Question

write an equation of the line that passes through the given point and has the given slope.
(10, -4); slope -\frac{2}{5}

Explanation:

Step1: Recall point - slope form

The point - slope form of a linear equation is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope of the line.
Here, $x_1 = 10$, $y_1=-4$ and $m =-\frac{2}{5}$.

Step2: Substitute values into point - slope form

Substitute $x_1 = 10$, $y_1=-4$ and $m =-\frac{2}{5}$ into the point - slope form:
$y-(-4)=-\frac{2}{5}(x - 10)$

Step3: Simplify the equation

Simplify the left - hand side: $y + 4=-\frac{2}{5}(x - 10)$
Distribute the $-\frac{2}{5}$ on the right - hand side: $y+4=-\frac{2}{5}x+\frac{2\times10}{5}$
Simplify $\frac{2\times10}{5}$: $\frac{20}{5} = 4$
So, $y+4=-\frac{2}{5}x + 4$
Subtract 4 from both sides: $y=-\frac{2}{5}x+4 - 4$
Simplify: $y=-\frac{2}{5}x$

We can also write it in standard form ($Ax+By = C$) by multiplying both sides by 5: $5y=-2x$, or $2x + 5y=0$

Answer:

The equation of the line is $y =-\frac{2}{5}x$ (or $2x + 5y=0$)