Linear Equations in Two Variables

MATH 109 - PrecalculusS. Rook

Overview

• Section 1.3 in the textbook:– Graphing a linear equation by using its slope– Finding the slope of a line– Writing Linear Equations– Parallel and Perpendicular Lines

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Graphing a Linear Equation by Using its Slope

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Graphing Lines in Slope-Intercept Form

• Slope-Intercept Form of a line: y = mx + b where m is the slope and the y-intercept is (0, b)

• To graph a line using its slope and y-intercept:– Solve the equation for y if necessary– Plot the y-intercept (0, b)– Use rise over run with the slope to get 2 or 3

more points

Other Lines

• Two other type of lines:– When a linear equation is only in 1 variable (x or y)– Vertical lines have the form x = a (a is a constant):• m = Ø (undefined)• To sketch, find x = a on the x-axis and draw a vertical

line– Horizontal lines have the form y = b (b is a constant)• m = 0• To sketch, find y = b on the y-axis and draw a horizontal

line

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Graphing a Linear Equation by Using its Slope (Example)

Ex 1: Sketch the graph of the linear equation using its slope and y-intercept:

a)

b)

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2 xy

5 yx

Graphing Other Lines (Example)

Ex 2: Sketch the line:

a) y + 3 = 2

b) x = 4

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Finding the Slope of a Line

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Definition and Properties of Slope

• Slope (m): the ratio of the change in y (Δ y) and the change in x (Δ x)– Quantifies (puts a numerical value on) the “steepness” of a

line

• Given 2 points on a line, we can find its slope:

12

12

xx

yy

x

ym

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Sign of the Slope of a Line

• To determine the sign of the slope, examine the graph of the line from left to right:– Positive if the line rises– Negative if the line drops

Finding the Slope of a Line (Example)

Ex 3: Find the slope of the line:

a) Through (-2, 5) and (4, -5)

b) Vertical line through (3, 11)

c) Through (0, 8) and (1, 10)

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Writing Linear Equations

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Point Slope Formula & Standard Form of a Line

• Given the slope and a point on a line, we can construct its equation

• Point-slope formula: y – y1 = m(x – x1)– (x1, y1) is any point– x and y are variables– Very similar to the slope formula– Could also use y = mx + b

• Standard form: a linear equation in the form Ax + By = C where A, B, and C are constants– Variables to the left and constants on the right– NO fractions or decimals

Writing an Equation in Standard Form

Ex 4: Write the linear equation of the line in standard form:

a) Through (-4, -8) with a slope of -¼

b) Through (2, 0) with a slope of 1

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Equations of Lines when Given Two Points

• We know how to find the slope of a line given two points

• Proceed as before – Pick one of the two points to use in the point-

slope formula

Writing the Equation of a Line Given Two Points (Example)

Ex 5: Write the equation of the line in slope-intercept form:

a) Through (4, 7) and (8, 9)

b) Through (2, 1) and (-3, 6)

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Equations of Vertical and Horizontal Lines

• Recall the slopes of vertical and horizontal lines:– Slope of a vertical line is undefined– Slope of a horizontal line is zero

• Use the slope along with the given point to construct the equation of the line:– If the line is vertical, use the x-coordinate of the

given point– If the line is horizontal, use the y-coordinate of the

given point

Equations of Vertical and Horizontal Lines (Example)

Ex 6: Write the equation of the line:

a) Through (-2, 5) with undefined slope

b) Through (4, 13) with a slope of zero

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Parallel and Perpendicular Lines

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Slopes of Parallel and Perpendicular Lines

• Parallel lines: two lines that have the SAME slope

• Perpendicular lines: two lines that have OPPOSITE RECIPROCAL slopes– i.e. the product of the slopes is -1

Classifying Lines as Parallel, Perpendicular, or Neither (Example)

Ex 7: Determine whether the two lines are parallel, perpendicular, or neither:

a) 3x + y = -4 and 6y – 2x = 12

b) y = 2x + 1 and y = -2x – 3

c) 3y – 15x = -6 and y = 5x + 2 21

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Equations of Parallel and Perpendicular Lines

• Given the equation of a line, we want to find the equation of a second line that is parallel or perpendicular to the first

• Slope is not explicitly given– Put the equation of the given line in slope-intercept form

• Determine the appropriate slope based on whether the second line is to be parallel or perpendicular to the first

• Use the slope and the given point in the point-slope formula or y = mx + b

• A vertical line is parallel to another vertical line and a horizontal line is perpendicular to a vertical line– Vice versa for a horizontal line

Equations of Parallel and Perpendicular Lines (Example)

Ex 8: Write the equation of the line in standard form if possible:

a) Perpendicular to y = -3 and passes through (1, -5)

b) Parallel to y = ½x + 5 and passes through (4, 2)

c) Perpendicular to 3x + y = -1 and passes through (-1, -1)

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Summary

• After studying these slides, you should be able to:– Graph a line using its slope and y-intercept– Graph vertical or horizontal lines– Find the slope of a line– Write linear equations– Classify two lines as being parallel, perpendicular, or neither– Write equations involving parallel and perpendicular lines

• Additional Practice– See the list of suggested problems for 1.3

• Next lesson– Functions (Section 1.4)

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