Finite Math Examples

$x = 7 (y - 5)$ , $4 y = 3 x + 5$

Step 1

Move all of the variables to the left side of each equation.

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Step 1.1

Subtract $7 (y - 5)$ from both sides of the equation.

$x - 7 (y - 5) = 0$

$4 y = 3 x + 5$

Step 1.2

Simplify each term.

$x - 7 y + 35 = 0$

$4 y = 3 x + 5$

Step 1.3

Subtract $35$ from both sides of the equation.

$x - 7 y = - 35$

$4 y = 3 x + 5$

Step 1.4

Subtract $3 x$ from both sides of the equation.

$x - 7 y = - 35$

$4 y - 3 x = 5$

Step 1.5

Reorder $4 y$ and $- 3 x$ .

$x - 7 y = - 35$

$- 3 x + 4 y = 5$

$x - 7 y = - 35$

$- 3 x + 4 y = 5$

Step 2

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 3

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

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Step 3.1

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

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

Step 3.2

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

$1 \cdot 4 - (- 3 \cdot - 7)$

Step 3.3

Simplify the determinant.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $4$ by $1$ .

$4 - (- 3 \cdot - 7)$

Step 3.3.1.2

Multiply $- (- 3 \cdot - 7)$ .

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Step 3.3.1.2.1

Multiply $- 3$ by $- 7$ .

$4 - 1 \cdot 21$

Step 3.3.1.2.2

Multiply $- 1$ by $21$ .

$4 - 21$

Step 3.3.2

Subtract $21$ from $4$ .

$- 17$

$D = - 17$

Step 4

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

Step 5

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

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Step 5.1

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

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

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 $- 35$ by $4$ .

$- 140 - 5 \cdot - 7$

Step 5.2.2.1.2

Multiply $- 5$ by $- 7$ .

$- 140 + 35$

Step 5.2.2.2

Add $- 140$ and $35$ .

$- 105$

$D_{x} = - 105$

Step 5.3

Use the formula to solve for $x$ .

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

Step 5.4

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

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

Step 5.5

Dividing two negative values results in a positive value.

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

Step 6

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

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Step 6.1

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 6.2

Find the determinant.

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Step 6.2.1

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

$1 \cdot 5 - (- 3 \cdot - 35)$

Step 6.2.2

Simplify the determinant.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Multiply $5$ by $1$ .

$5 - (- 3 \cdot - 35)$

Step 6.2.2.1.2

Multiply $- (- 3 \cdot - 35)$ .

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Step 6.2.2.1.2.1

Multiply $- 3$ by $- 35$ .

$5 - 1 \cdot 105$

Step 6.2.2.1.2.2

Multiply $- 1$ by $105$ .

$5 - 105$

Step 6.2.2.2

Subtract $105$ from $5$ .

$- 100$

$D_{y} = - 100$

Step 6.3

Use the formula to solve for $y$ .

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

Step 6.4

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

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

Step 6.5

Dividing two negative values results in a positive value.

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

Step 7

List the solution to the system of equations.

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

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