Finite Math Examples

$- x + 7 y = 35$ , $3 x - 4 y = - 5$

Step 1

Represent the system of equations in matrix format.

$[\begin{array}{c} - 1 & 7 \\ 3 & - 4 \end{array}] [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 35 \\ - 5 \end{array}]$

Step 2

Find the determinant of the coefficient matrix $[\begin{array}{c} - 1 & 7 \\ 3 & - 4 \end{array}]$ .

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Step 2.1

Write $[\begin{array}{c} - 1 & 7 \\ 3 & - 4 \end{array}]$ in determinant notation.

$| \begin{array}{c} - 1 & 7 \\ 3 & - 4 \end{array} |$

Step 2.2

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

$- - 4 - 3 \cdot 7$

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 $- 1$ by $- 4$ .

$4 - 3 \cdot 7$

Step 2.3.1.2

Multiply $- 3$ by $7$ .

$4 - 21$

Step 2.3.2

Subtract $21$ from $4$ .

$- 17$

$D = - 17$

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

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Step 4.1

Replace column $1$ of the coefficient matrix that corresponds to the $x$ -coefficients of the system with $[\begin{array}{c} 35 \\ - 5 \end{array}]$ .

$| \begin{array}{c} 35 & 7 \\ - 5 & - 4 \end{array} |$

Step 4.2

Find the determinant.

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Step 4.2.1

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

$35 \cdot - 4 - (- 5 \cdot 7)$

Step 4.2.2

Simplify the determinant.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $35$ by $- 4$ .

$- 140 - (- 5 \cdot 7)$

Step 4.2.2.1.2

Multiply $- (- 5 \cdot 7)$ .

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Step 4.2.2.1.2.1

Multiply $- 5$ by $7$ .

$- 140 - - 35$

Step 4.2.2.1.2.2

Multiply $- 1$ by $- 35$ .

$- 140 + 35$

Step 4.2.2.2

Add $- 140$ and $35$ .

$- 105$

$D_{x} = - 105$

Step 4.3

Use the formula to solve for $x$ .

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

Step 4.4

Substitute $- 17$ for $D$ and $- 105$ for $D_{x}$ in the formula.

$x = \frac{- 105}{- 17}$

Step 4.5

Dividing two negative values results in a positive value.

$x = \frac{105}{17}$

Step 5

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

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Step 5.1

Replace column $2$ of the coefficient matrix that corresponds to the $y$ -coefficients of the system with $[\begin{array}{c} 35 \\ - 5 \end{array}]$ .

$| \begin{array}{c} - 1 & 35 \\ 3 & - 5 \end{array} |$

Step 5.2

Find the determinant.

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Step 5.2.1

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

$- - 5 - 3 \cdot 35$

Step 5.2.2

Simplify the determinant.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

Multiply $- 1$ by $- 5$ .

$5 - 3 \cdot 35$

Step 5.2.2.1.2

Multiply $- 3$ by $35$ .

$5 - 105$

Step 5.2.2.2

Subtract $105$ from $5$ .

$- 100$

$D_{y} = - 100$

Step 5.3

Use the formula to solve for $y$ .

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

Step 5.4

Substitute $- 17$ for $D$ and $- 100$ for $D_{y}$ in the formula.

$y = \frac{- 100}{- 17}$

Step 5.5

Dividing two negative values results in a positive value.

$y = \frac{100}{17}$

Step 6

List the solution to the system of equations.

$x = \frac{105}{17}$

$y = \frac{100}{17}$

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