Finite Math Examples

$x + 7 y = 18$ , $3 x - 5 y = 15$

Step 1

Subtract $7 y$ from both sides of the equation.

$x = 18 - 7 y$

$3 x - 5 y = 15$

Step 2

Replace all occurrences of $x$ with $18 - 7 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x - 5 y = 15$ with $18 - 7 y$ .

$3 (18 - 7 y) - 5 y = 15$

$x = 18 - 7 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (18 - 7 y) - 5 y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$3 \cdot 18 + 3 (- 7 y) - 5 y = 15$

$x = 18 - 7 y$

Step 2.2.1.1.2

Multiply $3$ by $18$ .

$54 + 3 (- 7 y) - 5 y = 15$

$x = 18 - 7 y$

Step 2.2.1.1.3

Multiply $- 7$ by $3$ .

$54 - 21 y - 5 y = 15$

$x = 18 - 7 y$

$54 - 21 y - 5 y = 15$

$x = 18 - 7 y$

Step 2.2.1.2

Subtract $5 y$ from $- 21 y$ .

$54 - 26 y = 15$

$x = 18 - 7 y$

$54 - 26 y = 15$

$x = 18 - 7 y$

$54 - 26 y = 15$

$x = 18 - 7 y$

$54 - 26 y = 15$

$x = 18 - 7 y$

Step 3

Solve for $y$ in $54 - 26 y = 15$ .

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Step 3.1

Move all terms not containing $y$ to the right side of the equation.

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Step 3.1.1

Subtract $54$ from both sides of the equation.

$- 26 y = 15 - 54$

$x = 18 - 7 y$

Step 3.1.2

Subtract $54$ from $15$ .

$- 26 y = - 39$

$x = 18 - 7 y$

$- 26 y = - 39$

$x = 18 - 7 y$

Step 3.2

Divide each term in $- 26 y = - 39$ by $- 26$ and simplify.

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Step 3.2.1

Divide each term in $- 26 y = - 39$ by $- 26$ .

$\frac{- 26 y}{- 26} = \frac{- 39}{- 26}$

$x = 18 - 7 y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 26$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 26 y}{- 26} = \frac{- 39}{- 26}$

$x = 18 - 7 y$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 39}{- 26}$

$x = 18 - 7 y$

$y = \frac{- 39}{- 26}$

$x = 18 - 7 y$

$y = \frac{- 39}{- 26}$

$x = 18 - 7 y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $- 39$ and $- 26$ .

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Step 3.2.3.1.1

Factor $- 13$ out of $- 39$ .

$y = \frac{- 13 \cdot 3}{- 26}$

$x = 18 - 7 y$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $- 13$ out of $- 26$ .

$y = \frac{- 13 \cdot 3}{- 13 \cdot 2}$

$x = 18 - 7 y$

Step 3.2.3.1.2.2

Cancel the common factor.

$y = \frac{- 13 \cdot 3}{- 13 \cdot 2}$

$x = 18 - 7 y$

Step 3.2.3.1.2.3

Rewrite the expression.

$y = \frac{3}{2}$

$x = 18 - 7 y$

$y = \frac{3}{2}$

$x = 18 - 7 y$

$y = \frac{3}{2}$

$x = 18 - 7 y$

$y = \frac{3}{2}$

$x = 18 - 7 y$

$y = \frac{3}{2}$

$x = 18 - 7 y$

$y = \frac{3}{2}$

$x = 18 - 7 y$

Step 4

Replace all occurrences of $y$ with $\frac{3}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = 18 - 7 y$ with $\frac{3}{2}$ .

$x = 18 - 7 (\frac{3}{2})$

$y = \frac{3}{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $18 - 7 (\frac{3}{2})$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Multiply $- 7 (\frac{3}{2})$ .

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Step 4.2.1.1.1.1

Combine $- 7$ and $\frac{3}{2}$ .

$x = 18 + \frac{- 7 \cdot 3}{2}$

$y = \frac{3}{2}$

Step 4.2.1.1.1.2

Multiply $- 7$ by $3$ .

$x = 18 + \frac{- 21}{2}$

$y = \frac{3}{2}$

$x = 18 + \frac{- 21}{2}$

$y = \frac{3}{2}$

Step 4.2.1.1.2

Move the negative in front of the fraction.

$x = 18 - \frac{21}{2}$

$y = \frac{3}{2}$

$x = 18 - \frac{21}{2}$

$y = \frac{3}{2}$

Step 4.2.1.2

To write $18$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 18 \cdot \frac{2}{2} - \frac{21}{2}$

$y = \frac{3}{2}$

Step 4.2.1.3

Combine $18$ and $\frac{2}{2}$ .

$x = \frac{18 \cdot 2}{2} - \frac{21}{2}$

$y = \frac{3}{2}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{18 \cdot 2 - 21}{2}$

$y = \frac{3}{2}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $18$ by $2$ .

$x = \frac{36 - 21}{2}$

$y = \frac{3}{2}$

Step 4.2.1.5.2

Subtract $21$ from $36$ .

$x = \frac{15}{2}$

$y = \frac{3}{2}$

$x = \frac{15}{2}$

$y = \frac{3}{2}$

$x = \frac{15}{2}$

$y = \frac{3}{2}$

$x = \frac{15}{2}$

$y = \frac{3}{2}$

$x = \frac{15}{2}$

$y = \frac{3}{2}$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(\frac{15}{2}, \frac{3}{2})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{15}{2}, \frac{3}{2})$

Equation Form:

$x = \frac{15}{2}, y = \frac{3}{2}$

Step 7