Problem 29 - College Algebra with Modeling and Visualization

Question

Use Cramer's rule to solve the system of linear equations. $$ \begin{array}{r} 7 x+4 y=23 \\ 11 x-5 y=70 \end{array} $$

Step-by-Step Solution

Answer

The solution is \(x = 5\) and \(y = -3\).

Step 1: Write the equations in standard form

The given equations are already in standard form:1. \(7x + 4y = 23\)2. \(11x - 5y = 70\)

Step 2: Set up the determinants

Cramer's Rule uses determinants to solve linear equations. We set up three determinants: the coefficient determinant \(D\), the determinant \(D_x\) replacing the first column with the constants, and \(D_y\) replacing the second column with the constants. Thus:\[D = \begin{vmatrix} 7 & 4 \ 11 & -5 \end{vmatrix}\]\[D_x = \begin{vmatrix} 23 & 4 \ 70 & -5 \end{vmatrix}\]\[D_y = \begin{vmatrix} 7 & 23 \ 11 & 70 \end{vmatrix}\]

Step 3: Calculate the determinant D

Calculate \(D\) using cross multiplication:\[D = (7)(-5) - (4)(11) = -35 - 44 = -79\]

Step 4: Calculate the determinant D_x

Calculate \(D_x\) using cross multiplication:\[D_x = (23)(-5) - (4)(70) = -115 - 280 = -395\]

Step 5: Calculate the determinant D_y

Calculate \(D_y\) using cross multiplication:\[D_y = (7)(70) - (23)(11) = 490 - 253 = 237\]

Step 6: Solve for x and y using Cramer's Rule

According to Cramer's Rule, solve for \(x\) and \(y\):\[x = \frac{D_x}{D} = \frac{-395}{-79} = 5\]\[y = \frac{D_y}{D} = \frac{237}{-79} = -3\]

Key Concepts

System of Linear Equations

A system of linear equations is a set of two or more linear equations that have common variables. In our example, the system consists of two equations with two variables, \(x\) and \(y\):

• \(7x + 4y = 23\)
• \(11x - 5y = 70\)

Each equation represents a straight line when plotted on a graph. The solution to the system is the point where the lines intersect. This point will satisfy both equations at the same time. In algebraic terms, the solution set \((x, y)\) gives values that make both equations true simultaneously.

Determinants

Determinants are scalar values that are calculated from a square matrix. They are particularly useful in solving systems of linear equations using methods like Cramer's Rule. The determinant of a 2x2 matrix \(\begin{vmatrix} a & b \ c & d \end{vmatrix}\) is found using the formula:\[D = ad - bc\]In Cramer's Rule, there are three main determinants to consider:

• The coefficient determinant \(D\), which is formed from the coefficients of the variables in the system.
• The determinant \(D_x\), where the first column of \(D\) is replaced with the constants from the equations, representing the right-hand side.
• The determinant \(D_y\), where the second column of \(D\) is replaced with the constants from the equations.

By calculating these determinants, one can determine the values of \(x\) and \(y\) in the given equations.

Cross Multiplication

Cross multiplication is a simple and effective way of calculating the determinant of a 2x2 matrix. In the context of Cramer's Rule, it is used to find the determinants \(D\), \(D_x\), and \(D_y\). The process involves multiplying diagonally:

• For \(D\), we calculate \(7\times(-5)\) and subtract \(4\times11\), resulting in \(-35 - 44 = -79\).
• For \(D_x\), multiply \(23\times(-5)\) minus \(4\times70\), giving \(-115 - 280 = -395\).
• For \(D_y\), the calculation is \(7\times70\) minus \(23\times11\), resulting in \(490 - 253 = 237\).

This cross multiplication step is crucial as it helps you compute the values directly required to solve for \(x\) and \(y\).

Algebraic Solution Methods

Algebraic solution methods involve finding solutions to equations using algebraic techniques rather than numerical or graphical ones. Cramer's Rule is one such method specifically used for solving a system of linear equations with an equal number of equations and variables. Once the determinants \(D\), \(D_x\), and \(D_y\) are calculated:

• \(x\) is found using the formula \(x = \frac{D_x}{D}\)
• \(y\) is determined by \(y = \frac{D_y}{D}\)

This method is powerful because it directly uses the determinants to give solutions without needing to rearrange the equations or use substitution or elimination. However, Cramer’s Rule is applicable only when \(D\) is non-zero, ensuring that the system of equations has a unique solution.