Linear Algebra Examples

$3 x - y = 9$ , $- 6 x + 2 y = - 18$

Step 1

Write the system of equations in matrix form.

$[\begin{array}{c} 3 & - 1 & 9 \\ - 6 & 2 & - 18 \end{array}]$

Step 2

Row reduce to eliminate one of the variables.

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Step 2.1

Multiply each element of $R_{1}$ by $\frac{1}{3}$ to make the entry at $1, 1$ a $1$ .

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Step 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{3}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{3}{3} & \frac{- 1}{3} & \frac{9}{3} \\ - 6 & 2 & - 18 \end{array}]$

Step 2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{1}{3} & 3 \\ - 6 & 2 & - 18 \end{array}]$

Step 2.2

Perform the row operation $R_{2} = R_{2} + 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.2.1

Perform the row operation $R_{2} = R_{2} + 6 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{1}{3} & 3 \\ - 6 + 6 \cdot 1 & 2 + 6 (- \frac{1}{3}) & - 18 + 6 \cdot 3 \end{array}]$

Step 2.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{1}{3} & 3 \\ 0 & 0 & 0 \end{array}]$

Step 3

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

$x - \frac{y}{3} = 3$

Step 4

Solve the equation for $y$ .

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Step 4.1

Subtract $x$ from both sides of the equation.

$- \frac{y}{3} = 3 - x$

Step 4.2

Multiply both sides of the equation by $- 3$ .

$- 3 (- \frac{y}{3}) = - 3 (3 - x)$

Step 4.3

Simplify both sides of the equation.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Simplify $- 3 (- \frac{y}{3})$ .

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Step 4.3.1.1.1

Cancel the common factor of $3$ .

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Step 4.3.1.1.1.1

Move the leading negative in $- \frac{y}{3}$ into the numerator.

$- 3 \frac{- y}{3} = - 3 (3 - x)$

Step 4.3.1.1.1.2

Factor $3$ out of $- 3$ .

$3 (- 1) \frac{- y}{3} = - 3 (3 - x)$

Step 4.3.1.1.1.3

Cancel the common factor.

$3 \cdot - 1 \frac{- y}{3} = - 3 (3 - x)$

Step 4.3.1.1.1.4

Rewrite the expression.

$- - y = - 3 (3 - x)$

Step 4.3.1.1.2

Multiply.

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Step 4.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 y = - 3 (3 - x)$

Step 4.3.1.1.2.2

Multiply $y$ by $1$ .

$y = - 3 (3 - x)$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Simplify $- 3 (3 - x)$ .

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Step 4.3.2.1.1

Apply the distributive property.

$y = - 3 \cdot 3 - 3 (- x)$

Step 4.3.2.1.2

Multiply.

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Step 4.3.2.1.2.1

Multiply $- 3$ by $3$ .

$y = - 9 - 3 (- x)$

Step 4.3.2.1.2.2

Multiply $- 1$ by $- 3$ .

$y = - 9 + 3 x$

Step 4.4

Reorder $- 9$ and $3 x$ .

$y = 3 x - 9$

Step 5

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

$(x, 3 x - 9)$

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.

$X = [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} x \\ 3 x - 9 \end{array}]$