Algebra Examples

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

Step 1

Write as a linear system of equations.

$x + 3 y = 17$

$y = x - 4$

Step 2

Solve the system of equations.

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Step 2.1

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

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Step 2.1.1

Replace all occurrences of $y$ in $x + 3 y = 17$ with $x - 4$ .

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

$y = x - 4$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

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

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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.

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

$y = x - 4$

Step 2.1.2.1.1.2

Multiply $3$ by $- 4$ .

$x + 3 x - 12 = 17$

$y = x - 4$

$x + 3 x - 12 = 17$

$y = x - 4$

Step 2.1.2.1.2

Add $x$ and $3 x$ .

$4 x - 12 = 17$

$y = x - 4$

$4 x - 12 = 17$

$y = x - 4$

$4 x - 12 = 17$

$y = x - 4$

$4 x - 12 = 17$

$y = x - 4$

Step 2.2

Solve for $x$ in $4 x - 12 = 17$ .

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Step 2.2.1

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

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Step 2.2.1.1

Add $12$ to both sides of the equation.

$4 x = 17 + 12$

$y = x - 4$

Step 2.2.1.2

Add $17$ and $12$ .

$4 x = 29$

$y = x - 4$

$4 x = 29$

$y = x - 4$

Step 2.2.2

Divide each term in $4 x = 29$ by $4$ and simplify.

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Step 2.2.2.1

Divide each term in $4 x = 29$ by $4$ .

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

$y = x - 4$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

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

$y = x - 4$

Step 2.2.2.2.1.2

Divide $x$ by $1$ .

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

$y = x - 4$

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

$y = x - 4$

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

$y = x - 4$

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

$y = x - 4$

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

$y = x - 4$

Step 2.3

Replace all occurrences of $x$ with $\frac{29}{4}$ in each equation.

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Step 2.3.1

Replace all occurrences of $x$ in $y = x - 4$ with $\frac{29}{4}$ .

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

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

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

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

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

Step 2.3.2.1.2

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

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

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

Step 2.3.2.1.3

Combine the numerators over the common denominator.

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

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

Step 2.3.2.1.4

Simplify the numerator.

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Step 2.3.2.1.4.1

Multiply $- 4$ by $4$ .

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

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

Step 2.3.2.1.4.2

Subtract $16$ from $29$ .

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

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

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

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

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

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

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

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

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

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

Step 2.4

List all of the solutions.

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