Algebra Examples

$2 x + y = 5$ $x - 3 y = 2$

Step 1

Graph $2 x + y = 5$ .

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Step 1.1

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

$y = 5 - 2 x$

Step 1.2

Rewrite in slope-intercept form.

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Step 1.2.1

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 1.2.2

Reorder $5$ and $- 2 x$ .

$y = - 2 x + 5$

Step 1.3

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

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Step 1.3.1

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

$m = - 2$

$b = 5$

Step 1.3.2

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

Slope: $- 2$

y-intercept: $(0, 5)$

Slope: $- 2$

y-intercept: $(0, 5)$

Step 1.4

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 1.4.1

Reorder $5$ and $- 2 x$ .

$y = - 2 x + 5$

Step 1.4.2

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

$\begin{array}{c} x & y \\ 0 & 5 \\ 1 & 3 \end{array}$

Step 1.5

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

Slope: $- 2$

y-intercept: $(0, 5)$

$\begin{array}{c} x & y \\ 0 & 5 \\ 1 & 3 \end{array}$

Slope: $- 2$

y-intercept: $(0, 5)$

$\begin{array}{c} x & y \\ 0 & 5 \\ 1 & 3 \end{array}$

Step 2

Graph $x - 3 y = 2$ .

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Step 2.1

Solve for $y$ .

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Step 2.1.1

Subtract $x$ from both sides of the equation.

$- 3 y = 2 - x$

Step 2.1.2

Divide each term in $- 3 y = 2 - x$ by $- 3$ and simplify.

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Step 2.1.2.1

Divide each term in $- 3 y = 2 - x$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{2}{- 3} + \frac{- x}{- 3}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $- 3$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{2}{- 3} + \frac{- x}{- 3}$

Step 2.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{2}{- 3} + \frac{- x}{- 3}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{2}{3} + \frac{- x}{- 3}$

Step 2.1.2.3.1.2

Dividing two negative values results in a positive value.

$y = - \frac{2}{3} + \frac{x}{3}$

Step 2.2

Rewrite in slope-intercept form.

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Step 2.2.1

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 2.2.2

Reorder $- \frac{2}{3}$ and $\frac{x}{3}$ .

$y = \frac{x}{3} - \frac{2}{3}$

Step 2.2.3

Reorder terms.

$y = \frac{1}{3} x - \frac{2}{3}$

Step 2.3

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

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Step 2.3.1

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

$m = \frac{1}{3}$

$b = - \frac{2}{3}$

Step 2.3.2

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

Slope: $\frac{1}{3}$

y-intercept: $(0, - \frac{2}{3})$

Slope: $\frac{1}{3}$

y-intercept: $(0, - \frac{2}{3})$

Step 2.4

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 2.4.1

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

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Step 2.4.1.1

Reorder $- \frac{2}{3}$ and $\frac{x}{3}$ .

$y = \frac{x}{3} - \frac{2}{3}$

Step 2.4.1.2

Reorder terms.

$y = \frac{1}{3} x - \frac{2}{3}$

Step 2.4.2

Find the x-intercept.

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Step 2.4.2.1

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

$0 = \frac{1}{3} x - \frac{2}{3}$

Step 2.4.2.2

Solve the equation.

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Step 2.4.2.2.1

Rewrite the equation as $\frac{1}{3} x - \frac{2}{3} = 0$ .

$\frac{1}{3} x - \frac{2}{3} = 0$

Step 2.4.2.2.2

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

$\frac{x}{3} - \frac{2}{3} = 0$

Step 2.4.2.2.3

Add $\frac{2}{3}$ to both sides of the equation.

$\frac{x}{3} = \frac{2}{3}$

Step 2.4.2.2.4

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

$x = 2$

Step 2.4.2.3

x-intercept(s) in point form.

x-intercept(s): $(2, 0)$

Step 2.4.3

Find the y-intercept.

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Step 2.4.3.1

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

$y = \frac{1}{3} \cdot (0) - \frac{2}{3}$

Step 2.4.3.2

Solve the equation.

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Step 2.4.3.2.1

Multiply $\frac{1}{3}$ by $0$ .

$y = \frac{1}{3} \cdot 0 - \frac{2}{3}$

Step 2.4.3.2.2

Remove parentheses.

$y = \frac{1}{3} \cdot (0) - \frac{2}{3}$

Step 2.4.3.2.3

Simplify $\frac{1}{3} \cdot (0) - \frac{2}{3}$ .

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Step 2.4.3.2.3.1

Multiply $\frac{1}{3}$ by $0$ .

$y = 0 - \frac{2}{3}$

Step 2.4.3.2.3.2

Subtract $\frac{2}{3}$ from $0$ .

$y = - \frac{2}{3}$

Step 2.4.3.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{2}{3})$

Step 2.4.4

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

$\begin{array}{c} x & y \\ 0 & - \frac{2}{3} \\ 2 & 0 \end{array}$

Step 2.5

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

Slope: $\frac{1}{3}$

y-intercept: $(0, - \frac{2}{3})$

$\begin{array}{c} x & y \\ 0 & - \frac{2}{3} \\ 2 & 0 \end{array}$

Slope: $\frac{1}{3}$

y-intercept: $(0, - \frac{2}{3})$

$\begin{array}{c} x & y \\ 0 & - \frac{2}{3} \\ 2 & 0 \end{array}$

Step 3

Plot each graph on the same coordinate system.

$2 x + y = 5$

$x - 3 y = 2$

Step 4