# Section 1.3 – Linear Equations in Two Variables

Section Details.
• The general equation of a line and finding the slope of a line
• Equation for a line including the slope-intercept form and point-slope form
• Finding the equation of a line and graphing a line
• Equations and slopes for horizontal and vertical lines
• Parallel and Perpendicular lines

### Practice Problems

Directions. The following are review problems for this section. It is recommended that you work the problems, and then click "Solution" to check your answer. If you do not understand how to solve a problem, you can click "Video" to learn how to solve it.
1. Find an equation of the line through the points $$(5,4)$$ and $$(-10,-2)$$.

Answer: $$y=\dfrac{2}{5}x+2$$

Solution: Slope $$m=\dfrac{4-(-2)}{5-(-10)}=\dfrac{6}{15}=\dfrac{2}{5}$$

$$y-4=\dfrac{2}{5}(x-5)$$  OR  $$y-(-2)=\dfrac{2}{5}(x-(-10))$$

Simplifies to $$y=\dfrac{2}{5}x+2$$.

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2. Find an equation of the line through the points $$(5,4)$$ and $$(5,-2)$$.

Answer: $$x=5$$

Solution:
Slope $$m=\dfrac{4-(-2)}{5-5}=\dfrac{6}{0}$$ which is undefined

Undefined slope means the line is vertical and all x values are the same, so our equation is $$x=5$$.

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3. Write an equation of a line a) parallel to and b) perpendicular to the line $$5+6x-4y=0$$ and passing through the point $$(3,2)$$.

Answer: a) $$y-2=\dfrac{3}{2}(x-3)$$, b) $$y-2=-\dfrac{2}{3}(x-3)$$

Solution: First, we need to know the slope of our given line.

$5+6x-4y=0$

$5+6x=4y$

$\dfrac{5}{4}+\dfrac{3}{2}x=y$

a) The slope of the given line is $$\dfrac{3}{2}$$, so the parallel line for a) will also have a slope of $$m=\dfrac{3}{2}$$.

$y-2=\dfrac{3}{2}(x-3)$

b) The perpendicular slope will be $$m_{\perp}=-\dfrac{2}{3}$$, so the equation of the perpendicular line will be

$y-2=-\dfrac{2}{3}(x-3)$

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4. Write the equation of the line parallel $$5x-4y=8$$ passing through the point $$(3,-2)$$.

Point-Slope Form: $$\displaystyle y-(-2)=\frac{5}{4}(x-3)$$
Slope-Intercept Form: $$\displaystyle y=\frac{5}{4}x-\frac{23}{4}$$
Standard Form: $$\displaystyle 5x-4y=23$$

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