Algebra Examples

f (x) = \frac{3 x^{2} - 9 x + 6}{5 x - 10}

$f (x) = \frac{3 x^{2} - 9 x + 6}{5 x - 10}$

Step 1

Rewrite the function as an equation.

y = \frac{3 (x - 1)}{5}

$y = \frac{3 (x - 1)}{5}$

Step 2

Rewrite in slope-intercept form.

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Step 2.1

The slope-intercept form is

y = m x + b

$y = m x + b$ , where

m

$m$ is the slope and

b

$b$ is the y-intercept.

y = m x + b

$y = m x + b$

Step 2.2

Write in

y = m x + b

$y = m x + b$ form.

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Step 2.2.1

Apply the distributive property.

y = \frac{3 x + 3 \cdot - 1}{5}

$y = \frac{3 x + 3 \cdot - 1}{5}$

Step 2.2.2

Multiply

3

$3$ by

- 1

$- 1$ .

y = \frac{3 x - 3}{5}

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

Step 2.2.3

Split the fraction

\frac{3 x - 3}{5}

$\frac{3 x - 3}{5}$ into two fractions.

y = \frac{3 x}{5} + \frac{- 3}{5}

$y = \frac{3 x}{5} + \frac{- 3}{5}$

Step 2.2.4

Move the negative in front of the fraction.

y = \frac{3 x}{5} - \frac{3}{5}

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

Step 2.2.5

Reorder terms.

y = \frac{3}{5} x - \frac{3}{5}

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

y = \frac{3}{5} x - \frac{3}{5}

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

y = \frac{3}{5} x - \frac{3}{5}

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

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

$m$ and

b

$b$ using the form

y = m x + b

$y = m x + b$ .

m = \frac{3}{5}

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

b = - \frac{3}{5}

$b = - \frac{3}{5}$

Step 3.2

The slope of the line is the value of

m

$m$ , and the y-intercept is the value of

b

$b$ .

Slope:

\frac{3}{5}

$\frac{3}{5}$

y-intercept:

(0, - \frac{3}{5})

$(0, - \frac{3}{5})$

Slope:

\frac{3}{5}

$\frac{3}{5}$

y-intercept:

(0, - \frac{3}{5})

$(0, - \frac{3}{5})$

Step 4

Any line can be graphed using two points. Select two

x

$x$ values, and plug them into the equation to find the corresponding

y

$y$ values.

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Step 4.1

Write in

y = m x + b

$y = m x + b$ form.

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Step 4.1.1

Apply the distributive property.

y = \frac{3 x + 3 \cdot - 1}{5}

$y = \frac{3 x + 3 \cdot - 1}{5}$

Step 4.1.2

Multiply

3

$3$ by

- 1

$- 1$ .

y = \frac{3 x - 3}{5}

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

Step 4.1.3

Split the fraction

\frac{3 x - 3}{5}

$\frac{3 x - 3}{5}$ into two fractions.

y = \frac{3 x}{5} + \frac{- 3}{5}

$y = \frac{3 x}{5} + \frac{- 3}{5}$

Step 4.1.4

Move the negative in front of the fraction.

y = \frac{3 x}{5} - \frac{3}{5}

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

Step 4.1.5

Reorder terms.

y = \frac{3}{5} x - \frac{3}{5}

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

y = \frac{3}{5} x - \frac{3}{5}

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

Step 4.2

Find the x-intercept.

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Step 4.2.1

To find the x-intercept(s), substitute in

0

$0$ for

y

$y$ and solve for

x

$x$ .

0 = \frac{3}{5} x - \frac{3}{5}

$0 = \frac{3}{5} x - \frac{3}{5}$

Step 4.2.2

Solve the equation.

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Step 4.2.2.1

Rewrite the equation as

\frac{3}{5} x - \frac{3}{5} = 0

$\frac{3}{5} x - \frac{3}{5} = 0$ .

\frac{3}{5} x - \frac{3}{5} = 0

$\frac{3}{5} x - \frac{3}{5} = 0$

Step 4.2.2.2

Combine

\frac{3}{5}

$\frac{3}{5}$ and

x

$x$ .

\frac{3 x}{5} - \frac{3}{5} = 0

$\frac{3 x}{5} - \frac{3}{5} = 0$

Step 4.2.2.3

Add

\frac{3}{5}

$\frac{3}{5}$ to both sides of the equation.

\frac{3 x}{5} = \frac{3}{5}

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

Step 4.2.2.4

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

3 x = 3

$3 x = 3$

Step 4.2.2.5

Divide each term in

3 x = 3

$3 x = 3$ by

3

$3$ and simplify.

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Step 4.2.2.5.1

Divide each term in

3 x = 3

$3 x = 3$ by

3

$3$ .

\frac{3 x}{3} = \frac{3}{3}

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

Step 4.2.2.5.2

Simplify the left side.

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Step 4.2.2.5.2.1

Cancel the common factor of

3

$3$ .

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Step 4.2.2.5.2.1.1

Cancel the common factor.

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

Step 4.2.2.5.2.1.2

Divide

$x$ by

$1$ .

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

Step 4.2.2.5.3

Simplify the right side.

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Step 4.2.2.5.3.1

Divide

$3$ by

$3$ .

$x = 1$

Step 4.2.3

x-intercept(s) in point form.

x-intercept(s):

$(1, 0)$

x-intercept(s):

$(1, 0)$

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

$y$ .

$y = \frac{3}{5} \cdot (0) - \frac{3}{5}$

Step 4.3.2

Solve the equation.

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Step 4.3.2.1

Multiply

$\frac{3}{5}$ by

$0$ .

$y = \frac{3}{5} \cdot 0 - \frac{3}{5}$

Step 4.3.2.2

Remove parentheses.

$y = \frac{3}{5} \cdot (0) - \frac{3}{5}$

Step 4.3.2.3

Simplify

$\frac{3}{5} \cdot (0) - \frac{3}{5}$ .

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Step 4.3.2.3.1

Multiply

$\frac{3}{5}$ by

$0$ .

$y = 0 - \frac{3}{5}$

Step 4.3.2.3.2

Subtract

$\frac{3}{5}$ from

$0$ .

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

Step 4.3.3

y-intercept(s) in point form.

y-intercept(s):

$(0, - \frac{3}{5})$

y-intercept(s):

$(0, - \frac{3}{5})$

Step 4.4

Create a table of the

$x$ and

$y$ values.

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

Step 5

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

Slope:

$\frac{3}{5}$

y-intercept:

$(0, - \frac{3}{5})$

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

Step 6