Linear Algebra Examples

$y = - x + 3$ , $y = 2 x - 1$

Step 1

Eliminate the equal sides of each equation and combine.

$- x + 3 = 2 x - 1$

Step 2

Solve $- x + 3 = 2 x - 1$ for $x$ .

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Step 2.1

Move all terms containing $x$ to the left side of the equation.

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Step 2.1.1

Subtract $2 x$ from both sides of the equation.

$- x + 3 - 2 x = - 1$

Step 2.1.2

Subtract $2 x$ from $- x$ .

$- 3 x + 3 = - 1$

Step 2.2

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

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Step 2.2.1

Subtract $3$ from both sides of the equation.

$- 3 x = - 1 - 3$

Step 2.2.2

Subtract $3$ from $- 1$ .

$- 3 x = - 4$

Step 2.3

Divide each term in $- 3 x = - 4$ by $- 3$ and simplify.

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Step 2.3.1

Divide each term in $- 3 x = - 4$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 4}{- 3}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 3$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 4}{- 3}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 3}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{4}{3}$

Step 3

Evaluate $y$ when $x = \frac{4}{3}$ .

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Step 3.1

Substitute $\frac{4}{3}$ for $x$ .

$y = 2 (\frac{4}{3}) - 1$

Step 3.2

Simplify $2 (\frac{4}{3}) - 1$ .

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Step 3.2.1

Multiply $2 (\frac{4}{3})$ .

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Step 3.2.1.1

Combine $2$ and $\frac{4}{3}$ .

$y = \frac{2 \cdot 4}{3} - 1$

Step 3.2.1.2

Multiply $2$ by $4$ .

$y = \frac{8}{3} - 1$

Step 3.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{8}{3} - 1 \cdot \frac{3}{3}$

Step 3.2.3

Combine $- 1$ and $\frac{3}{3}$ .

$y = \frac{8}{3} + \frac{- 1 \cdot 3}{3}$

Step 3.2.4

Combine the numerators over the common denominator.

$y = \frac{8 - 1 \cdot 3}{3}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $- 1$ by $3$ .

$y = \frac{8 - 3}{3}$

Step 3.2.5.2

Subtract $3$ from $8$ .

$y = \frac{5}{3}$

Step 4

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

$(\frac{4}{3}, \frac{5}{3})$

Step 5

The result can be shown in multiple forms.

Point Form:

$(\frac{4}{3}, \frac{5}{3})$

Equation Form:

$x = \frac{4}{3}, y = \frac{5}{3}$

Step 6