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

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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.

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 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.