Question

its your turn
formulate the equation of the line in the form ax + by = c

1. whose slope is $-\frac{2}{3}$ and a y-intercept of $-5$
2. whose slope is 2 passing through $(-2, 4)$
3. passing through the points $(-2, -5)$ and $(1, -1)$

Explanation:

Problem 1:

Step1: Recall slope - intercept form

The slope - intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. Given $m=-\frac{2}{3}$ and $b = - 5$, we substitute these values into the slope - intercept form:
$y=-\frac{2}{3}x-5$

Step2: Convert to $ax + by=c$ form

Multiply each term by 3 to eliminate the fraction:
$3y=-2x - 15$
Then, add $2x$ to both sides of the equation:
$2x+3y=-15$

Step1: Use point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope. Given $m = 2$ and $(x_1,y_1)=(-2,4)$, we substitute these values into the point - slope form:
$y - 4=2(x+2)$

Step2: Simplify the equation

Expand the right - hand side: $y - 4=2x + 4$
Subtract $2x$ from both sides and add 4 to both sides:
$-2x+y=8$ (or multiply both sides by - 1 to get $2x - y=-8$)

Step1: Calculate the slope

The slope $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. For the points $(-2,-5)$ and $(1,-1)$, we have $x_1=-2,y_1=-5,x_2 = 1,y_2=-1$.
$m=\frac{-1-(-5)}{1-(-2)}=\frac{-1 + 5}{1 + 2}=\frac{4}{3}$

Step2: Use point - slope form

Using the point $(1,-1)$ and $m=\frac{4}{3}$ in the point - slope form $y - y_1=m(x - x_1)$:
$y+1=\frac{4}{3}(x - 1)$

Step3: Convert to $ax+by = c$ form

Multiply each term by 3 to eliminate the fraction:
$3y+3 = 4x-4$
Subtract $4x$ from both sides and subtract 3 from both sides:
$-4x+3y=-7$ (or multiply both sides by - 1 to get $4x-3y = 7$)

Answer:

$2x + 3y=-15$

Problem 2: