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1.3 Linear Equations in Two Variables

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Objective Write linear equations in two variables. Use slope to identify parallel and perpendicular lines. Use slope and linear equations in two variables to model and solve real-life problems.

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The Slope-intercept Form of the Equation of a Line The graph of the equation y = mx+b is a line whose slope is m and whose y- intercept is (0, b)

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A vertical line has an equation of the form x = a. The equation of a vertical line cannot be written in the form y = mx + b because the slope of a vertical line is undefined.

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Finding the Slope of a Line Passing Through Two Points The slope m of the nonvertical line through

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Example 1 Finding the Slope of a Line Through Two Points Find the slope of the line passing through each pair of points. a. (-5, -6) and (2,8)b. (3, 4) and (3, 1)

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Writing Linear Equations in Two Variables Point-Slope Form of the Equation of a Line The equation of the line with slope m passing through the point

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Example 2Using the Point-Slope Find the slope-intercept form of the equation of the line that has a slope of 2 and passes through the point (3, -7).

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From the point-slope form we can determine the two-point form of the equation of a line.

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Parallel and Perpendicular Lines 1.Two distinct nonvertical lines are parallel if and only if their slopes are equal. That is

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Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. That is,

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Example 3Finding Parallel and Perpendicular Lines Find the slope-intercept form of the equation of the line that passes through the point (-4, 1) and is (a) parallel to and (b) perpendicular to the line

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Applications In real-life problems, the slope of a line can be interpreted as either a ratio or rate of change. If the x- and y-axis have the same unit of measure, then the slope has no units and is a ratio. If the x-axis and y-axis have different units of measure, then the slope is a rate or rate of change.

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Example 4Using slope as a ratio When driving down a mountainside, you notice warning signs indicating that the road has a grade of twelve percent. This means that the slope of the road is - 12/100. Approximate the amount of horizontal change in your position if you note from elevation markers that you have descended 2000 feet vertically.

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Example 5 The following are the slopes of lines representing annual sales y in terms of time x in years. Use the slopes to interpret any change in annual sales for a 1-year increase in time a.The line has a slope of m = 120. b.The line has a slope of m = 0. c.The line has a slope of m = -35.

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a.The line has a slope of m = 120. Sales increasing 120 units per year. b.The line has a slope of m = 0. No change in sales. c.The line has a slope of m = -35. Sales decreasing 35 units per year.

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Example 6Straight Line Depreciation A person purchases a car for $25,290. After 11 years, the car will have to be replaced. Its value at that time is expected to be $1200. Write a linear equation giving the value V of the car during the 11 years in which it will be used.

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Example 7Predicting Sales per Share A business purchases a piece of equipment for $34,200. After 15 years, the equipment will have to be replaced. Its value at that time is expected to be $1500. Write a linear equation giving the value V of the equipment during the 15 years in which it will be used. Use the equation to estimate the value of the equipment after 7 years.

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Summary of Equations of Lines

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