Algebra Examples

$2 x - y = 5$ $x + 4 y = 7$

Step 1

Graph $2 x - y = 5$ .

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Step 1.1

Solve for $y$ .

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Step 1.1.1

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

$- y = 5 - 2 x$

Step 1.1.2

Divide each term in $- y = 5 - 2 x$ by $- 1$ and simplify.

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Step 1.1.2.1

Divide each term in $- y = 5 - 2 x$ by $- 1$ .

$\frac{- y}{- 1} = \frac{5}{- 1} + \frac{- 2 x}{- 1}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{5}{- 1} + \frac{- 2 x}{- 1}$

Step 1.1.2.2.2

Divide $y$ by $1$ .

$y = \frac{5}{- 1} + \frac{- 2 x}{- 1}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Divide $5$ by $- 1$ .

$y = - 5 + \frac{- 2 x}{- 1}$

Step 1.1.2.3.1.2

Move the negative one from the denominator of $\frac{- 2 x}{- 1}$ .

$y = - 5 - 1 \cdot (- 2 x)$

Step 1.1.2.3.1.3

Rewrite $- 1 \cdot (- 2 x)$ as $- (- 2 x)$ .

$y = - 5 - (- 2 x)$

Step 1.1.2.3.1.4

Multiply $- 2$ by $- 1$ .

$y = - 5 + 2 x$

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 $x + 4 y = 7$ .

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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.

$4 y = 7 - x$

Step 2.1.2

Divide each term in $4 y = 7 - x$ by $4$ and simplify.

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Step 2.1.2.1

Divide each term in $4 y = 7 - x$ by $4$ .

$\frac{4 y}{4} = \frac{7}{4} + \frac{- x}{4}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $4$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{7}{4} + \frac{- x}{4}$

Step 2.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{7}{4} + \frac{- x}{4}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Move the negative in front of the fraction.

$y = \frac{7}{4} - \frac{x}{4}$

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 $\frac{7}{4}$ and $- \frac{x}{4}$ .

$y = - \frac{x}{4} + \frac{7}{4}$

Step 2.2.3

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

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Step 2.2.3.1

Reorder terms.

$y = - (\frac{1}{4} x) + \frac{7}{4}$

Step 2.2.3.2

Remove parentheses.

$y = - \frac{1}{4} x + \frac{7}{4}$

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 $y = m x + b$ .

$m = - \frac{1}{4}$

$b = \frac{7}{4}$

Step 2.3.2

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

Slope: $- \frac{1}{4}$

y-intercept: $(0, \frac{7}{4})$

Slope: $- \frac{1}{4}$

y-intercept: $(0, \frac{7}{4})$

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 $\frac{7}{4}$ and $- \frac{x}{4}$ .

$y = - \frac{x}{4} + \frac{7}{4}$

Step 2.4.1.2

Reorder terms.

$y = - (\frac{1}{4} x) + \frac{7}{4}$

Step 2.4.1.3

Remove parentheses.

$y = - \frac{1}{4} x + \frac{7}{4}$

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 $x$ .

$0 = - \frac{1}{4} x + \frac{7}{4}$

Step 2.4.2.2

Solve the equation.

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Step 2.4.2.2.1

Rewrite the equation as $- \frac{1}{4} x + \frac{7}{4} = 0$ .

$- \frac{1}{4} x + \frac{7}{4} = 0$

Step 2.4.2.2.2

Combine $x$ and $\frac{1}{4}$ .

$- \frac{x}{4} + \frac{7}{4} = 0$

Step 2.4.2.2.3

Subtract $\frac{7}{4}$ from both sides of the equation.

$- \frac{x}{4} = - \frac{7}{4}$

Step 2.4.2.2.4

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

$- x = - 7$

Step 2.4.2.2.5

Divide each term in $- x = - 7$ by $- 1$ and simplify.

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Step 2.4.2.2.5.1

Divide each term in $- x = - 7$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 7}{- 1}$

Step 2.4.2.2.5.2

Simplify the left side.

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Step 2.4.2.2.5.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 7}{- 1}$

Step 2.4.2.2.5.2.2

Divide $x$ by $1$ .

$x = \frac{- 7}{- 1}$

Step 2.4.2.2.5.3

Simplify the right side.

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Step 2.4.2.2.5.3.1

Divide $- 7$ by $- 1$ .

$x = 7$

Step 2.4.2.3

x-intercept(s) in point form.

x-intercept(s): $(7, 0)$

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 $y$ .

$y = - \frac{1}{4} \cdot 0 + \frac{7}{4}$

Step 2.4.3.2

Solve the equation.

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Step 2.4.3.2.1

Remove parentheses.

$y = - \frac{1}{4} \cdot 0 + \frac{7}{4}$

Step 2.4.3.2.2

Simplify $- \frac{1}{4} \cdot 0 + \frac{7}{4}$ .

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Step 2.4.3.2.2.1

Multiply $- \frac{1}{4} \cdot 0$ .

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Step 2.4.3.2.2.1.1

Multiply $0$ by $- 1$ .

$y = 0 (\frac{1}{4}) + \frac{7}{4}$

Step 2.4.3.2.2.1.2

Multiply $0$ by $\frac{1}{4}$ .

$y = 0 + \frac{7}{4}$

Step 2.4.3.2.2.2

Add $0$ and $\frac{7}{4}$ .

$y = \frac{7}{4}$

Step 2.4.3.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{7}{4})$

Step 2.4.4

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

$\begin{array}{c} x & y \\ 0 & \frac{7}{4} \\ 7 & 0 \end{array}$

Step 2.5

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

Slope: $- \frac{1}{4}$

y-intercept: $(0, \frac{7}{4})$

$\begin{array}{c} x & y \\ 0 & \frac{7}{4} \\ 7 & 0 \end{array}$

Slope: $- \frac{1}{4}$

y-intercept: $(0, \frac{7}{4})$

$\begin{array}{c} x & y \\ 0 & \frac{7}{4} \\ 7 & 0 \end{array}$

Step 3

Plot each graph on the same coordinate system.

$2 x - y = 5$

$x + 4 y = 7$

Step 4