Linear Algebra Examples

$2 x - 8 y = 4$ , $x - 4 y = 2$

Step 1

Write the system of equations in matrix form.

$[\begin{array}{c} 2 & - 8 & 4 \\ 1 & - 4 & 2 \end{array}]$

Step 2

Row reduce to eliminate one of the variables.

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Step 2.1

Multiply each element of $R_{1}$ by $\frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

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Step 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{2}{2} & \frac{- 8}{2} & \frac{4}{2} \\ 1 & - 4 & 2 \end{array}]$

Step 2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 1 & - 4 & 2 \end{array}]$

Step 2.2

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.2.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 1 - 1 & - 4 + 4 & 2 - 2 \end{array}]$

Step 2.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 4 & 2 \\ 0 & 0 & 0 \end{array}]$

Step 3

Use the result matrix to declare the final solutions to the system of equations.

$x - 4 y = 2$

Step 4

Solve the equation for $y$ .

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Step 4.1

Subtract $x$ from both sides of the equation.

$- 4 y = 2 - x$

Step 4.2

Divide each term in $- 4 y = 2 - x$ by $- 4$ and simplify.

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Step 4.2.1

Divide each term in $- 4 y = 2 - x$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{2}{- 4} + \frac{- x}{- 4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{2}{- 4} + \frac{- x}{- 4}$

Step 4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{2}{- 4} + \frac{- x}{- 4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Cancel the common factor of $2$ and $- 4$ .

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Step 4.2.3.1.1.1

Factor $2$ out of $2$ .

$y = \frac{2 (1)}{- 4} + \frac{- x}{- 4}$

Step 4.2.3.1.1.2

Cancel the common factors.

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Step 4.2.3.1.1.2.1

Factor $2$ out of $- 4$ .

$y = \frac{2 \cdot 1}{2 \cdot - 2} + \frac{- x}{- 4}$

Step 4.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot 1}{2 \cdot - 2} + \frac{- x}{- 4}$

Step 4.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{1}{- 2} + \frac{- x}{- 4}$

Step 4.2.3.1.2

Move the negative in front of the fraction.

$y = - \frac{1}{2} + \frac{- x}{- 4}$

Step 4.2.3.1.3

Dividing two negative values results in a positive value.

$y = - \frac{1}{2} + \frac{x}{4}$

Step 5

The solution is the set of ordered pairs that makes the system true.

$(x, - \frac{1}{2} + \frac{x}{4})$

Step 6

Decompose a solution vector by re-arranging each equation represented in the row-reduced form of the augmented matrix by solving for the dependent variable in each row yields the vector equality.

$X = [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} x \\ - \frac{1}{2} + \frac{x}{4} \end{array}]$