Finite Math Examples

$5 x - 6 y = 9$ , $10 x - 12 y = 18$

Step 1

Solve for $x$ in $5 x - 6 y = 9$ .

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Step 1.1

Add $6 y$ to both sides of the equation.

$5 x = 9 + 6 y$

$10 x - 12 y = 18$

Step 1.2

Divide each term in $5 x = 9 + 6 y$ by $5$ and simplify.

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Step 1.2.1

Divide each term in $5 x = 9 + 6 y$ by $5$ .

$\frac{5 x}{5} = \frac{9}{5} + \frac{6 y}{5}$

$10 x - 12 y = 18$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{9}{5} + \frac{6 y}{5}$

$10 x - 12 y = 18$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{5} + \frac{6 y}{5}$

$10 x - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

$10 x - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

$10 x - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

$10 x - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

$10 x - 12 y = 18$

Step 2

Replace all occurrences of $x$ with $\frac{9}{5} + \frac{6 y}{5}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $10 x - 12 y = 18$ with $\frac{9}{5} + \frac{6 y}{5}$ .

$10 (\frac{9}{5} + \frac{6 y}{5}) - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $10 (\frac{9}{5} + \frac{6 y}{5}) - 12 y$ .

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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.

$10 (\frac{9}{5}) + 10 (\frac{6 y}{5}) - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

Step 2.2.1.1.2

Cancel the common factor of $5$ .

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Step 2.2.1.1.2.1

Factor $5$ out of $10$ .

$5 (2) (\frac{9}{5}) + 10 (\frac{6 y}{5}) - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

Step 2.2.1.1.2.2

Cancel the common factor.

$5 \cdot (2 (\frac{9}{5})) + 10 (\frac{6 y}{5}) - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

Step 2.2.1.1.2.3

Rewrite the expression.

$2 \cdot 9 + 10 (\frac{6 y}{5}) - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

$2 \cdot 9 + 10 (\frac{6 y}{5}) - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

Step 2.2.1.1.3

Multiply $2$ by $9$ .

$18 + 10 (\frac{6 y}{5}) - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

Step 2.2.1.1.4

Cancel the common factor of $5$ .

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Step 2.2.1.1.4.1

Factor $5$ out of $10$ .

$18 + 5 (2) (\frac{6 y}{5}) - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

Step 2.2.1.1.4.2

Cancel the common factor.

$18 + 5 \cdot (2 (\frac{6 y}{5})) - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

Step 2.2.1.1.4.3

Rewrite the expression.

$18 + 2 (6 y) - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

$18 + 2 (6 y) - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

Step 2.2.1.1.5

Multiply $6$ by $2$ .

$18 + 12 y - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

$18 + 12 y - 12 y = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

Step 2.2.1.2

Combine the opposite terms in $18 + 12 y - 12 y$ .

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Step 2.2.1.2.1

Subtract $12 y$ from $12 y$ .

$18 + 0 = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

Step 2.2.1.2.2

Add $18$ and $0$ .

$18 = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

$18 = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

$18 = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

$18 = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

$18 = 18$

$x = \frac{9}{5} + \frac{6 y}{5}$

Step 3

Remove any equations from the system that are always true.

$x = \frac{9}{5} + \frac{6 y}{5}$

Step 4