Question

1. write an equation in standard form for a line with a slope of \\(\frac{-3}{5}\\) that passes through (-1, -4).
2. write two equations in point slope form for a line that passes through (4, 5) and (2, 1).
3. write an equation in slope-intercept form for a line that passes through (0, -1) and (-2, 3).
4. write an equation in standard form for a line with a y-intercept of -1 and an x-intercept of 4.

Explanation:

Problem 14:

Step1: Use point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $m$ is the slope and $(x_1,y_1)$ is a point on the line. Here, $m =-\frac{3}{5}$ and $(x_1,y_1)=(-1,-4)$. So we have $y-(-4)=-\frac{3}{5}(x - (-1))$, which simplifies to $y + 4=-\frac{3}{5}(x + 1)$.

Step2: Convert to standard form

The standard form of a line is $Ax+By = C$, where $A$, $B$, and $C$ are integers and $A\geq0$.
Start with $y + 4=-\frac{3}{5}(x + 1)$. Multiply both sides by 5 to get rid of the fraction: $5(y + 4)=- 3(x + 1)$.
Expand both sides: $5y+20=-3x - 3$.
Add $3x$ to both sides: $3x + 5y+20=-3$.
Subtract 20 from both sides: $3x+5y=-23$.

Step1: Find 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 $(4,5)$ and $(2,1)$, $m=\frac{1 - 5}{2 - 4}=\frac{-4}{-2}=2$.

Step2: Write point - slope form using $(4,5)$

The point - slope form is $y - y_1=m(x - x_1)$. Using $(x_1,y_1)=(4,5)$ and $m = 2$, we get $y - 5=2(x - 4)$.

Step3: Write point - slope form using $(2,1)$

Using $(x_1,y_1)=(2,1)$ and $m = 2$, we get $y - 1=2(x - 2)$.

Step1: Find the slope

The slope $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For the points $(0,-1)$ and $(-2,3)$, $m=\frac{3-(-1)}{-2 - 0}=\frac{4}{-2}=-2$ (Note: There was a mistake in the original calculation. The correct slope is - 2).

Step2: Use slope - intercept form

The slope - intercept form is $y=mx + b$, where $b$ is the y - intercept. We know that when $x = 0$, $y=-1$, so $b=-1$. And $m=-2$. So the equation is $y=-2x-1$.
(If we start from point - slope form: Using the point $(0,-1)$ and $m=-2$, the point - slope form is $y-(-1)=-2(x - 0)$, which simplifies to $y + 1=-2x$, and then $y=-2x - 1$)

Answer:

$3x + 5y=-23$

Problem 15: