1. Warmup
1. Find the equation of a circle with center (-4,1) and radius 3
2. Find an equation of a circle with center at the origin passing through P(4, -7)
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2. Answers 1. (x+4)2 + (y-1)2 =9 2. x2 + y2 = 65
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3. 2.3 Linear Equations in Two Variables
Digital Lesson
2.3 Linear Equations in Two Variables

4. The slope of a line is a number, m, which measures its steepness.y x 2 -2
m is undefined
m = 2
m = 1 2
m = 0
m = - 1 4
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Slope of a Line

5. The slope of the line passing through the two points (x1, y1) and (x2, y2) is given by the formula
y2 – y1
x2 – x1
m =
, (x1 ≠ x2 ). x y
(x1, y1)
(x2, y2)
x2 – x1
y2 – y1
change in y
change in x
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Slope Formula

6. Example: Find the slope of the line passing through the points (2, 3) and (4, 5).
y2 – y1
x2 – x1
m =
5 – 3
4 – 2 = = 2 = 1 x y
(4, 5)
(2, 3) 2 2
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Example: Find Slope

7. Example: Graph the line y = 2x – 4.
(1, -2) 2
(0, - 4) 1
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Example: y=mx+b

8. A linear equation written in the form y – y1 = m(x – x1) is in point-slope form.
The graph of this equation is a line with slope m passing through the point (x1, y1).
Example: The graph of the equation y – 3 = - (x – 4) is a line of slope m = - passing through the point (4, 3). 1 2 x y 4 8
m = - 1 2
(4, 3)
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Point-Slope Form

9. Example: Point-Slope Form
Example: Write the slope-intercept form for the equation of the line through the point (-2, 5) with a slope of 3.
y – y1 = m(x – x1)
Point-slope form
y – y1 = 3(x – x1)
Let m = 3.
y – 5 = 3(x – (-2))
Let (x1, y1) = (-2, 5).
y – 5 = 3(x + 2)
Simplify.
y = 3x + 11
Slope-intercept form
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Example: Point-Slope Form

10. Example: Slope-Intercept Form
Example: Write the slope-intercept form for the equation of the line through the points (4, 3) and (-2, 5). 2 1
5 – 3
-2 – 4
= - 6 3
Calculate the slope.
m =
y – y1 = m(x – x1)
Point-slope form
Use m = - and the point (4, 3).
y – 3 = - (x – 4) 1 3
Slope-intercept form
y = - x + 13 3 1
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Example: Slope-Intercept Form

11. Example: Parallel Lines
Two lines are parallel if they have the same slope.
If the lines have slopes m1 and m2, then the lines are parallel whenever m1 = m2. x y
y = 2x + 4
(0, 4)
Example: The lines y = 2x – 3 and y = 2x + 4 have slopes m1 = 2 and m2 = 2.
y = 2x – 3
(0, -3)
The lines are parallel.
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Example: Parallel Lines

12. Example: Perpendicular Lines
Two lines are perpendicular if their slopes are negative reciprocals of each other.
If two lines have slopes m1 and m2, then the lines are perpendicular whenever x y 1 m1
m2= -
or m1m2 = -1.
y = 3x – 1
y = x + 4 1 3
Example: The lines y = 3x – 1 and y = - x + 4 have slopes m1 = 3 and m2 = 1 3
(0, 4)
(0, -1)
The lines are perpendicular.
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Example: Perpendicular Lines