Finite Math Examples

$- x + y = 4$ , $13 x + 3 y = 68$

Step 1

Add $x$ to both sides of the equation.

$y = 4 + x$

$13 x + 3 y = 68$

Step 2

Replace all occurrences of $y$ with $4 + x$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $13 x + 3 y = 68$ with $4 + x$ .

$13 x + 3 (4 + x) = 68$

$y = 4 + x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $13 x + 3 (4 + x)$ .

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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.

$13 x + 3 \cdot 4 + 3 x = 68$

$y = 4 + x$

Step 2.2.1.1.2

Multiply $3$ by $4$ .

$13 x + 12 + 3 x = 68$

$y = 4 + x$

$13 x + 12 + 3 x = 68$

$y = 4 + x$

Step 2.2.1.2

Add $13 x$ and $3 x$ .

$16 x + 12 = 68$

$y = 4 + x$

$16 x + 12 = 68$

$y = 4 + x$

$16 x + 12 = 68$

$y = 4 + x$

$16 x + 12 = 68$

$y = 4 + x$

Step 3

Solve for $x$ in $16 x + 12 = 68$ .

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Step 3.1

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

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Step 3.1.1

Subtract $12$ from both sides of the equation.

$16 x = 68 - 12$

$y = 4 + x$

Step 3.1.2

Subtract $12$ from $68$ .

$16 x = 56$

$y = 4 + x$

$16 x = 56$

$y = 4 + x$

Step 3.2

Divide each term in $16 x = 56$ by $16$ and simplify.

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Step 3.2.1

Divide each term in $16 x = 56$ by $16$ .

$\frac{16 x}{16} = \frac{56}{16}$

$y = 4 + x$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $16$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{16 x}{16} = \frac{56}{16}$

$y = 4 + x$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{56}{16}$

$y = 4 + x$

$x = \frac{56}{16}$

$y = 4 + x$

$x = \frac{56}{16}$

$y = 4 + x$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $56$ and $16$ .

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Step 3.2.3.1.1

Factor $8$ out of $56$ .

$x = \frac{8 (7)}{16}$

$y = 4 + x$

Step 3.2.3.1.2

Cancel the common factors.

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Step 3.2.3.1.2.1

Factor $8$ out of $16$ .

$x = \frac{8 \cdot 7}{8 \cdot 2}$

$y = 4 + x$

Step 3.2.3.1.2.2

Cancel the common factor.

$x = \frac{8 \cdot 7}{8 \cdot 2}$

$y = 4 + x$

Step 3.2.3.1.2.3

Rewrite the expression.

$x = \frac{7}{2}$

$y = 4 + x$

$x = \frac{7}{2}$

$y = 4 + x$

$x = \frac{7}{2}$

$y = 4 + x$

$x = \frac{7}{2}$

$y = 4 + x$

$x = \frac{7}{2}$

$y = 4 + x$

$x = \frac{7}{2}$

$y = 4 + x$

Step 4

Replace all occurrences of $x$ with $\frac{7}{2}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = 4 + x$ with $\frac{7}{2}$ .

$y = 4 + \frac{7}{2}$

$x = \frac{7}{2}$

Step 4.2

Simplify $y = 4 + \frac{7}{2}$ .

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Remove parentheses.

$y = 4 + \frac{7}{2}$

$x = \frac{7}{2}$

$y = 4 + \frac{7}{2}$

$x = \frac{7}{2}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $4 + \frac{7}{2}$ .

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Step 4.2.2.1.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = 4 \cdot \frac{2}{2} + \frac{7}{2}$

$x = \frac{7}{2}$

Step 4.2.2.1.2

Combine $4$ and $\frac{2}{2}$ .

$y = \frac{4 \cdot 2}{2} + \frac{7}{2}$

$x = \frac{7}{2}$

Step 4.2.2.1.3

Combine the numerators over the common denominator.

$y = \frac{4 \cdot 2 + 7}{2}$

$x = \frac{7}{2}$

Step 4.2.2.1.4

Simplify the numerator.

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Step 4.2.2.1.4.1

Multiply $4$ by $2$ .

$y = \frac{8 + 7}{2}$

$x = \frac{7}{2}$

Step 4.2.2.1.4.2

Add $8$ and $7$ .

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

$x = \frac{7}{2}$

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

$x = \frac{7}{2}$

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

$x = \frac{7}{2}$

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

$x = \frac{7}{2}$

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

$x = \frac{7}{2}$

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

$x = \frac{7}{2}$

Step 5

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

$(\frac{7}{2}, \frac{15}{2})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{7}{2}, \frac{15}{2})$

Equation Form:

$x = \frac{7}{2}, y = \frac{15}{2}$

Step 7