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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 2 Using 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**

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 3 Finding 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 4 Using 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 The line has a slope of m = 120. The line has a slope of m = 0. c. The line has a slope of m = -35.

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**The line has a slope of m = 120.**

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

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**Example 6 Straight 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 $ 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 7 Predicting 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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Graph a linear equation Graph: 2x – 3y = -12 Solve for y so the equation looks like y = mx + b - 3y = -2x – 12 Subtract 2x to both sides. y = x + 4 Divide.

Graph a linear equation Graph: 2x – 3y = -12 Solve for y so the equation looks like y = mx + b - 3y = -2x – 12 Subtract 2x to both sides. y = x + 4 Divide.

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