Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section 7.1 The Rectangular Coordinate System and Linear Equations in Two Variables Math in Our World

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Learning Objectives o Plot points in a rectangular coordinate system. o Graph linear equations. o Find the intercepts of a linear equation. o Find the slope of a line. o Graph linear equations in slope-intercept form. o Graph horizontal and vertical lines. o Find linear equations that describe situations in our world.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Rectangular Coordinate System The foundation of graphing in math 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.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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. Negative coordinates correspond to negative numbers on the axes, so a point like (–5, –4) is plotted by starting at the origin, moving five units left and four units down.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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).

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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)

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Linear Equations in Two Variables An equation of the form ax + by = c, where a, b, and c are real numbers, is called a linear equation in two variables. Consider the equation y = 2x + 6. If we choose a pair of numbers to substitute into the equation for x and y, the resulting equation is either true or false. For example, for x = 4 and y = 14, the equation is 14 = 2(4) + 6, which is a true statement. We call the pair (4, 14) a solution to the equation, and say the pair of numbers satisfies the equation.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Linear Equations in Two Variables Here’s a list of some pairs that satisfy the equation y = 2x + 6. (0, 6) (1, 8) (2, 10) (3, 12) (4, 14) If we plot the points corresponding to these pairs, all of the points appear to line up in a straight line pattern. If we connect these points plotted with a line, the result is called the graph of the equation. The graph is a way to geometrically represent every pair of numbers that is a solution to the equation.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. EXAMPLE 3 Graphing a Linear Equation in Two Variables Graph x + 2y = 5 SOLUTION Only two points are necessary to find the graph of a line, but it’s a good idea to find three. To find pairs of numbers that make the equation true, we will choose some numbers to substitute in for x, then solve the resulting equation to find the associated y.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. EXAMPLE 3 Graphing a Linear Equation in Two Variables SOLUTION CONTINUED In this case, we chose x = –1, x = 1, and x = 5, but any three will do. Three points on the graph are (–1, 3), (1, 2), and (5, 0). We plot those three points and draw a straight line through them.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Intercepts The point where a graph crosses the x axis is called the x intercept. The point where a graph crosses the y axis is called the y intercept. Every point on the x axis has y coordinate zero, and every point on the y axis has x coordinate zero, so we get the following rules. Finding Intercepts To find the x intercept, substitute zero for y and solve the equation for x. To find the y intercept, substitute zero for x and solve the equation for y.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. EXAMPLE 4 Finding Intercepts Find the intercepts of 2x – 3y = 6, and use them to draw the graph. SOLUTION To find the x intercept, let y = 0 and solve for x. The x intercept has the coordinates (3, 0). To find the y intercept, let x = 0 and solve for y. The y intercept has the coordinates (0, –2).

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. EXAMPLE 4 Finding Intercepts SOLUTION CONTINUED Now we plot the points (3, 0) and (0, –2), and draw a straight line through them. (It would still be a good idea to find one additional point to check your work. If the three points don’t line up, there must be a mistake.)

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slope 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. The slope of a line (designated by m) is where (x 1, y 1 ) and (x 2, y 2 ) are two points on the line.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slope 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. When finding slope, it doesn’t matter which of the two points you choose to call (x 1, y 1 ) and which you call (x 2, y 2 ). But the order of the subtraction in the numerator and denominator has to be consistent.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. EXAMPLE 6 Finding Slope Given the Equation of a Line Find the slope of the line 5x – 3y = 15. SOLUTION Find the coordinates of any two points on the line. In this case, we choose the intercepts, which are (3, 0) and (0, – 5). Then substitute into the slope formula.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slope-Intercept Form Start with the equation 5x – 3y = 15 from Example 6 and solve the equation for y: 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. Notice that the coefficient of x is 5/3, which is the same as the slope of the line, as found in Example 6.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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. Horizontal and Vertical Lines

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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 can 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 0.30 times the number of miles: 0.30x. Adding the upfront cost of $5.50, the total cost is given by y = 0.30x Now let’s evaluate for x = 6, x = 8.5 and x = 12 miles,

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slope and Rate of Change The slope of any line is the rate of change of y with respect to x.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. EXAMPLE 10 Using Rate of Change to Model With a Linear Equation The United States Census Bureau uses demographic information to set a poverty threshold that is used to determine how many Americans are living in poverty based on annual income. For an individual on her own, the poverty threshold was $4,190 in 1980, and has increased at the rate of about $220 per year since then. Write a linear equation that describes the poverty threshold in dollars in terms of years after Then use your equation to estimate the poverty threshold in 2010, and the year that it will pass $15,000 per year.

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. EXAMPLE 10 Using Rate of Change to Model With a Linear Equation The problem gives us two key pieces of information: the poverty threshold was $4,190 at time zero (we’re told to use years after 1980, and 1980 is zero years after 1980) and the threshold is changing at the rate of +$220 per year (positive because the distance is increasing). The rate of change is the slope of a line describing the poverty threshold, and the threshold at time zero is the y intercept. Let y = the poverty threshold and x = number of years after y = 220x + 4,190. To estimate the threshold in 2010, we need to substitute in x = 30 because 2010 is 30 years after y = 220(30) + 4,190 = $10,790 SOLUTION

Copyright © 2015 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. EXAMPLE 10 Using Rate of Change to Model With a Linear Equation To find the year when the threshold is expected to pass $15,000 per year, we need to substitute in 15,000 for y and solve for x: SOLUTION CONTINUED So the threshold should pass $15,000 about 49 years after 1980, which is 2029.