Problem 95 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use Cramer's Rule to solve the system of equations. $$\left\\{\begin{array}{r} -5 x+3 y=-14 \\ 7 x-2 y=\quad 2 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations using Cramer's Rule is \(x = 20/31\) and \(y = -88/31\).

Step 1: Formulate the Coefficient Matrix and the Constants Vector

First, identify the coefficients of the variables and the constants from the two equations. For the given system of equations:\[-5x + 3y = -14\]\[7x - 2y = 2\]The coefficient matrix \(A\) is:\[\begin{bmatrix} -5 & 3 \ 7 & -2 \end{bmatrix}\]And the constants vector \(B\) is:\[\begin{bmatrix} -14 \ 2 \end{bmatrix}\]

Step 2: Compute the Determinant of the Coefficient Matrix

The determinant of a 2x2 matrix:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]is given by \(ad - bc\). Using this formula, we can calculate the determinant of the coefficient matrix \(A\), denoted by \(Det(A)\), as:\[-5*-2 - 3*7 = -10 - 21 = -31\]

Step 3: Compute the Matrix for x and its Determinant

Create a new matrix \(A_x\) by replacing the first column in \(A\) with the constants vector:\[\begin{bmatrix} -14 & 3 \ 2 & -2 \end{bmatrix}\]Then compute its determinant (using the same formula as in step 2):\[-14*-2 - 3*2 = -20\]

Step 4: Compute the Matrix for y and its Determinant

Create a new matrix \(A_y\) by replacing the second column in \(A\) with the constants vector:\[\begin{bmatrix} -5 & -14 \ 7 & 2 \end{bmatrix}\]Then compute its determinant (using the same formula as in step 2):\[-5*2 - 7*-14 = -10 + 98 = 88\]

Step 5: Solve for x and y Using Cramer's Rule

Cramer's rule states that the solution for x is the ratio of the determinant of the matrix for x and the determinant of the coefficient matrix. Same for y. So, we have:\[x = Det(A_x) / Det(A) = -20 / -31 = 20/31\]\[y = Det(A_y) / Det(A) = 88 / -31 = -88/31\]

Key Concepts

System of Equations

A system of equations is a set of equations with multiple variables that you aim to solve simultaneously. In our example, the system consists of two equations:

• -5x + 3y = -14
• 7x - 2y = 2

Each equation involves two variables, "x" and "y," and the goal is to find values for these variables that satisfy both equations.
The solution to a system of equations represents the point where the graphs of the equations intersect on a coordinate system. This means at that point, both equations are true.
Understanding systems of equations is essential because they model real-world situations where multiple conditions must be met simultaneously.

Determinant

The determinant is a special number that is calculated from a square matrix. In the case of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is found using the formula:\[Determinant = ad - bc\]For the given system of equations, we extract the coefficient matrix and find its determinant. If the determinant is zero, the system of equations may not have a unique solution. However, a non-zero determinant, like in our example \(-31\), indicates that there is a unique solution.
Understanding the determinant is important for solving systems algebraically, particularly through methods such as Cramer's Rule.

Coefficient Matrix

The coefficient matrix in a system of equations is a compact way to represent the coefficients of the variables in the equations. For instance, in our equations:

• -5x + 3y = -14
• 7x - 2y = 2

The coefficient matrix \(A\) is:\[\begin{bmatrix} -5 & 3 \ 7 & -2 \end{bmatrix}\]This matrix is used to perform calculations, such as finding the determinant, and to eventually solve for the variables using methods like Cramer's Rule. The rows of the matrix represent each equation, while the columns represent each variable's coefficients.
A solid understanding of coefficient matrices enables you to transition from equations written in a standard format to a matrix form that can be manipulated mathematically.

Solution of Linear Equations

The solution to a set of linear equations is the values of the variables that make all equations true simultaneously. In our system, using Cramer's Rule, we solved for the variables "x" and "y" by:

• Calculating separate determinants for modified matrices (replacing one column with the constant vector each time).
• Dividing these determinants by the original coefficient determinant.

The solutions are:\[x = \frac{-20}{-31} = \frac{20}{31}\]\[y = \frac{88}{-31} = -\frac{88}{31}\]This means that when \(x = \frac{20}{31}\) and \(y = -\frac{88}{31}\), both equations are satisfied.
Understanding how to find solutions using various methods, such as Cramer's Rule, provides valuable techniques for addressing more complex systems of equations in mathematics.