Question

practice and problem solving
graph the line with the given slope that contains the given point.

1. slope = $-\frac{1}{2}$; $(-3, 4)$ \t 21. slope = $\frac{3}{5}$; $(1, -2)$ \t 22. slope = 4; $(-1, 0)$

write an equation in

Explanation:

For Problem 20:

Step1: Recall point - slope form

The point - slope form of a linear equation 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, $m =-\frac{1}{2}$ and $(x_1,y_1)=(-3,4)$.

Step2: Substitute values into point - slope form

Substitute $m =-\frac{1}{2}$, $x_1=-3$ and $y_1 = 4$ into $y - y_1=m(x - x_1)$. We get $y - 4=-\frac{1}{2}(x - (-3))$, which simplifies to $y - 4=-\frac{1}{2}(x + 3)$.

Step3: (Optional) Convert to slope - intercept form (for graphing)

Expand the right - hand side: $y-4=-\frac{1}{2}x-\frac{3}{2}$. Then add 4 to both sides. $y=-\frac{1}{2}x-\frac{3}{2}+4=-\frac{1}{2}x+\frac{5}{2}$. To graph the line, plot the point $(-3,4)$. Then use the slope $-\frac{1}{2}$ (which means from the point $(-3,4)$, go down 1 unit and right 2 units, or up 1 unit and left 2 units) to find another point and draw the line through the two points.

For Problem 21:

Step1: Recall point - slope form

The point - slope form is $y - y_1=m(x - x_1)$, with $m=\frac{3}{5}$ and $(x_1,y_1)=(1,-2)$.

Step2: Substitute values

Substitute $m = \frac{3}{5}$, $x_1 = 1$ and $y_1=-2$ into the formula: $y-(-2)=\frac{3}{5}(x - 1)$, which simplifies to $y + 2=\frac{3}{5}(x - 1)$.

Step3: (Optional) Convert to slope - intercept form

Expand: $y + 2=\frac{3}{5}x-\frac{3}{5}$. Subtract 2 from both sides: $y=\frac{3}{5}x-\frac{3}{5}-2=\frac{3}{5}x-\frac{3}{5}-\frac{10}{5}=\frac{3}{5}x-\frac{13}{5}$. To graph, plot the point $(1,-2)$ and use the slope $\frac{3}{5}$ (up 3 units, right 5 units or down 3 units, left 5 units) to find another point and draw the line.

For Problem 22:

Step1: Recall point - slope form

The point - slope form is $y - y_1=m(x - x_1)$, where $m = 4$ and $(x_1,y_1)=(-1,0)$.

Step2: Substitute values

Substitute $m = 4$, $x_1=-1$ and $y_1 = 0$ into the formula: $y-0=4(x-(-1))$, which simplifies to $y = 4(x + 1)$.

Step3: (Optional) Convert to slope - intercept form

Expand: $y=4x + 4$. To graph, plot the point $(-1,0)$. Then use the slope 4 (up 4 units, right 1 unit or down 4 units, left 1 unit) to find another point and draw the line.

For writing the equation in point - slope form (if that's the intended question for the "Write an equation in point..." part):

Problem 20 (point - slope form):

Answer:

$y - 4=-\frac{1}{2}(x + 3)$ (or $y=-\frac{1}{2}x+\frac{5}{2}$ for slope - intercept)

Problem 21 (point - slope form):