Question

example 1
write an equation of the line that passes through the given point and has the given slope.

1. (4, 2); slope \\(\frac{1}{2}\\)
2. (3, -2); slope \\(\frac{1}{3}\\)
3. (6, 4); slope \\(-\frac{3}{4}\\)
4. (-5, 4); slope -3
5. (4, 3); slope \\(\frac{1}{2}\\)
6. (1, -5); slope \\(-\frac{3}{2}\\)

Explanation:

Let's solve these problems one by one using the point - slope form of a line's equation, which is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope of the line. Then we can simplify it to the slope - intercept form $y=mx + b$ (or keep it in point - slope form as needed).

Problem 1: Point $(4,2)$ and slope $\frac{1}{2}$

Step 1: Substitute into point - slope form

We know that $x_1 = 4$, $y_1=2$ and $m=\frac{1}{2}$. Substituting into $y - y_1=m(x - x_1)$, we get:
$y - 2=\frac{1}{2}(x - 4)$

Step 2: Simplify the equation

Distribute $\frac{1}{2}$ on the right - hand side:
$y - 2=\frac{1}{2}x-2$
Add 2 to both sides of the equation:
$y=\frac{1}{2}x$

Problem 2: Point $(3,-2)$ and slope $\frac{1}{3}$

Step 1: Substitute into point - slope form

Here, $x_1 = 3$, $y_1=-2$ and $m=\frac{1}{3}$. Substituting into $y - y_1=m(x - x_1)$:
$y-(-2)=\frac{1}{3}(x - 3)$
$y + 2=\frac{1}{3}(x - 3)$

Step 2: Simplify the equation

Distribute $\frac{1}{3}$:
$y + 2=\frac{1}{3}x-1$
Subtract 2 from both sides:
$y=\frac{1}{3}x-3$

Problem 3: Point $(6,4)$ and slope $-\frac{3}{4}$

Step 1: Substitute into point - slope form

With $x_1 = 6$, $y_1 = 4$ and $m=-\frac{3}{4}$, we have:
$y - 4=-\frac{3}{4}(x - 6)$

Step 2: Simplify the equation

Distribute $-\frac{3}{4}$:
$y - 4=-\frac{3}{4}x+\frac{18}{4}$
Simplify $\frac{18}{4}=\frac{9}{2}$
$y - 4=-\frac{3}{4}x+\frac{9}{2}$
Add 4 (which is $\frac{8}{2}$) to both sides:
$y=-\frac{3}{4}x+\frac{9 + 8}{2}=-\frac{3}{4}x+\frac{17}{2}$

Problem 4: Point $(-5,4)$ and slope $-3$

Step 1: Substitute into point - slope form

Here, $x_1=-5$, $y_1 = 4$ and $m=-3$. Substituting into $y - y_1=m(x - x_1)$:
$y - 4=-3(x+5)$

Step 2: Simplify the equation

Distribute $-3$:
$y - 4=-3x-15$
Add 4 to both sides:
$y=-3x-11$

Problem 5: Point $(4,3)$ and slope $\frac{1}{2}$

Step 1: Substitute into point - slope form

We have $x_1 = 4$, $y_1 = 3$ and $m=\frac{1}{2}$. Substituting into $y - y_1=m(x - x_1)$:
$y - 3=\frac{1}{2}(x - 4)$

Step 2: Simplify the equation

Distribute $\frac{1}{2}$:
$y - 3=\frac{1}{2}x-2$
Add 3 to both sides:
$y=\frac{1}{2}x + 1$

Problem 6: Point $(1,-5)$ and slope $-\frac{3}{2}$

Step 1: Substitute into point - slope form

With $x_1 = 1$, $y_1=-5$ and $m=-\frac{3}{2}$, we get:
$y-(-5)=-\frac{3}{2}(x - 1)$
$y + 5=-\frac{3}{2}(x - 1)$

Step 2: Simplify the equation

Distribute $-\frac{3}{2}$:
$y + 5=-\frac{3}{2}x+\frac{3}{2}$
Subtract 5 (which is $\frac{10}{2}$) from both sides:
$y=-\frac{3}{2}x+\frac{3-10}{2}=-\frac{3}{2}x-\frac{7}{2}$

Answer:

s:

1. $y=\frac{1}{2}x$
2. $y=\frac{1}{3}x - 3$
3. $y=-\frac{3}{4}x+\frac{17}{2}$
4. $y=-3x - 11$
5. $y=\frac{1}{2}x + 1$
6. $y=-\frac{3}{2}x-\frac{7}{2}$