Linear Algebra Examples

$- 11 = x - 2 y$

Step 1

Solve for $y$ .

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Step 1.1

Rewrite the equation as $x - 2 y = - 11$ .

$x - 2 y = - 11$

Step 1.2

Subtract $x$ from both sides of the equation.

$- 2 y = - 11 - x$

Step 1.3

Divide each term in $- 2 y = - 11 - x$ by $- 2$ and simplify.

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Step 1.3.1

Divide each term in $- 2 y = - 11 - x$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{- 11}{- 2} + \frac{- x}{- 2}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $- 2$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{- 11}{- 2} + \frac{- x}{- 2}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 11}{- 2} + \frac{- x}{- 2}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{11}{2} + \frac{- x}{- 2}$

Step 1.3.3.1.2

Dividing two negative values results in a positive value.

$y = \frac{11}{2} + \frac{x}{2}$

Step 2

Rewrite in slope-intercept form.

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Step 2.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 2.2

Reorder $\frac{11}{2}$ and $\frac{x}{2}$ .

$y = \frac{x}{2} + \frac{11}{2}$

Step 2.3

Reorder terms.

$y = \frac{1}{2} x + \frac{11}{2}$

Step 3

Use the slope-intercept form to find the slope and y-intercept.

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Step 3.1

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m = \frac{1}{2}$

$b = \frac{11}{2}$

Step 3.2

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope: $\frac{1}{2}$

y-intercept: $(0, \frac{11}{2})$

Slope: $\frac{1}{2}$

y-intercept: $(0, \frac{11}{2})$

Step 4

Any line can be graphed using two points. Select two $x$ values, and plug them into the equation to find the corresponding $y$ values.

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Step 4.1

Write in $y = m x + b$ form.

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Step 4.1.1

Reorder $\frac{11}{2}$ and $\frac{x}{2}$ .

$y = \frac{x}{2} + \frac{11}{2}$

Step 4.1.2

Reorder terms.

$y = \frac{1}{2} x + \frac{11}{2}$

Step 4.2

Find the x-intercept.

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Step 4.2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{1}{2} x + \frac{11}{2}$

Step 4.2.2

Solve the equation.

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Step 4.2.2.1

Rewrite the equation as $\frac{1}{2} x + \frac{11}{2} = 0$ .

$\frac{1}{2} x + \frac{11}{2} = 0$

Step 4.2.2.2

Combine $\frac{1}{2}$ and $x$ .

$\frac{x}{2} + \frac{11}{2} = 0$

Step 4.2.2.3

Subtract $\frac{11}{2}$ from both sides of the equation.

$\frac{x}{2} = - \frac{11}{2}$

Step 4.2.2.4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = - 11$

Step 4.2.3

x-intercept(s) in point form.

x-intercept(s): $(- 11, 0)$

Step 4.3

Find the y-intercept.

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Step 4.3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{1}{2} \cdot (0) + \frac{11}{2}$

Step 4.3.2

Solve the equation.

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Step 4.3.2.1

Multiply $\frac{1}{2}$ by $0$ .

$y = \frac{1}{2} \cdot 0 + \frac{11}{2}$

Step 4.3.2.2

Remove parentheses.

$y = \frac{1}{2} \cdot (0) + \frac{11}{2}$

Step 4.3.2.3

Simplify $\frac{1}{2} \cdot (0) + \frac{11}{2}$ .

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Step 4.3.2.3.1

Multiply $\frac{1}{2}$ by $0$ .

$y = 0 + \frac{11}{2}$

Step 4.3.2.3.2

Add $0$ and $\frac{11}{2}$ .

$y = \frac{11}{2}$

Step 4.3.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{11}{2})$

Step 4.4

Create a table of the $x$ and $y$ values.

$\begin{array}{c} x & y \\ - 11 & 0 \\ 0 & \frac{11}{2} \end{array}$

Step 5

Graph the line using the slope and the y-intercept, or the points.

Slope: $\frac{1}{2}$

y-intercept: $(0, \frac{11}{2})$

$\begin{array}{c} x & y \\ - 11 & 0 \\ 0 & \frac{11}{2} \end{array}$

Step 6