Section 7.1 Math in Our World The Rectangular Coordinate System and Linear Equations in Two Variables

Learning Objectives Plot points in a rectangular coordinate system. Graph linear equations. Find the slope of a line. Graph linear equations in slope-intercept form. Graph horizontal and vertical lines. Find linear equations that describe real-world situations.

Rectangular Coordinate System The foundation of graphing in mathematics is a system for locating data points using a pair of perpendicular number lines. We call each one an axis. The horizontal line is called the x axis, and the vertical line is called the y axis. The point where the two intersect is called the origin. Collectively, they form what is known as a rectangular coordinate system, sometimes called the Cartesian plane. The two axes divide the plane into four regions called quadrants. They are numbered using Roman numerals I, II, III, and IV.

Rectangular Coordinate System The location of each point is given by a pair of numbers called the coordinates, and are written as (x, y), where the first number describes a number on the x-axis and the second describes a number on the y-axis. The coordinates of the origin are (0, 0). A point P whose x coordinate is 2 and whose y coordinate is 5 is written as P = (2, 5). It is plotted by starting at the origin and moving two units right and five units up.

EXAMPLE 1 Plotting Points Plot the points (5, –3), (0, 4), (–3, –2), (–2, 0), and (2, 6). SOLUTION To plot each point, start at the origin and move left or right according to the x value, and then up or down according to the y value.

Identifying Coordinates Given a point on the plane, its coordinates can be found by drawing a vertical line back to the x axis and a horizontal line back to the y axis. For example, the coordinates of point C shown are (–3, 4).

EXAMPLE 2 Finding the Coordinates of Points Find the coordinates of each point shown on the plane. SOLUTION A = (1, 4) B = (–2, 6) C = (–5, 0) D = (0, 3) E = (–4, –4)

Slope The slope of a line (designated by m) is where (x1, y1) and (x2, y2) are two points on the line. The “slope” can be defined as the “rise” (vertical height) divided by the “run” (horizontal distance) or as the change in y with respect to the change in x.

EXAMPLE 5 Finding the Slope of a Line Find the slope of a line passing through the points (2, 3) and (5, 8). SOLUTION Designate the points as follows Substitute into the formula That means the line is rising 5 feet vertically for every 3 feet horizontally.

Slope When finding slope, it doesn’t matter which of the two points you choose to call (x1, y1) and which you call (x2, y2). But the order of the subtraction in the numerator and denominator has to be consistent If the line goes “uphill” from left to right, the slope will be positive. If a line goes “downhill” from left to right, the slope will be negative. The slope of a vertical line is undefined. The slope of a horizontal line is 0.

Slope-Intercept Form The slope-intercept form for an equation in two variables is y = mx + b, where m is the slope and (0, b) is the y intercept. If we start with the equation 5x – 3y = 15 from Example 6 and solve the equation for y, Notice that the coefficient of x is 5/3, which is the same as the slope of the line, as found in Example 6.

EXAMPLE 7 Using Slope-Intercept Form to Draw a Graph Graph the line SOLUTION The slope is 5/3 and the y intercept is (0, –6). Starting at the point (0, –6), we move vertically upward 5 units for the rise, and move horizontally 3 units right for the run. That gives us second point (3, –1). Then draw a line through these points. To check, notice that (3, –1) satisfies the equation.

Horizontal and Vertical Lines Think about what the equation y = 3 says in words: that the y coordinate is always 3. This is a line whose height is always 3, which is a horizontal line. Similarly, an equation like x = – 6 is a vertical line with every point having x coordinate – 6.

EXAMPLE 8 Graphing Vertical and Horizontal Lines Graph each line: (a) x = 5 and (b) y = – 3. SOLUTION (a) The graph of x = 5 is a vertical line with every point having x coordinate 5. We draw it so that it passes through 5 on the x axis. (b) The graph of y = – 3 is a horizontal line with every point having y coordinate 3. We draw it so that it passes through – 3 on the y axis.

EXAMPLE 9 Finding a Linear Equation Describing Cab Fare The standard fare for a taxi in one city is $5.50, plus $0.30 per mile. Write a linear equation that describes the cost of a cab ride in terms of the length of the ride in miles. Then use your equation to find the cost of a 6-mile ride, an 8.5-mile ride, and a 12-mile ride.

EXAMPLE 9 Finding a Linear Equation Describing Cab Fare SOLUTION The first quantity that varies in this situation is the length of the trip, so we will assign variable x to number of miles. The corresponding quantity that changes is the cost, so we will let y = the cost of the ride. Since each mile costs $0.30, the total mileage cost is 0.30x. Adding the upfront cost of $5.50, the total cost is given by y = 0.30x + 5.50. Now let’s evaluate for x = 6, x = 8.5 and x = 12 miles,

Slope and Rate of Change The slope of any line tells us the rate at which y changes with respect to x.

EXAMPLE 10 Finding a Linear Equation Describing Distance After a brisk bike ride, you take a break and set out for home. Let’s say you start out 15 miles from home and decide to relax on the way home and ride at 9 miles per hour. Write a linear equation that describes your distance from home in terms of hours, and use it to find how long it will take you to reach home.

EXAMPLE 10 Finding a Linear Equation Describing Distance SOLUTION In this case, we know two key pieces of information: at time zero (when you start out for home) the distance is 15, and the rate at which that distance is changing is – 9 miles per hour (negative because the distance is decreasing). The rate is the slope of a line describing distance, and the distance when time is zero is the y intercept. Let y = distance and x = hours after starting for home. y = – 9x + 15.

EXAMPLE 10 Finding a Linear Equation Describing Distance SOLUTION You reach home when the distance (y) is zero, so substitute in y = 0 and solve for x: It will take 1 hour and 40 minutes to get home.