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Point-Slope Form of a Line
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We can also find the equation of a line when given the slope and any point (not the y-intercept), and there are two methods to do so. The following video will use a single example to show how to use both methods to find the equation of a line with a given slope and single point.

Point-Slope Form of a Line

These are the two methods to finding the equation of a line when given a point and the slope:

1. Substitution method = plug in the slope and the (x, y) point values into $$y = mx + b$$ and solve for b. Use the m given in the problem, and the b that was just solved for, to create the equation $$y = mx + b$$.
2. Point-slope form $$= y − y1 = m ( x − x1 )$$, where $$(x1, y1)$$ is the point given and m is the slope given. The "x" and the "y" stay as variables.

### Practice Problems

1. Find the equation of the line that passes through the point $$(1, 4)$$ and has a slope of $$12$$.
(
Solution
Solution:
$${\text{y}} = 12{\text{x}} - 8$$
)
2. Find the equation of the line that passes through the point $$(1, 4)$$and has a slope of $$2$$.
(
Solution
Solution:
$${\text{y}} = 2{\text{x}} + 2$$
Details:
We will use the point-slope form to find the equation of the line. Point-slope form is the following:

$${\text{y}}{-}{\text{y}}_1={\text{m}}({\text{x}}-{\text{x}}_1)$$

We will use the point $$({\color{Red}1}, {\color{Purple}4})$$ and the slope of $${\color{DarkOrange}2}$$ and substitute them in to $${\text{y}}{-}{\color{Purple}{\text{y}}_1}={\color{DarkOrange}{\text{m}}}({\text{x}}-{\color{Red}{\text{x}}_1})$$:

$${\text{y}}{-}{\color{Purple}4}={\color{DarkOrange}2}({\text{x}}-{\color{Red}1})$$

Then distribute on the right side:

$${\text{y}}{-}{\color{Purple}4}={\color{DarkOrange}2}{\text{x}}-{\color{DarkOrange}2}(1)$$

Which simplifies to the following:

$${\text{y}}{-}{4}=2{\text{x}}{-}{2}$$

$${\text{y}}{-}{4}{\color{Red}+4} = 2{\text{x}} - 2 {\color{Red}+4}$$

Which gives us the equation:

$$y = 2x +{\color{Red}2}$$
)
3. Find the equation of the line that passes through the point $$(27, 4)$$ and has a slope of $$-\dfrac{2}{9}$$.
(
Solution
Solution:
$${\text{y}} =-\dfrac{2}{9}{\text{x}}+10$$
)
4. Find the equation of the line that passes through the point $$(-11,2)$$ and has a slope of $$-\dfrac{5}{11}$$. (
Solution
Solution:
$${\text{y}}=-\dfrac{5}{11}{\text{x}}-3$$
)
5. Find the equation of the line that passes through the point $$(10, 6)$$ and has a slope of $$\dfrac{1}{5}$$. What is the y-intercept of the line? (
Solution
Solution:
$$(0,4)$$
Details:
We will use the point-slope form to find the equation of the line. Point-slope form is the following:

$${\text{y}}{-}{\text{y}_{1}}=\text{m}\left ( {\text{x}}{-}{\text{x}_{1}} \right )$$

We will use the point $$({\color{Red}10}, {\color{Purple}6})$$ and the slope of $${\color{DarkOrange} \dfrac{1}{5}}$$ and substitute them in to the following:

$${\text{y}}{-}{\color{Purple} {\text{y}_{1}}}={\color{DarkOrange} \text{m}}\left ( {\text{x}}{-}{\color{Red} {\text{x}_{1}}} \right )$$

Then distribute on the right side:

$${\text{y}}{-}{\color{Purple}6}={\color{Orange}\dfrac{1}{5}}{\text{x}}-{\color{Orange}\dfrac{1}{5}}(10)$$

Which simplifies to the following:

$${\text{y}}{-}6={\color{Orange}\dfrac{1}{5}}{\text{x}}-2$$

$${\text{y}}{-}6{\color{Red}+6}={\color{Orange}\dfrac{1}{5}}{\text{x}}-2{\color{Red}+6}$$

Which gives us the equation:

$$\text{y}={\color{DarkOrange} \dfrac{1}{5}}\text{x}+{\color{Red} 4}$$

The equation is in slope-intercept form, $$y=mx+b$$, so we can see that the y-intercept is at $$4$$ or the point $$(0,4)$$.
)
6. Find the equation of the line that passes through the point $$(3, 29)$$ and has a slope of $$6$$. What is the y-intercept of the line? (
Solution
Solution:
$$(0,11)$$
)

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