Problem 52 - Algebra 2

Question

Use Cramer's Rule to solve each system of equations. $$ \begin{array}{l}{2 x+5 y=35} \\ {7 x-4 y=-28}\end{array} $$

Step-by-Step Solution

Answer

The solution is \(x = 0\) and \(y = 7\).

Step 1: Identify the Coefficients

First, identify the coefficients from the given system of equations: 1. From the first equation, the coefficients are 2 (for \(x\)) and 5 (for \(y\)). 2. From the second equation, the coefficients are 7 (for \(x\)) and -4 (for \(y\)).

Step 2: Create the Coefficient Determinant

Form the determinant \(D\) using the coefficients of \(x\) and \(y\):\[D = \begin{vmatrix} 2 & 5 \ 7 & -4 \end{vmatrix}\]Calculate \(D\) by the formula:\[D = (2)(-4) - (5)(7) = -8 - 35 = -43\]

Step 3: Create Determinant for x

Now form the determinant \(D_x\) by replacing the \(x\) coefficients with the constants from the equations (35 and -28):\[D_x = \begin{vmatrix} 35 & 5 \ -28 & -4 \end{vmatrix} \]Calculate \(D_x\):\[D_x = (35)(-4) - (5)(-28) = -140 + 140 = 0\]

Step 4: Create Determinant for y

Form the determinant \(D_y\) by replacing the \(y\) coefficients with the constants:\[D_y = \begin{vmatrix} 2 & 35 \ 7 & -28 \end{vmatrix} \]Calculate \(D_y\):\[D_y = (2)(-28) - (35)(7) = -56 - 245 = -301\]

Step 5: Find Solutions for x and y

Now use Cramer's Rule to find \(x\) and \(y\):\[ x = \frac{D_x}{D} = \frac{0}{-43} = 0 \]\[ y = \frac{D_y}{D} = \frac{-301}{-43} = 7 \]

Step 6: Verify the Solution

Substitute \( x = 0 \) and \( y = 7 \) back into the original equations to verify the solution:1. For the first equation: \( 2(0) + 5(7) = 35 \Rightarrow 35 = 35 \) which holds true.2. For the second equation: \( 7(0) - 4(7) = -28 \Rightarrow -28 = -28 \) which also holds true.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same set of unknowns. In linear systems, like the one given in the exercise, each equation represents a straight line. Solving the system essentially means finding the point where these lines intersect, which is the common solution for all equations.

In our example, we have:

• The first equation: \( 2x + 5y = 35 \)
• The second equation: \( 7x - 4y = -28 \)

These equations together form a system in two variables, \( x \) and \( y \). The solution is the values of \( x \) and \( y \) that satisfy both equations simultaneously. Systems of equations can be tackled using several methods, such as substitution, elimination, or graphical solutions, each offering unique insights into the interplay of the equations.

Determinants

Determinants are critical in solving systems of linear equations using Cramer's Rule. They are numerical values that can be computed from a square matrix and are used to express different properties of the matrix. For a 2x2 matrix \( A = \begin{vmatrix} a & b \ c & d \end{vmatrix} \), the determinant is calculated as follows:

• \( \text{Det}(A) = ad - bc \)

In our solution process, the determinant is used to evaluate the consistency and solvability of the system.

We compute three specific determinants:

• The main determinant \( D \) from the coefficients, which helps us know if a unique solution exists.
• Determinant \( D_x \), substituting the constants into the column of \( x \).
• Determinant \( D_y \), substituting into the column of \( y \).

These calculated values are then used in Cramer's Rule to solve the system for \( x \) and \( y \).

Linear Algebra

Linear Algebra is the branch of mathematics concerning linear equations and their representations through matrices and vector spaces. It provides tools for modeling many natural phenomena and real-life situations. In the case of solving systems of equations like our exercise, linear algebra helps translate the algebraic expressions into matrix forms.

In this exercise, you encountered concepts like:

• Matrices - A rectangular array of numbers that represents coefficients of linear equations.
• Linear transformations - Describe how solutions to equations change along with variations in input values.

Linear Algebra offers frameworks like Cramer's Rule, which leverages determinants to provide exact solutions to these systems by systematically replacing columns in coefficients matrices and computing determinants.

Solution Verification

Once the system of equations is solved, it is crucial to verify the solution for accuracy. Verification involves plugging the obtained values back into the original equations to ensure both equations hold true.

For the solution \( x = 0 \) and \( y = 7 \), verification was performed by substituting these values:

• For \( 2x + 5y = 35 \): Substitute \( x = 0 \, \text{and} \, y = 7 \) to check if the equation results in \( 35 = 35 \).
• For \( 7x - 4y = -28 \): Again, substitute to ensure the equation reads \( -28 = -28 \).

Successful verification means our calculated values satisfy the system, confirming the correctness of the solution. This step is essential, especially when solutions are re-calculated or when interpreting results in practical applications.