Problem 74 Solve each problem using a syste... [FREE SOLUTION] (2024)

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

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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Problem 74 Solve each problem using a syste... [FREE SOLUTION] (2024)

FAQs

How to solve percent equations? ›

Percent problems have three parts: the percent, the base (or whole), and the amount. Any of those parts may be the unknown value to be found. To solve percent problems, you can use the equation, Percent ⋅ Base = Amount , and solve for the unknown numbers.

How do you determine how many solutions a system will have? ›

A system of two equations can be classified as follows: If the slopes are the same but the y-intercepts are different, the system has no solution. If the slopes are different, the system has one solution. If the slopes are the same and the y-intercepts are the same, the system has infinitely many solutions.

How many solutions does this system of linear equations have? ›

A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line).

How to solve an equation? ›

To solve any equation we need to perform arithmetic operations, to separate the variable, such as: Adding the same number on both sides. Subtracting same number on both sides. Multiplying with the same number on both sides.

How to find the solution set of a system? ›

To solve a system of equations by graphing, graph all the equations in the system. The point(s) at which all the lines intersect are the solutions to the system. Graph of System Since the two lines intersect at the point (1, 1), this point is a solution to the system.

What is the percent solution formula? ›

To determine the weight per cent of a solution, divide the mass of solute by mass of the solution (solute and solvent together) and multiply by 100 to obtain per cent. Calculate how many grams of NaOH are required to make a 30.0% solution by using De-ionized water as the solvent.

How do you solve percents step by step? ›

If we have to calculate percent of a number, divide the number by the whole and multiply by 100. Hence, the percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”.

What does no solution look like? ›

An equation has no solution if solving it leads to a false statement, such as 0 = 5. There are three ways to tell if a system of equations has no solution: 1) Graphically: there is no point at which all of the functions intersect. 2) Algebraically: solving the system leads to a false statement, such as 0 = 5.

What is the formula for no solution? ›

Condition for No Solution:

Considering the pair of linear equations by two variables u and v. Therefore a1, b1, c1, a2, b2, c2 are real numbers. If (a1/a2) = (b1/b2) ≠ (c1/c2), then this will result in no solution.

How do systems of equations work? ›

system of equations, In algebra, two or more equations to be solved together (i.e., the solution must satisfy all the equations in the system). For a system to have a unique solution, the number of equations must equal the number of unknowns. Even then a solution is not guaranteed.

What does a system of equations look like? ›

Geometrically, a system of two equations is represented by the intersection of two lines, because the representation of a linear equation of two variables is a line in a plane. There are two methods to solve a system of linear equations, by substitution and by elimination.

What does the slope-intercept form look like? ›

The equation of the line is written in the slope-intercept form, which is: y = mx + b, where m represents the slope and b represents the y-intercept.

Which system of equations is inconsistent? ›

Inconsistent System

Let both the lines to be parallel to each other, then there exists no solution because the lines never intersect. Algebraically, for such a case, a1/a2 = b1/b2 ≠ c1/c2, and the pair of linear equations in two variables is said to be inconsistent.

What is the formula for the system of equations? ›

A system of equations represents two sets of x and y values, using forms such as slope-intercept form, represented as y = mx + b, or standard form, represented as Ax + By = C. Solving for one variable will assist in finding the value for the other.

How to find the solution to the system of equations without graphing? ›

To solve a system of linear equations without graphing, you can use the substitution method. This method works by solving one of the linear equations for one of the variables, then substituting this value for the same variable in the other linear equation and solving for the other variable.

How do you find the solution to a system of equations on a graph? ›

To solve a system of linear equations graphically we graph both equations in the same coordinate system. The solution to the system will be in the point where the two lines intersect. The two lines intersect in (-3, -4) which is the solution to this system of equations.

When a system of equations has a solution? ›

Concept: A system of equation will have a unique solution when the determinant of coefficients does not equals to zero, i.e. Here, a1, b1... are the coefficients in each equation.

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