Problem 61 - Algebra 2

Question

Use Cramer's Rule to solve each system of equations. $$ \begin{array}{l}{5 x+7 y=1} \\ {3 x+5 y=3}\end{array} $$

Step-by-Step Solution

Answer

The solution is \((-4, 3)\).

Step 1: Write the System of Equations

Identify the given equations: \(5x + 7y = 1\) and \(3x + 5y = 3\).

Step 2: Construct the Coefficients Matrix

Build the matrix from the coefficients of the variables: \[A = \begin{bmatrix} 5 & 7 \ 3 & 5 \end{bmatrix}\]

Step 3: Determine the Determinant of Matrix A

Calculate the determinant of matrix \(A\): \[\det(A) = (5)(5) - (7)(3) = 25 - 21 = 4\]

Step 4: Construct Matrices for Numerators

Create matrices for the numerators to solve for \(x\) and \(y\). For \(x\): Replace the first column of \(A\) with the constants from the equations: \[A_x = \begin{bmatrix} 1 & 7 \ 3 & 5 \end{bmatrix}\] For \(y\): Replace the second column of \(A\) with the constants from the equations: \[A_y = \begin{bmatrix} 5 & 1 \ 3 & 3 \end{bmatrix}\]

Step 5: Determine the Determinants of Modified Matrices

Calculate determinants needed for Cramer's Rule.For \(A_x\): \[\det(A_x) = (1)(5) - (7)(3) = 5 - 21 = -16\]For \(A_y\): \[\det(A_y) = (5)(3) - (1)(3) = 15 - 3 = 12\]

Step 6: Apply Cramer's Rule to Solve for Variables

Use Cramer's Rule equations to find \(x\) and \(y\). \[x = \frac{\det(A_x)}{\det(A)} = \frac{-16}{4} = -4\] \[y = \frac{\det(A_y)}{\det(A)} = \frac{12}{4} = 3\]

Step 7: Write the Solution as an Ordered Pair

The solution to the system of equations, given as the ordered pair \((x, y)\), is \((-4, 3)\).

Key Concepts

System of Equations

A system of equations is a set of two or more equations with the same variables. The goal is to find the values of these variables that satisfy all the equations simultaneously. In this particular exercise, we have a system of two linear equations:

• 5x + 7y = 1
• 3x + 5y = 3

Each equation represents a straight line on a graph. The solution of the system is the intersection point of these lines, where both conditions are met. To find this solution, we use mathematical techniques, such as substitution, elimination, or—in this case—Cramer's Rule. By understanding the behavior of each line, you can predict how they interact and determine the set of solutions, which in this case is given as an ordered pair (x, y).

Determinant of a Matrix

The determinant of a matrix is a special number that gives us important information about a matrix. It can help us determine if a matrix has an inverse or if a system of equations has a unique solution.
For a 2x2 matrix like the one in our problem, the determinant is calculated using the formula: \[det(A) = ad - bc\]where \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix arranged as follows:

• a b
• c d

In this exercise, the determinant of the matrix \(A\) is calculated using the coefficients of the variables from the original system. A non-zero determinant, like the one we obtained (4), indicates that the matrix is invertible and the system of equations has a unique solution. Calculating determinants is crucial in applying Cramer's Rule or finding the inverse of a matrix.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. It provides a systematic way to solve systems of linear equations, like the ones presented in our exercise. Using concepts from linear algebra, we can manipulate equations and matrices to make calculations easier and more efficient.
In the context of Cramer's Rule, we use linear algebra to form matrix representations of systems and solve for variables systematically. This discipline includes operations involving matrices, such as finding determinants, inverses, and performing matrix multiplication, which are key concepts for solving linear systems. Linear algebra helps us visualize and understand the geometric properties of linear equations, making it a powerful tool for problem-solving in various fields, including engineering, computer science, and economics.