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

Learning Objectives  Plot points in a rectangular coordinate system.  Graph linear equations.  Find the intercepts of a linear equation.  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)

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. Let’s look at the example 3x + y = 6. If we choose a pair of numbers to substitute in for x and y, the resulting equation will be either true or false. For example, for x = 1 and y = 3, the equation becomes 3(1) + 3 = 6, which is a true statement. We call the pair (1, 3) a solution to the equation, and say the pair satisfies the equation.

Linear Equations in Two Variables Listed below are a handful of pairs that satisfy the equation 3x + y = 6. (0, 6) (1, 3) (2, 0) (3, – 3) (4, – 6) 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 with a line, the result is called the graph of the equation. The graph is a geometric representation of every pair of numbers that is a solution to the equation.

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.

EXAMPLE 3 Graphing a Linear Equation in Two Variables SOLUTION 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.

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 0 for y and solve the equation for x. To find the y intercept, substitute 0 for x and solve the equation for y.

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).

EXAMPLE 4 Finding Intercepts SOLUTION 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.)

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.

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 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

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.

Slope-Intercept Form If we 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.

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.

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

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 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 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 SOLUTION

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