Problem 2 - Algebra 2

Question

Use Cramer's Rule to solve each system. $$ \left\\{\begin{aligned} 2 x+y &=7 \\\\-2 x+5 y &=-1 \end{aligned}\right. $$

Step-by-Step Solution

Answer

Using Cramer's Rule, the solution to the system is \( x = 3 \) and \( y = 1 \).

Step 1: Write down the system of equations

Identify the system of linear equations to solve. In this case, the system is: \[\begin{aligned} 2x + y &= 7 \ -2x + 5y &= -1 \end{aligned}\]

Step 2: Find the determinant of the coefficient matrix

Construct the coefficient matrix A for the system and calculate its determinant, \( \Delta \). The coefficient matrix A is: \[A = \begin{bmatrix} 2 & 1 \ -2 & 5 \end{bmatrix}\]. The determinant \( \Delta \) is calculated as \[\Delta = (2)(5) - (-2)(1) = 10 + 2 = 12\].

Step 3: Find determinants for the numerator of Cramer's Rule

Find the determinants \( \Delta_x \) and \( \Delta_y \) by replacing the respective columns in A with the constants from the right side of the equations.For \( \Delta_x \):\[A_x = \begin{bmatrix} 7 & 1 \ -1 & 5 \end{bmatrix} \], so \[\Delta_x = (7)(5) - (-1)(1) = 35 + 1 = 36 \].For \( \Delta_y \):\[A_y = \begin{bmatrix} 2 & 7 \ -2 & -1 \end{bmatrix} \], so \[\Delta_y = (2)(-1) - (-2)(7) = -2 + 14 = 12 \].

Step 4: Solve for the variables using Cramer's Rule

Use Cramer's Rule to find the value of the variables. For \( x \):\[x = \frac{\Delta_x}{\Delta}\]\[x = \frac{36}{12} = 3\]For \( y \):\[y = \frac{\Delta_y}{\Delta}\]\[y = \frac{12}{12} = 1\]

Key Concepts

Determinants

In linear algebra, the determinant is a special number that can be computed from the elements of a square matrix. It provides important information about the matrix, including whether the matrix is invertible, and the volume scaling factor when the matrix is viewed as a transformation.

The determinant of a 2x2 matrix like
\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]
can be calculated using the formula
\[\Delta = ad - bc.\]
This is a straightforward computation that just involves multiplication and subtraction. In a system of linear equations, determinants play a crucial role in Cramer's Rule, where they are used to solve for the variables in the system by replacing the columns of the coefficient matrix with constants.

One thing to remember about determinants is that if the determinant of a matrix is zero, the system of equations it represents does not have a unique solution; this might indicate that the equations have either no solution or infinitely many solutions.

Coefficient Matrix

The coefficient matrix is a matrix consisting of the coefficients of the variables from a system of linear equations. This matrix is fundamental when applying methods like Cramer's Rule to solve systems.

Given a system of equations like
\[\begin{aligned} ax + by &= e \ cx + dy &= f \end{aligned}\]
the coefficient matrix would be
\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\].
In the context of Cramer's Rule, finding the determinant of this matrix (as long as it's square and therefore, has a determinant) is the initial step in the process. This matrix should not be confused with the augmented matrix, which also includes the constants (e and f in the case above) from the right-hand side of the equations. Ensuring accuracy when constructing the coefficient matrix is critical, as any error here will propagate through the calculations.

System of Linear Equations

A system of linear equations is a set of equations where each is linear, which means that the exponents of the variables are all one. In a system, you are looking for the values of the variables that make all of the equations true simultaneously.

For example, the system given in the exercise
\[\begin{aligned} 2x + y &= 7 \ -2x + 5y &= -1 \end{aligned}\]
contains two equations with two variables, x and y. Solving a system like this involves finding the point of intersection where both equations hold true. There are several methods to solve such systems, including graphing, substitution, elimination, and matrix methods like Cramer's Rule.

Understanding the concept of a 'solution' to such systems is also important: in two dimensions, it's the x and y value where the lines cross; in three dimensions, it's where the planes intersect. Depending on the system, there could be one solution, no solution (if the lines are parallel), or infinitely many solutions (if the two lines are, in fact, the same line).