Question

question
what is the equation of the line that passes through the point (6, 4) and has a slope of \\(\frac{2}{3}\\)?

Explanation:

Step1: Recall point - slope form

The point - slope form of a linear equation is given by $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope of the line.
Here, $x_1 = 6$, $y_1=4$ and $m=\frac{2}{3}$.

Step2: Substitute values into point - slope form

Substitute $x_1 = 6$, $y_1 = 4$ and $m=\frac{2}{3}$ into the point - slope formula:
$y - 4=\frac{2}{3}(x - 6)$

Step3: Simplify the equation

First, distribute the $\frac{2}{3}$ on the right - hand side:
$y-4=\frac{2}{3}x-4$
Then, add 4 to both sides of the equation:
$y=\frac{2}{3}x-4 + 4$
$y=\frac{2}{3}x$
Or, if we want to write it in standard form $Ax+By = C$ (where $A$, $B$, and $C$ are integers and $A\geq0$), we can rewrite $y=\frac{2}{3}x$ as $2x-3y = 0$ (by multiplying both sides by 3 to get rid of the fraction: $3y = 2x$, then rearranging to $2x-3y=0$). But the slope - intercept form ($y=mx + b$) is also a valid form of the equation of a line.

Answer:

The equation of the line is $y=\frac{2}{3}x$ (or $2x - 3y=0$)