Section 2.2 Linear Equations

Graphing Linear Equations Definition 1: A function whose graph is a line is a linear function. Definition 2: You can represent a linear function with a linear equation. Definition 3: Because the value of y depends on the value of x, y is called the dependent variable and x is called the independent variable.

Examples 1 – 2 Graph each equation. y = 2/3x + 3 y = 3/4x

Examples 3 – 4 Graph each equation. x + y = -2 y = -1/2x + 1/2

Graphing Linear Equations Definition 4: The y-intercept of a line is the point at which the line crosses the y-axis. Definition 5: The x-intercept of a line is the point at which the line crosses the x-axis. Definition 6: The standard form of an equation is Ax + By = C.

Example 5 Transportation: The equation 3x + 2y = 120 models the number of passengers who sit in a train car, where x is the number of adults and y is the number of children. Graph the equation. Explain what the x- and y-intercepts represent. Describe the domain and the range.

Example 5 Continued Suppose the train system buys new train cars with molded plastic seats. The model changes to x + y = 40. Graph the equation and interpret the x- and y-intercepts.

TOTD The school glee club needs a total of $4500 for a trip to Omaha, Nebraska. To make money, members are selling baseball caps for $4.50 and sweatshirts for $12.50. Construct and equation and graph the equation, where x is the number of baseball caps and y is the number of sweatshirts sold. Explain the meaning of the x- and y-intercepts in terms of fund-raising.

Slope Definition 7: The slope of a nonvertical line is the ratio of the vertical change to a corresponding horizontal change.

Examples of Slopes

Examples 6 – 8 Find the slope of the line through each pair of points. (3, 2) and (-9, 6) (-2, -2) and (4, 2) (0, -3) and (7, -9)

Writing Equations of Lines Definition 8: When you know the slope and a point on a line, you can use the point-slope form to write the equation of a line.

Examples 9 – 10 Write in slope-intercept form the equation of each line. m = -1/2; point: (8, -1) slope 2; through (4, -2)

Example 11 Write in slope-intercept form the equation of each line. slope 5/6; through (5, 6)

TOTD Find the slope of the line through each pair of points. (1, 6) and (8, -1) Write the equation of the line. Graph. slope = 3; (1, 5)

Examples 12 – 13 Write in slope-intercept form the equation of the line through each pair of point. (1, 5) and (4, -1) (5, 0) and (-3, 2)

Examples 14 – 15 Write in slope-intercept form the equation of the line through each pair of point. (-2, -1) and (-10, 17) (5, 1) and (-4, -3)

Writing Equations of Lines Definition 9: Another form of the equation of a line is slope-intercept form, which you can use to find the slope by examining the equation.

Examples 16 – 19 Find the slope of each line. 4x + 3y = 7 3x + 2y = 1 Ax + By = C

Summary: Equations of a line

TOTD Write in point-slope form the equation of the line through each pair of points. (-10, 3) and (-2, -5)

An Assortment of Lines Horizontal Line Vertical Line

An Assortment of Lines Perpendicular Lines Parallel Lines

Examples 20 – 21 Find the given lines. Find the equation of a line that is perpendicular to y = 3/4x + 2 and passes through the point (0,4). Find the equation that goes through the point (-1, 3) and is perpendicular to the line y = 5x – 3.

Examples 22 – 23 Find the given lines. Find the equation of a line that is parallel to y = 2/3x + 5/8 and passes through the point (2,1). Find the equation that is vertical and passes through the point (5, -3)

TOTD Write an equation for each line. Then graph the line. Through (-2, 1) and parallel to y = -3x + 1