Use the equation to complete parts (a) through (c). Build the equation on your Equation Mat using and simplify it as much as possible. Record your steps and what you see when you have simplified the equation fully. Draw a picture of your final mat. Have each member of your team test a different value for x in the original equation, such as x = 0, x = 1, x = −5, x = 10, etc. What happens in each case? Are any solutions to this equation? If so, how many? SPECIAL CASES, Part Two Use the equation to complete parts (a) through (c). Build the equation on your Equation Mat using and simplify it as much as possible. Record your steps and what you see when you have simplified the equation fully. Draw a picture of your final mat. Have each member of your team test a different value for x in the equation, such as x = 0, x = 1, x = −5, x = 10, etc. What happens? Is there a pattern to the results you get from the equation? Did you find any values for x that satisfied the equation in part (a)? When is an imbalance of units left on the mat (such as 2 = 0), what does this mean? Is x = 0 a solution to the equation? Keeping these special cases in mind, continue to develop your equation-solving strategies by visualizing and solving each equation below. Remember to build each equation on your mat, if needed, to simplify as much as possible, and to solve for x. Identify if one number is the solution, if any number is the solution, or if there is no solution. Record your steps. Additional Challenge: For each of the parts below, create your own equation (involving at least three steps) that has the given solution. x = –2 x = no solution all numbers Investigation 13: Choosing a Solving Strategy
Checking a Solution To check a solution to an equation, substitute the solution into the equation and verify that it makes the two sides of the equation equal. For example, to verify that x = 10 is a solution to the equation 3(x − 5) = 15, substitute 10 into the equation for x and then verify that the two sides of the equation are equal. As shown at right, x = 10 is a solution to the equation 3(x − 5) = 15. What happens when you do this check if your answer is incorrect? For example, try substituting x = 2 into the same equation. The result shows that x = 2 is not a solution to this equation. Investigation 9: Solving a Word Problem Math Notes Solving Problems with the 5D Process The 5D Process is an organized method to problems. The D’s stand for Describe/Draw, Define, Do, Decide, and Declare. An example of this work is shown below. Problem: The base of a rectangle is 13 centimeters longer than the height. If the perimeter is 58 centimeters, find the base and the height of the rectangle. Describe/Draw: The shape is a rectangle and we are looking at the perimeter. Investigation 10: a Table to Write Equations from Word Problems Math Notes Defining a Variable When you write an equation, it is important to define the variable carefully. You need to be clear about what you are talking about so that someone else looking at your work understands what the variable represents. This step is an important habit to develop because it is an important step in solving many different math problems.