Linear functions are a powerful tool in mathematics and are used in many different areas. Linear functions are equations that have the form f(x) = mx + b, where m and b are constants. Linear functions have many important properties, such as being able to represent straight lines on a graph. In this lesson, we will look at how to identify, graph, and solve linear functions.

## Identifying Linear Functions

The first step in working with linear functions is to be able to identify them. In order for a function to be linear, it must have the form f(x) = mx + b, where m and b are constants. To verify that a function is linear, we can substitute different values for x and see if the result is always the same. For example, if we substitute x = 3 and x = 5 into the equation f(x) = 3x + 2, we get f(3) = 11 and f(5) = 17. Both of these results are the same, so the equation is linear.

## Graphing Linear Functions

Once a linear function has been identified, it can be graphed. This is done by plotting points on a coordinate plane. For example, if the equation is f(x) = 3x + 2, we can plot the points (0,2), (1,5), (2,8), (3,11), and (4,14). We can then use these points to draw the line that represents the equation. This line will be a straight line, since linear functions always result in straight lines when graphed.

## Solving Linear Functions

Finally, linear functions can be solved to find the value of x. This is done by substituting the given values into the equation and solving for x. For example, if we have the equation f(x) = 3x + 2 and we know that f(x) = 11, we can substitute 11 in for f(x) and solve for x. Doing this gives us x = 3, which is the value of x that makes the equation true.

Linear functions are powerful tools in mathematics and are used to represent many different types of data. They can be identified by verifying that they have the form f(x) = mx + b, where m and b are constants. They can be graphed by plotting points on a coordinate plane and drawing the resulting line. Finally, linear functions can be solved to find the value of x by substituting the given values into the equation and solving for x.

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