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 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

Guided Practice Example 1 Graph the solution set for the equation -3x + y = : Graphing the Set of All Solutions y = 3x – 2

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 – y = 3 x

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 ✔

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

Distance remaining in miles Time in hours Guided Practice: Example 3, continued 3.Plot the ordered pairs on a coordinate plane. td

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

Distance remaining in miles Time in hours Guided Practice: Example 3, continued td

Introduction Recall that the graph of an equation, such as y = x + 1, is the complete set of solutions for that equation. The values for y are the result of a process being done to x. To make a general statement, we call the process by a letter, such as f, and we can call the results of that process “f of x.” We write “f of x” as f(x). The process f is a function; in a function, every element of the domain is paired with exactly one element of the range. That is, for every value of x, there is exactly one value of y. Both y and f(x) represent the outcome of the function f on x. So, for the equation y = x + 1, adding 1 to x is the function f. Therefore, f(x) is x + 1 and y = f(x) : Intersecting Graphs

Introduction, continued Writing equations this way is called function notation. Function notation is a way to name a function using f(x) instead of y. Using function notation, we can graph more than one function at a time. A set of more than one equations is called a system. If we call one function f and another g, then we can graph y = f(x) and y = g(x) on the same coordinate plane. By graphing the functions f and g on the same plane, we can more easily see the functions’ differences or similarities : Intersecting Graphs

Introduction, continued Functions can be named using any letter, though f and g are used often. In this lesson, we will work with pairs of equations, f(x) and g(x), attempting to find a solution set where f(x) = g(x). In other words, we want to find all values for x where f(x) and g(x) are the same : Intersecting Graphs

Key Concepts Graphing Solutions of Functions In the graph of the functions f(x) and g(x), the set of solutions for f(x) = g(x) will be where the two graphs intersect. If f(x) = g(x) = b for a particular value of x, say a, then the point (a, b) will be on the curve defined by f and will also fall on the curve defined by g. Since (a, b) falls on both curves, they must intersect at point (a, b) : Intersecting Graphs

Key Concepts, continued In the graphs of f(x) and g(x), look for the point(s) where the curves intersect. Substitute the x-value from the point into the functions to see if it is, or is close to, a solution. Note that it is possible for a system of equations to intersect at more than one point, at only one point, or to not intersect at all : Intersecting Graphs

Key Concepts, continued : Intersecting Graphs Points of Intersection More than oneOneNone Both points of intersection are solutions to the system. The point of intersection is the system’s only solution. There is no solution to the system.

Key Concepts, continued Using a Table of Values to Find Solutions Making a table of values for a system of two equations means listing the inputs for each equation and then comparing the outputs. List the values of x to substitute into the first column. List the first equation in the second column and the corresponding outputs, which are the resulting values after substituting in the x-values chosen in the first column : Intersecting Graphs

Key Concepts, continued List the second equation in the third column and the corresponding outputs, which are the resulting values after substituting in the x-values chosen in the first column. If the difference between the outputs is 0, the value of x that was substituted is the x-coordinate of the solution, and the corresponding output is the y-coordinate of the solution : Intersecting Graphs

Guided Practice Example 4 Use a graph to approximate the solutions for the following system of equations. f(x) = 2 x g(x) = x : Intersecting Graphs

Guided Practice: Example 4, continued 1.Graph f(x) = 2 x and g(x) = x + 2 on the same coordinate plane : Intersecting Graphs

Guided Practice: Example 4, continued 2.Approximate the values for x where f(x) = g(x). From the graph, –2 and 2 should be good estimates : Intersecting Graphs

Guided Practice Example 5 Use a table of values to approximate the solutions for the following system of equations: f(x) = 3 x g(x) = 2 x : Intersecting Graphs

Guided Practice: Example 5, continued 1.Create a table of values : Intersecting Graphs xf(x) = 3 x g(x) = 2 x + 1 – The output values are the same when x = 1. This tells us the curves f and g intersect at x = 1.