Question

1. a. use the point - slope formula to write an equation of the line that is parallel to 3x + y = 4 and passes through the point (-1, 0). b. use the slope - intercept form to graph the line found in part (a). verify that the line passes through the point (-1, 0). 4. a. sketch a line perpendicular to the y - axis passing through the point (-3, 4). b. write an equation of the line perpendicular to the y - axis passing through the point (-3, 4). see the graph from part (a).

Explanation:

Step1: Rewrite the given line in slope - intercept form

Rewrite $3x + y=4$ as $y=-3x + 4$. The slope of this line is $m=-3$. Parallel lines have the same slope.

Step2: Use the point - slope formula

The point - slope formula is $y - y_1=m(x - x_1)$. Given the point $(-1,0)$ and $m=-3$, we substitute $x_1=-1$, $y_1 = 0$ and $m=-3$ into the formula: $y-0=-3(x + 1)$.

Step3: Simplify the equation

$y=-3x - 3$.

Step4: For graphing in part (b)

The slope - intercept form is $y=-3x - 3$, where the slope $m=-3=\frac{-3}{1}$ and the $y$-intercept $b=-3$. Start at the point $(0,-3)$ (the $y$-intercept), then use the slope. For every 1 unit we move to the right along the $x$-axis, we move 3 units down along the $y$-axis. To verify it passes through $(-1,0)$, substitute $x=-1$ into $y=-3x - 3$, we get $y=-3\times(-1)-3=3 - 3=0$.

Step5: For part 4a

A line perpendicular to the $y$-axis is a horizontal line. To sketch it, find the point $(-3,4)$ on the coordinate plane and draw a horizontal line passing through it.

Step6: For part 4b

The equation of a horizontal line passing through the point $(x_0,y_0)$ is of the form $y = y_0$. For the point $(-3,4)$, the equation of the line is $y = 4$.

Answer:

3a. $y=-3x - 3$
3b. Graph with $y$-intercept at $(0,-3)$ and slope of $-3$, verified to pass through $(-1,0)$
4a. Sketch a horizontal line through $(-3,4)$
4b. $y = 4$