Finite Math Examples

$x + 2 y = 7$ , $2 x - y = 4$

Step 1

Represent the system of equations in matrix format.

$[\begin{array}{c} 1 & 2 \\ 2 & - 1 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 7 \\ 4 \end{array}]$

Step 2

Find the determinant of the coefficient matrix $[\begin{array}{c} 1 & 2 \\ 2 & - 1 \end{array}]$ .

Tap for more steps...

Step 2.1

Write $[\begin{array}{c} 1 & 2 \\ 2 & - 1 \end{array}]$ in determinant notation.

$| \begin{array}{c} 1 & 2 \\ 2 & - 1 \end{array} |$

Step 2.2

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot - 1 - 2 \cdot 2$

Step 2.3

Simplify the determinant.

Tap for more steps...

Step 2.3.1

Simplify each term.

Tap for more steps...

Step 2.3.1.1

Multiply $- 1$ by $1$ .

$- 1 - 2 \cdot 2$

Step 2.3.1.2

Multiply $- 2$ by $2$ .

$- 1 - 4$

Step 2.3.2

Subtract $4$ from $- 1$ .

$- 5$

$D = - 5$

Step 3

Since the determinant is not $0$ , the system can be solved using Cramer's Rule.

Step 4

Find the value of $x$ by Cramer's Rule, which states that $x = \frac{D_{x}}{D}$ .

Tap for more steps...

Step 4.1

Replace column $1$ of the coefficient matrix that corresponds to the $x$ -coefficients of the system with $[\begin{array}{c} 7 \\ 4 \end{array}]$ .

$| \begin{array}{c} 7 & 2 \\ 4 & - 1 \end{array} |$

Step 4.2

Find the determinant.

Tap for more steps...

Step 4.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$7 \cdot - 1 - 4 \cdot 2$

Step 4.2.2

Simplify the determinant.

Tap for more steps...

Step 4.2.2.1

Simplify each term.

Tap for more steps...

Step 4.2.2.1.1

Multiply $7$ by $- 1$ .

$- 7 - 4 \cdot 2$

Step 4.2.2.1.2

Multiply $- 4$ by $2$ .

$- 7 - 8$

Step 4.2.2.2

Subtract $8$ from $- 7$ .

$- 15$

$D_{x} = - 15$

Step 4.3

Use the formula to solve for $x$ .

$x = \frac{D_{x}}{D}$

Step 4.4

Substitute $- 5$ for $D$ and $- 15$ for $D_{x}$ in the formula.

$x = \frac{- 15}{- 5}$

Step 4.5

Divide $- 15$ by $- 5$ .

$x = 3$

Step 5

Find the value of $y$ by Cramer's Rule, which states that $y = \frac{D_{y}}{D}$ .

Tap for more steps...

Step 5.1

Replace column $2$ of the coefficient matrix that corresponds to the $y$ -coefficients of the system with $[\begin{array}{c} 7 \\ 4 \end{array}]$ .

$| \begin{array}{c} 1 & 7 \\ 2 & 4 \end{array} |$

Step 5.2

Find the determinant.

Tap for more steps...

Step 5.2.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$1 \cdot 4 - 2 \cdot 7$

Step 5.2.2

Simplify the determinant.

Tap for more steps...

Step 5.2.2.1

Simplify each term.

Tap for more steps...

Step 5.2.2.1.1

Multiply $4$ by $1$ .

$4 - 2 \cdot 7$

Step 5.2.2.1.2

Multiply $- 2$ by $7$ .

$4 - 14$

Step 5.2.2.2

Subtract $14$ from $4$ .

$- 10$

$D_{y} = - 10$

Step 5.3

Use the formula to solve for $y$ .

$y = \frac{D_{y}}{D}$

Step 5.4

Substitute $- 5$ for $D$ and $- 10$ for $D_{y}$ in the formula.

$y = \frac{- 10}{- 5}$

Step 5.5

Divide $- 10$ by $- 5$ .

$y = 2$

Step 6

List the solution to the system of equations.

$x = 3$

$y = 2$