Finite Math Examples

2 x - 5 y = 10

$2 x - 5 y = 10$ ,

4 x - 10 y = 20

$4 x - 10 y = 20$

Step 1

Represent the system of equations in matrix format.

[\begin{matrix} 2 & - 5 4 & - 10 \end{matrix}] [\begin{matrix} x y \end{matrix}] = [\begin{matrix} 1020 \end{matrix}]

$[\begin{array}{c} 2 & - 5 \\ 4 & - 10 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 10 \\ 20 \end{array}]$

Step 2

Find the determinant of the coefficient matrix

[\begin{matrix} 2 & - 5 4 & - 10 \end{matrix}]

$[\begin{array}{c} 2 & - 5 \\ 4 & - 10 \end{array}]$ .

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Step 2.1

Write

[\begin{matrix} 2 & - 5 4 & - 10 \end{matrix}]

$[\begin{array}{c} 2 & - 5 \\ 4 & - 10 \end{array}]$ in determinant notation.

∣ ∣ ∣ \begin{matrix} 2 & - 5 4 & - 10 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 2 & - 5 \\ 4 & - 10 \end{array} |$

Step 2.2

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$2 \cdot - 10 - 4 \cdot - 5$

Step 2.3

Simplify the determinant.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply

$2$ by

$- 10$ .

$- 20 - 4 \cdot - 5$

Step 2.3.1.2

Multiply

$- 4$ by

$- 5$ .

$- 20 + 20$

Step 2.3.2

Add

$- 20$ and

$20$ .

$0$

$D = 0$

Step 3

Since the determinant is

$0$ , the system cannot be solved using Cramer's Rule.