Problem 19 - College Algebra

Question

Use Cramer's rule to solve each system or to determine that the system is inconsistent or contains dependent equations. $$ \begin{aligned}&3 x-4 y=4\\\&2 x+2 y=12\end{aligned} $$

Step-by-Step Solution

Answer

The solution of the system of equations is \( x = 2.8571 \) and \( y = 2 \)

Step 1: Calculation of determinant

Calculate the determinant for the matrix formed by the coefficients of x and y in the given equation. In this case, the determinant \( \Delta \) is given by: \( \Delta = (3 * 2) - (-4 * 2) = 6 + 8 = 14 \)

Step 2: Calculation of determinant of x

Replace the x coefficients in the matrix by the constants on the other side of the equation. The determinant Δx will then be calculated as \( Δx = (12 * 4) - (4 * 2) = 48 - 8 = 40 \)

Step 3: Solution for x

The solution for x can be found by substituting values of Δ and Δx into the formula \( x = \frac{\Delta x}{\Delta} \). So, \( x = \frac{40}{14} = 2.8571 \)

Step 4: Calculation of determinant for y

Replace the y coefficients in the matrix by the constants on the other side of the equation. The determinant Δy is then calculated as \( Δy = (3 * 12) - (4 * 2) = 36 - 8 = 28 \)

Step 5: Solution for y

The solution for y can be found by substituting values of Δ and Δy into the formula \( y = \frac{\Delta y}{\Delta} \). So, \( y = \frac{28}{14} = 2 \)

Key Concepts

Determinant Calculation

To understand Cramer's Rule, the determinant is a crucial component. With any system of linear equations, determinants help us find unique solutions. Here, we have coefficients in a 2x2 matrix: \\( \begin{bmatrix} 3 & -4 \ 2 & 2 \end{bmatrix} \). \
To calculate the determinant, use the formula: \( \Delta = ad - bc \), where \( a, b, c, \) and \( d \) are elements of the matrix: \

• \( a = 3 \)
• \( b = -4 \)
• \( c = 2 \)
• \( d = 2 \)

\Calculating further, \( \Delta = (3 \times 2) - (-4 \times 2) = 6 + 8 = 14 \). \
In Cramer's Rule, the non-zero determinant \( \Delta = 14 \) indicates a system that can have a unique solution, while a zero determinant would suggest no unique solution exists or the possibility of dependent equations. \
Use this key insight when solving any system!

System of Equations

A **system of equations** consists of two or more equations with the same set of unknowns. In Cramer's Rule, we specifically deal with linear equations. For example, our system is:\\[ 3x - 4y = 4 \] \\[ 2x + 2y = 12 \] \
Each equation describes a line on the coordinate plane. Finding a solution means determining the intersection point of these lines. \
With Cramer's Rule, solving for "x" and "y" involves substituting coefficients with constants in the determinant. \
For variables:\

• The \( \Delta x \) places constants where the \( x \) coefficients were.
\
• The \( \Delta y \) does the same for the \( y \) coefficients.

\Solving the system involves calculating these new determinants and dividing by the original \( \Delta \). This process directly yields the "x" and "y" values.

Dependent Equations

Dependent equations might appear in a system of equations when one equation is a derivative or scalar multiple of another. \
These equations essentially describe the same line, causing infinite solutions or no solutions if they never intersect with another line of the system. \
In our example, checking dependency means monitoring for zero determinants. \
If \\( \Delta = 0 \), then the lines coincide or are parallel, causing dependency. Therefore, no unique solution exists. However, our determinant \( \Delta = 14 \) was non-zero, implying no dependent equations. \
Detecting dependent equations is critical in determining whether further solution methods are necessary or viable.