Question

1. a. write an equation of the line with x-intercept (3, 0) and y-intercept (0, -4).

b. write an equation of a line parallel to the line $y - 5 = \frac{4}{3}(x - 2)$.
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Explanation:

Part a

Step1: Recall intercept form

The intercept form of a line is $\frac{x}{a}+\frac{y}{b} = 1$, where $a$ is the $x$-intercept and $b$ is the $y$-intercept.
Here, $a = 3$ and $b=- 4$.

Step2: Substitute values

Substitute $a = 3$ and $b=-4$ into the intercept form:
$\frac{x}{3}+\frac{y}{-4}=1$
Simplify to slope - intercept form ($y=mx + c$) or keep it in standard form. Let's convert to slope - intercept form.
$\frac{x}{3}-\frac{y}{4}=1$
Multiply through by 12 to clear the fractions:
$4x-3y = 12$
Or solve for $y$:
$-3y=-4x + 12$
$y=\frac{4}{3}x-4$

Part b

Step1: Recall slope of parallel lines

Parallel lines have the same slope. The given line is in point - slope form $y - y_1=m(x - x_1)$, where $m$ is the slope.
For the line $y - 5=\frac{4}{3}(x - 2)$, the slope $m=\frac{4}{3}$.

Step2: Write equation of parallel line

We can use the point - slope form $y - y_1=m(x - x_1)$ with any point $(x_1,y_1)$. Let's use the point $(0,0)$ (we can choose any point).
Substitute $m = \frac{4}{3}$, $x_1 = 0$ and $y_1=0$ into the point - slope form:
$y-0=\frac{4}{3}(x - 0)$
$y=\frac{4}{3}x$
(We could also use other points. For example, if we use the point $(1,1)$, the equation would be $y - 1=\frac{4}{3}(x - 1)$ which simplifies to $y=\frac{4}{3}x-\frac{1}{3}$)

Answer:

(Part a):
One form of the equation is $y=\frac{4}{3}x - 4$ (or $\frac{x}{3}-\frac{y}{4}=1$ or $4x-3y = 12$)