Introduction In an equation with one variable, x, the solution will be the value that makes the equation true. For example: 1 is the solution for the equation x = 1. 2 is the solution for the equation 2 x = 4. The solution of an equation with two variables x and y is the pair of values (x, y) that make the equation true. For example: (1, 2) is a solution to the equation y = 2x because the statement 2 = 2 is true. (1, 3) is not a solution for y = 2x because the statement 3 = 2 is false : Graphing the Set of All Solutions

Introduction, continued The pairs of values (x, y) are called ordered pairs, and the set of all ordered pairs that satisfy the equation is called the solution set. Each ordered pair in the solution set represents a point in the coordinate plane. When we plot these points, they will begin to form a curve. A curve is a graphical representation of the solution set for the equation. In the special case of a linear equation, the curve will be a straight line. A linear equation is an equation that can be written in the form ax + by = c, where a, b, and c are rational numbers. It can also be written as y = mx + b, in which m is the slope, and b is the y-intercept : Graphing the Set of All Solutions

Introduction, continued It is important to understand that the solution set for most equations is infinite; therefore, it is impossible to plot every point. There are several reasons the solution set is infinite; one reason is that there is always a number between any two numbers x 1 and x 2, and for that number there will be a y that satisfies the equation. So when we graph the solution set for an equation, we plot several points and then connect them with the appropriate curve. The curve that connects the points represents the infinite solution set to the equation : Graphing the Set of All Solutions

Key Concepts A solution to an equation with two variables is an ordered pair, written (x, y). Ordered pairs can be plotted in the coordinate plane. The path the plotted ordered pairs describe is called a curve. A curve may be without curvature, and therefore is a line : Graphing the Set of All Solutions

Key Concepts, continued An equation whose graph is a line is a linear equation. The solution set of an equation is infinite. When we graph the solution set of an equation, we connect the plotted ordered pairs with a curve that represents the complete solution set : Graphing the Set of All Solutions

Common Errors/Misconceptions believing the number of solutions an equation has is limited to points seen on the graph incorrectly evaluating the equation for different given values incorrectly plotting ordered pair solutions on a coordinate plane : Graphing the Set of All Solutions

Guided Practice Example 2 Graph the solution set for the equation y = 3 x : Graphing the Set of All Solutions

Guided Practice: Example 2, continued 1.Make a table. Choose at least 3 values for x and find the corresponding values of y using the equation : Graphing the Set of All Solutions xy –2 –

Guided Practice: Example 2, continued 2.Plot the ordered pairs in the coordinate plane : Graphing the Set of All Solutions

Guided Practice: Example 2, continued 3.Notice the points do not fall on a line. The solution set for y = 3 x is an exponential curve. Connect the points by drawing a curve through them. Use arrows at each end of the line to demonstrate that the curve continues indefinitely in each direction. This represents all of the solutions for the equation : Graphing the Set of All Solutions

Guided Practice: Example 2, continued : Graphing the Set of All Solutions ✔

: Graphing the Set of All Solutions Guided Practice: Example 2, continued 12

Guided Practice Example 3 The Russell family is driving 1,000 miles to the beach for vacation. They are driving at an average rate of 60 miles per hour. Write an equation that represents the distance remaining in miles and the time in hours they have been driving, until they reach the beach. They plan on stopping 4 times during the trip. Draw a graph that represents all of the possible distances and times they could stop on their drive : Graphing the Set of All Solutions

Guided Practice: Example 3, continued 1.Write an equation to represent the distance from the beach. Let d = 1000 – 60t, where d is the distance remaining in miles and t is the time in hours : Graphing the Set of All Solutions

Guided Practice: Example 3, continued 2.Make a table. Choose values for t and find the corresponding values of d. The trip begins at time 0. Let 0 = the first value of t. The problem states that the Russells plan to stop 4 times on their trip. Choose 4 additional values for t. Let’s use 2, 5, 10, and 15. Use the equation d = 1000 – 60t to find d for each value of t. Fill in the table : Graphing the Set of All Solutions

Guided Practice: Example 3, continued : Graphing the Set of All Solutions td

Guided Practice: Example 3, continued 3.Plot the ordered pairs on a coordinate plane : Graphing the Set of All Solutions Distance remaining in miles Time in hours

Guided Practice: Example 3, continued 4.Connect the points by drawing a line. Do not use arrows at each end of the line because the line does not continue in each direction. This represents all of the possible stopping points in distance and time : Graphing the Set of All Solutions

Guided Practice: Example 3, continued : Graphing the Set of All Solutions ✔ Distance remaining in miles Time in hours

: Graphing the Set of All Solutions Guided Practice: Example 3, continued 20