# A line passes through point (3, 7) and has a slope of 3/4. What is the equation of this line?

May 4, 2017

Point slope form: $y - 7 = \frac{3}{4} \left(x - 3\right)$

Slope intercept form: $y = \frac{3}{4} x + \frac{19}{4}$$\textcolor{w h i t e}{. .} \text{or} \textcolor{w h i t e}{. .}$$y = \frac{3}{4} x + 4 \frac{3}{4}$

#### Explanation:

Since you have one point and the slope, you can use the point slope formula, then solve for $y$ to get the slope intercept form, so that you can determine the y-intercept $\left(b\right)$. Then you can graph the resulting equation.

Point slope formula
$y - {y}_{1} = m \left(x - {x}_{1}\right)$, where ${x}_{1} , {y}_{1} = \left(3 , 7\right)$, and $m = \frac{3}{4}$ is the slope.

Substitute the given values into the formula.

$y - 7 = \frac{3}{4} \left(x - 3\right)$

Expand.

$y - 7 = \frac{3}{4} x - \frac{9}{4}$

Solve for $y$ to get the slope intercept form: $y = m x + b$, where $m = \frac{3}{4}$ is the slope and $b$ is the y-intercept.

$y - 7 = \frac{3}{4} x - \frac{9}{4}$

Add $7$ to both sides.

$y - \textcolor{red}{\cancel{\textcolor{b l a c k}{7}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{7}}} = \frac{3}{4} x - \frac{9}{4} + 7$

Simplify.

$y = \frac{3}{4} x - \frac{9}{4} + 7$

In order to add and subtract fractions, they must have the same denominator. $7 = \frac{7}{1}$ and can be multiplied by the equivalent fraction $\textcolor{red}{\frac{4}{4}}$ in order to get the same denominator as $\frac{9}{4}$.

$y = \frac{3}{4} x - \frac{9}{4} + \frac{7}{1} \times \textcolor{red}{\frac{4}{4}}$

Simplify.

$y = \frac{3}{4} x - \frac{9}{4} + \frac{28}{4}$

Simplify.

$y = \frac{3}{4} x + \frac{19}{4}$ or

$y = \frac{3}{4} x + 4 \frac{3}{4}$

The y-intercept is $\frac{19}{4}$

graph{y=3/4x+19/4 [-17.42, 14.6, -5.96, 10.06]}

May 4, 2017

$\text{answer :}$

$y = \frac{3}{4} x - \frac{37}{4}$

$\text{Or }$

$- 3 x + 4 y = - 37$

#### Explanation:

$\text{equation for solution is }$

$y - {y}_{1} = m \left(x - {x}_{1}\right)$

$\text{where } P \left({x}_{1} , {y}_{1}\right) = P \left(3 , 7\right)$

$y - 7 = \frac{3}{4} \left(x - 3\right)$

$y = \frac{3}{4} \left(x - 3\right) + 7$

$y = \frac{3}{4} x - \frac{9}{4} + 7$

$y = \frac{3}{4} x - \frac{37}{4}$

$\text{Or }$

$y - \frac{3}{4} x = - \frac{37}{4}$

$\frac{4 y - 3 x}{4} = - \frac{37}{4}$

$\frac{4 y - 3 x}{\cancel{4}} = - \frac{37}{\cancel{4}}$

$- 3 x + 4 y = - 37$