Find the x and y Intercepts y=6x-1/x-5

$y = 6 x - \frac{1}{x} - 5$

Step 1

Find the x-intercepts.

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

$0 = 6 x - \frac{1}{x} - 5$

Solve the equation.

Rewrite the equation as $6 x - \frac{1}{x} - 5 = 0$ .

$6 x - \frac{1}{x} - 5 = 0$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x, 1, 1$

The LCM of one and any expression is the expression.

$x$

Multiply each term in $6 x - \frac{1}{x} - 5 = 0$ by $x$ to eliminate the fractions.

Multiply each term in $6 x - \frac{1}{x} - 5 = 0$ by $x$ .

$6 x \cdot x - \frac{1}{x} x - 5 x = 0 x$

Simplify the left side.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$6 (x \cdot x) - \frac{1}{x} x - 5 x = 0 x$

Multiply $x$ by $x$ .

$6 x^{2} - \frac{1}{x} x - 5 x = 0 x$

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$6 x^{2} + \frac{- 1}{x} x - 5 x = 0 x$

Cancel the common factor.

$6 x^{2} + \frac{- 1}{x} x - 5 x = 0 x$

Rewrite the expression.

$6 x^{2} - 1 - 5 x = 0 x$

Simplify the right side.

Multiply $0$ by $x$ .

$6 x^{2} - 1 - 5 x = 0$

Solve the equation.

Factor by grouping.

Reorder terms.

$6 x^{2} - 5 x - 1 = 0$

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 6 \cdot - 1 = - 6$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

$6 x^{2} - 5 x - 1 = 0$

Rewrite $- 5$ as $1$ plus $- 6$

$6 x^{2} + (1 - 6) x - 1 = 0$

Apply the distributive property.

$6 x^{2} + 1 x - 6 x - 1 = 0$

Multiply $x$ by $1$ .

$6 x^{2} + x - 6 x - 1 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(6 x^{2} + x) - 6 x - 1 = 0$

Factor out the greatest common factor (GCF) from each group.

$x (6 x + 1) - (6 x + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $6 x + 1$ .

$(6 x + 1) (x - 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$6 x + 1 = 0$

$x - 1 = 0$

Set $6 x + 1$ equal to $0$ and solve for $x$ .

Set $6 x + 1$ equal to $0$ .

$6 x + 1 = 0$

Solve $6 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$6 x = - 1$

Divide each term in $6 x = - 1$ by $6$ and simplify.

Divide each term in $6 x = - 1$ by $6$ .

$\frac{6 x}{6} = \frac{- 1}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 1}{6}$

Divide $x$ by $1$ .

$x = \frac{- 1}{6}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{6}$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $(6 x + 1) (x - 1) = 0$ true.

$x = - \frac{1}{6}, 1$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{1}{6}, 0), (1, 0)$

Step 2

Find the y-intercepts.

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

$y = 6 (0) - \frac{1}{0} - 5$

The equation has an undefined fraction.

Undefined

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

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(- \frac{1}{6}, 0), (1, 0)$

y-intercept(s): $None$

Step 4

Find the x and y Intercepts y=6x-1/x-5

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