrix} \] 2. Solving the resulting equation for M: \ -M = -80 \implies M = 80 \ 3. Substituting \ M = 80 \ back into the first equation to solve for C: \ C + 80 = 187 \implies C = 107 \ . These operations simplified the original system into a much easier-to-solve form, allowing us to find that Jay Leno owns 107 cars and 80 motorcycles.

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