Algebra Examples

[\begin{array}{c} x + 3 y \\ y \end{array}] = [\begin{array}{c} 17 \\ x - 4 \end{array}]

$[\begin{array}{c} x + 3 y \\ y \end{array}] = [\begin{array}{c} 17 \\ x - 4 \end{array}]$

Step 1

Write as a linear system of equations.

x + 3 y = 17

$x + 3 y = 17$

y = x - 4

$y = x - 4$

Step 2

Solve the system of equations.

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Step 2.1

Replace all occurrences of

y

$y$ with

x - 4

$x - 4$ in each equation.

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Step 2.1.1

Replace all occurrences of

y

$y$ in

x + 3 y = 17

$x + 3 y = 17$ with

x - 4

$x - 4$ .

x + 3 (x - 4) = 17

$x + 3 (x - 4) = 17$

y = x - 4

$y = x - 4$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify

x + 3 (x - 4)

$x + 3 (x - 4)$ .

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

Apply the distributive property.

x + 3 x + 3 \cdot - 4 = 17

$x + 3 x + 3 \cdot - 4 = 17$

y = x - 4

$y = x - 4$

Step 2.1.2.1.1.2

Multiply

3

$3$ by

- 4

$- 4$ .

x + 3 x - 12 = 17

$x + 3 x - 12 = 17$

y = x - 4

$y = x - 4$

x + 3 x - 12 = 17

$x + 3 x - 12 = 17$

y = x - 4

$y = x - 4$

Step 2.1.2.1.2

Add

x

$x$ and

3 x

$3 x$ .

4 x - 12 = 17

$4 x - 12 = 17$

y = x - 4

$y = x - 4$

4 x - 12 = 17

$4 x - 12 = 17$

y = x - 4

$y = x - 4$

4 x - 12 = 17

$4 x - 12 = 17$

y = x - 4

$y = x - 4$

4 x - 12 = 17

$4 x - 12 = 17$

y = x - 4

$y = x - 4$

Step 2.2

Solve for

x

$x$ in

4 x - 12 = 17

$4 x - 12 = 17$ .

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Step 2.2.1

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 2.2.1.1

Add

12

$12$ to both sides of the equation.

4 x = 17 + 12

$4 x = 17 + 12$

y = x - 4

$y = x - 4$

Step 2.2.1.2

Add

17

$17$ and

12

$12$ .

4 x = 29

$4 x = 29$

y = x - 4

$y = x - 4$

4 x = 29

$4 x = 29$

y = x - 4

$y = x - 4$

Step 2.2.2

Divide each term in

4 x = 29

$4 x = 29$ by

4

$4$ and simplify.

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Step 2.2.2.1

Divide each term in

4 x = 29

$4 x = 29$ by

4

$4$ .

\frac{4 x}{4} = \frac{29}{4}

$\frac{4 x}{4} = \frac{29}{4}$

y = x - 4

$y = x - 4$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of

4

$4$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

\frac{4 x}{4} = \frac{29}{4}

$\frac{4 x}{4} = \frac{29}{4}$

y = x - 4

$y = x - 4$

Step 2.2.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{29}{4}

$x = \frac{29}{4}$

y = x - 4

$y = x - 4$

x = \frac{29}{4}

$x = \frac{29}{4}$

y = x - 4

$y = x - 4$

x = \frac{29}{4}

$x = \frac{29}{4}$

y = x - 4

$y = x - 4$

x = \frac{29}{4}

$x = \frac{29}{4}$

y = x - 4

$y = x - 4$

x = \frac{29}{4}

$x = \frac{29}{4}$

y = x - 4

$y = x - 4$

Step 2.3

Replace all occurrences of

x

$x$ with

\frac{29}{4}

$\frac{29}{4}$ in each equation.

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Step 2.3.1

Replace all occurrences of

x

$x$ in

y = x - 4

$y = x - 4$ with

\frac{29}{4}

$\frac{29}{4}$ .

y = (\frac{29}{4}) - 4

$y = (\frac{29}{4}) - 4$

x = \frac{29}{4}

$x = \frac{29}{4}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify

(\frac{29}{4}) - 4

$(\frac{29}{4}) - 4$ .

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Step 2.3.2.1.1

To write

- 4

$- 4$ as a fraction with a common denominator, multiply by

\frac{4}{4}

$\frac{4}{4}$ .

y = \frac{29}{4} - 4 \cdot \frac{4}{4}

$y = \frac{29}{4} - 4 \cdot \frac{4}{4}$

x = \frac{29}{4}

$x = \frac{29}{4}$

Step 2.3.2.1.2

Combine

- 4

$- 4$ and

\frac{4}{4}

$\frac{4}{4}$ .

y = \frac{29}{4} + \frac{- 4 \cdot 4}{4}

$y = \frac{29}{4} + \frac{- 4 \cdot 4}{4}$

x = \frac{29}{4}

$x = \frac{29}{4}$

Step 2.3.2.1.3

Combine the numerators over the common denominator.

y = \frac{29 - 4 \cdot 4}{4}

$y = \frac{29 - 4 \cdot 4}{4}$

x = \frac{29}{4}

$x = \frac{29}{4}$

Step 2.3.2.1.4

Simplify the numerator.

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Step 2.3.2.1.4.1

Multiply

- 4

$- 4$ by

4

$4$ .

y = \frac{29 - 16}{4}

$y = \frac{29 - 16}{4}$

x = \frac{29}{4}

$x = \frac{29}{4}$

Step 2.3.2.1.4.2

Subtract

16

$16$ from

29

$29$ .

y = \frac{13}{4}

$y = \frac{13}{4}$

x = \frac{29}{4}

$x = \frac{29}{4}$

y = \frac{13}{4}

$y = \frac{13}{4}$

x = \frac{29}{4}

$x = \frac{29}{4}$

y = \frac{13}{4}

$y = \frac{13}{4}$

x = \frac{29}{4}

$x = \frac{29}{4}$

y = \frac{13}{4}

$y = \frac{13}{4}$

x = \frac{29}{4}

$x = \frac{29}{4}$

y = \frac{13}{4}

$y = \frac{13}{4}$

x = \frac{29}{4}

$x = \frac{29}{4}$

Step 2.4

List all of the solutions.

y = \frac{13}{4}, x = \frac{29}{4}

$y = \frac{13}{4}, x = \frac{29}{4}$

y = \frac{13}{4}, x = \frac{29}{4}

$y = \frac{13}{4}, x = \frac{29}{4}$