Linear Lessons

Lesson 3: Linear Equations in Two Variables

​Slope of a Line

The slope is a ratio of the vertical movement of a line to the ​horizontal movement of the line. This is sometimes called the "rise over run."

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​The formula for the slope of a line (shown to the right) uses two points from the line. We subtract the y values and place them on the top of a fraction bar. Subtract the x values and place them on the bottom of the fraction bar. Then, we simplify. Make sure you place the same point's y and x values over one another or else you'll find the wrong slope.

​The Slope-Intercept Form

​The slope-intercept form of linear equations is the most commonly used form. The slope is written as m in the form and the y-intercept (or where the line hits the y-axis) is b. We can easily graph lines using this form by plotting the y-intercept as (0, b) and then counting the rise/run two times in order to have 3 points plotted. Then, draw the line through those points.

​Graphing Horizontal and Vertical Lines

​Horizontal lines are written in the form y = c. The slope is 0.

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​Vertical lines are written in the form x = a. It is an undefined slope.

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Multiple Choice

x = 9 is a ________________ line.

• vertical

• horizontal

Multiple Choice

Find the slope of the line that passes through (-2, -2) and (3, 1)

• $$\frac{3}{5}$$  

• $$\frac{5}{3}$$  

• $$-\frac{5}{3}$$  

• $$-\frac{3}{5}$$  

​Point-Slope Form

​When given two points, we can find the equation of the line using the point slope form. First, we find the slope between the two points using the slope formula from slide 2. Then, we plug in the slope and one of the points and solve for y. This will give us the linear equation in slope-intercept form.

​Parallel and Perpendicular Lines

​Two lines are parallel if and only if their slopes are equal.

​(If line 1 has a slope of 3 and line 2 is parallel to it, then line 2's slope must be 3)

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​Two lines are perpendicular if and only if their slopes are opposite reciprocals.

​(If line 1 has a slope of 1/2 and line 2 is perpendicular to it, then line 2 must have a slope of -2)

NOTE​

If two lines have the same slope AND the same y-intercept, then they are not parallel, but are the same line (they intersect at every point on the line)​

Multiple Choice

Line 1's equation is y = -2x + 1. What is the slope of line 2 if it is parallel to line 1?

• -2

• $$\frac{1}{2}$$  

• $$-\frac{1}{2}$$  

• 2