# How do you find the x and y intercepts of 3x-4y-10=0?

Mar 25, 2017

The intercepts are at $\left(3 \frac{1}{3} , 0\right) \mathmr{and} \left(0 , - 2 \frac{1}{2}\right)$

#### Explanation:

Finding the intercepts is a very easy and very useful skill to develop for drawing graphs. It is particularly handy for linear programming.

All along the $x$-axis, $y = 0$ All points are $\left(x , 0\right)$

All along the $y$-axis, $x = 0$ All points are $\left(y , 0\right)$

To find the $x$-intercept, set $y = 0$ and solve for $x$

To find the $y$-intercept, set $x = 0$ and solve for $y$

The only time this does not work is if the graph goes through the origin.

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$3 x - 4 y - 10 = 0 \text{ } \rightarrow 3 x - 4 y = 10$

For the $x$-int: $y = 0$
$\text{ "3x -4(0) = 10" "rarr3x=10" } x = 3 \frac{1}{3}$

For the $y$-int: $x = 0$
$\text{ "3(0) -4y = 10" "rarr-4y=10" } y = - \frac{5}{2}$

The intercepts are at $\left(3 \frac{1}{3} , 0\right) \mathmr{and} \left(0 , - 2 \frac{1}{2}\right)$

Mar 25, 2017

The x-intercept is $\left(\frac{10}{3} , 0\right) = \left(3 \frac{1}{3} , 0\right)$.

The y-intercept is $\left(0 , - \frac{5}{2}\right) = \left(0 , - 2 \frac{1}{2}\right)$

See the explanation for the process.

#### Explanation:

Find x and y intercepts. The x-intercept is the point where $y$ is $0$. The y-intercept is the point where $x$ is $0$.

Given

$3 x - 4 y - 10 = 0$

X-intercept

Substitute $0$ for $y$.

$3 x - 4 \left(0\right) - 10 = 0$

Simplify.

$3 x - 10 = 0$

Add $10$ to both sides.

$3 x \textcolor{red}{\cancel{\textcolor{b l a c k}{- 10}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{10}}} = 0 + 10$

Simplify.

$3 x = 10$

Divide both sides by $3$.

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3} \textcolor{b l a c k}{x}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}}} = \frac{10}{3}$

$x = \frac{10}{3}$

The x-intercept is $\left(\frac{10}{3} , 0\right) = \left(3 \frac{1}{3} , 0\right)$.

Y-intercept

Substitute $0$ for the $x$. Solve for $y$.

$3 \left(0\right) - 4 y - 10 = 0$

Simplify.

$- 4 y - 10 = 0$

Add $10$ to both sides.

$- 4 y = 10$

Divide both sides by $- 4$.

$y = \frac{10}{- 4}$

$y = - \frac{10}{4}$

Simplify.

$y = - \frac{5}{2}$

The y-intercept is $\left(0 , - \frac{5}{2}\right) = \left(0 , - 2 \frac{1}{2}\right)$

Use the graph below to check the x and y intercepts.

graph{3x-4y-10=0 [-16.02, 16, -8.01, 8.01]}