Special Cases and Applications Show Learning Objective(s) · Solve equations that have one solution, no solution, or an infinite number of solutions. · Solve application problems by using an equation in one variable. Introduction When you follow the steps to solve an equation, you try to isolate the variable. You have a solution when you get the equation x = some value. There are equations, however, that have no solution, and other equations that have an infinite number of solutions. How does this work? Algebraic Equations with No Solution Let’s apply the steps for solving an algebraic equation to the equation below.
This is not a solution! You did not find a value for x. Solving for x the way you know how, you arrive at the false statement 4 = 5. Surely 4 cannot be equal to 5! This may make sense when you consider the second line in the solution where like terms were combined. If you multiply a number by 2 and add 4 you would never get the same answer as when you multiply that same number by 2 and add 5. Since there is no value of x that will ever make this a true statement, the solution to the equation above is “no solution”. Be careful that you do not confuse the solution x = 0 with “no solution”. The solution x = 0 means that the value 0 satisfies the equation, so there is a solution. “No solution” means that there is no value, not even 0, which would satisfy the equation. Also, be careful not to make the mistake of thinking that the equation 4 = 5 means that 4 and 5 are values for x thatare solutions. If you substitute these values into the original equation, you’ll see that they do not satisfy the equation. This is because there is truly no solution—there are no values for x that will make the equation 12 + 2x – 8 = 7x + 5 – 5x true.
Algebraic Equations with an Infinite Number of Solutions You have seen that if an equation has no solution, you end up with a false statement instead of a value for x. You can probably guess that there might be a way you could end up with a true statement instead of a value for x.
You arrive at the true statement “3 = 3”. When you end up with a true statement like this, it means that the solution to the equation is “all real numbers”. Try substituting x = 0 into the original equation—you will get a true statement! Try This equation happens to have an infinite number of solutions. Any value for x that you can think of will make this equation true. When you think about the context of the problem, this makes sense—the equation x + 3 = 3 + x means “some number plus 3 is equal to 3 plus that same number.” We know that this is always true—it’s the commutative property of addition!
When solving an equation, multiplying both sides of the equation by zero is not a good choice. Multiplying both side of an equation by 0 will always result in an equation of 0 = 0, but an equation of 0 = 0 does not help you know what the solution to the original equation is.
In solving the algebraic equation 2(x – 5) = 2x + 10, you end up with −10 = 10. What does this mean? A) x = −10 and 10 B) There is no solution to the equation. C) You must have made a mistake in solving the equation. D) x = all real numbers Advanced Question How many solutions are there for the equation: A) There is one solution. B) There are two solutions. C) There are an infinite number of solutions. D) There are no solutions. The power of algebra is how it can help you model real situations in order to answer questions about them. This requires you to be able to translate real-world problems into the language of algebra, and then be able to interpret the results correctly. Let’s start by exploring a simple word problem that uses algebra for its solution. Amanda’s dad is twice as old as she is today. The sum of their ages is 66. Use an algebraic equation to find the ages of Amanda and her dad. One way to solve this problem is to use trial and error—you can pick some numbers for Amanda’s age, calculate her father’s age (which is twice Amanda’s age), and then combine them to see if they work in the equation. For example, if Amanda is 20, then her father would be 40 because he is twice as old as she is, but then their combined age is 60, not 66. What if she is 12? 15? 20? As you can see, picking random numbers is a very inefficient strategy! You can represent this situation algebraically, which provides another way to find the answer.
Let’s try a new problem. Consider that the rental fee for a landscaping machine includes a one-time fee plus an hourly fee. You could use algebra to create an expression that helps you determine the total cost for a variety of rental situations. An equation containing this expression would be useful for trying to stay within a fixed expense budget.
Using the information provided in the problem, you were able to create a general expression for this relationship. This means that you can find the rental cost of the machine for any number of hours! Let’s use this new expression to solve another problem.
It is often helpful to follow a list of steps to organize and solve application problems. Solving Application Problems Follow these steps to translate problem situations into algebraic equations you can solve. 1. Read and understand the problem. 2. Determine the constants and variables in the problem. 3. Write an equation to represent the problem. 4. Solve the equation. 5. Check your answer. 6. Write a sentence that answers the question in the application problem. Let’s try applying the problem-solving steps with some new examples.
Advanced Question Albert and Bryn are buying candy at the corner store. Albert buys 5 bags and 3 individual pieces; Bryn buys 3 bags and then eats 2 pieces of candy from one of the bags. Each bag has the same number of pieces of candy. After Bryn eats the 2 pieces, she has exactly half the number of pieces of candy as Albert. How many pieces of candy are in each bag? Pick the equation that could be used to solve the problem above. Use the variable b to represent the number of pieces of candy in one bag. A) B)
C) D) Summary Some equations are considered special cases. These are equations that have no solution and equations whose solution is the set of all real numbers. When you use the steps for solving an equation, and you get a false statement rather than a value for the variable, there is no solution. When you use the steps for solving an equation, have avoided multiplying both sides of the equation by zero, and you get a true statement rather than a value for the variable, the solution is all real numbers. Algebra is a powerful tool for modeling and solving real-world problems. What does 0 0 mean in a system of equations?0 = 0. Since 0 = 0 for any value of x, the system of equations has infinite solutions.
What does 0 0 mean in a solution?If you solve this your answer would be 0=0 this means the problem has an infinite number of solutions. For an answer to have no solution both answers would not equal each other.
Is 0 and no solution the same?Be careful that you do not confuse the solution x = 0 with “no solution”. The solution x = 0 means that the value 0 satisfies the equation, so there is a solution. “No solution” means that there is no value, not even 0, which would satisfy the equation.
What does it mean for a system to have no solution?If a system has no solution, it is said to be inconsistent . The graphs of the lines do not intersect, so the graphs are parallel and there is no solution.
|