Yuanxin (Amy) Yang Alcocer
Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.
View bio
Kathryn Boddie
Kathryn has taught high school or university mathematics for over 10 years. She has a Ph.D. in Applied Mathematics from the University of Wisconsin-Milwaukee, an M.S. in Mathematics from Florida State University, and a B.S. in Mathematics from the University of Wisconsin-Madison.
View bio
When solving equations in algebra, there are two cases in which the solutions do not make sense: infinite solutions and no solutions. Explore the definition of and differences between infinite and no solutions, and understand examples of each. Updated: 10/01/2021
When you're solving equations in algebra, it is kind of like a treasure hunt. You are looking for your x. You want to know where your x is, so you can go find your treasure. With most equations, you will get an answer letting you know where your treasure is located. For example, solving the equation x + 3 = 4 by subtracting 3 from both sides gives us x = 1 as our answer and location of our treasure.
But sometimes, an equation that you are trying to solve for gives you an answer that just doesn't make sense. It is these types of answers that we are going to discuss in this video lesson. It is important to understand these so you can spot them and identify the equations as unsolvable because they have an answer that doesn't make sense. We will go over the two possible cases where the answer doesn't make sense.
- Video
- Quiz
- Course
Infinite Solutions
The first is when we have what is called infinite solutions. This happens when all numbers are solutions. This situation means that there is no one solution. In terms of our treasure hunt, it means that we can't find the treasure because the location of the treasure can be anywhere. There is no x that marks the spot. Our x here marks the whole world, which doesn't help us.
The equation 2x + 3 = x + x + 3 is an example of an equation that has an infinite number of solutions. Let's see what happens when we solve it. We first combine our like terms. We see two x terms that we can combine to make 2x.
2x + 3 = 2x + 3.
Now we can subtract 3 from both sides: 2x = 2x. Hmm. This is an interesting situation; both sides are equal to each other. How many different values of x will make this equation true? Why, isn't it any number? Yes, and so we have our infinite solutions.
No Solutions
The next case is what is called no solutions. In this case, we have no answer. Our problem equation is a dud. In terms of helping us find our treasure, it actually leads us down the wrong path, to a dead end, so to speak. We think we are going somewhere, but in the end, this equation just laughs at us with an end that doesn't make sense.
Practice Problems - Equations with Infinite Solutions or No Solutions
In the following practice problems, students will practice solving equations to determine if there is one solution, infinitely many solutions, or no solution. They will also graph each side of the equation on the same graph to make a discovery.
Problems
Solve the following
equations to determine if there is one solution, infinitely many solutions, or no solution. Then, follow the instructions to make a graph.
1. x + 7 + 2x = x - 9. After solving, graph each side of the equation: y = x + 7 + 2x and y = x - 9.
2. 2x - x + 2 = -5x + 6x - 3. After solving, graph each side of the equation: y = 2x - x + 2 and y = -5x + 6x
-3.
3. x + 5 = 3x - 2x + 7 - 2. After solving, graph each side of the equation: y = x + 5 and y = 3x - 2x + 7 - 2.
Solutions
1. Combining like terms on each side, we have 3x + 7 = x - 9. Subtracting x from both sides and subtracting 7 from both sides, we have 2x = -16. Dividing by 2 gives x = -8. This equation has one solution. To graph, first combine like terms in the first equation to get y = 3x + 7 and y = x - 9.
The graphs intersect each other at the solution x = -8.
2. Combining like terms on each side, x + 2 = x - 3. Subtracting x from both sides of the equation gives 2 = -3. This equation has no solution. To graph, first combine like terms on each side to get the equations y = x + 2 and y = x - 3.
The graphs never intersect - they are parallel lines.
3. Combining like terms on each side, x + 5 = x + 5. Subtracting 5 from both sides of the equation gives x = x. This equation has infinitely many solutions. To graph, first combine like terms on each side to get the equations y = x + 5 and y = x + 5.
The graphs intersect everywhere - they are the same line.
As seen in the graphs above, the solutions to an equation can be viewed as the intersections of the graphs of the left and right sides.
Register to view this lesson
Are you a student or a teacher?
Unlock Your Education
See for yourself why 30 million people use Study.com
Become a Study.com member and start learning now.Become a Member
Already a member? Log In
Back
Resources created by teachers for teachers
Over 30,000 video lessons & teaching resources‐all in one place.
Video lessons
Quizzes & Worksheets
Classroom Integration
Lesson Plans
I would definitely recommend Study.com to my colleagues. It’s like a teacher waved a magic wand and did the work for me. I feel like it’s a lifeline.
Back
Create an account to start this course today
Used by over 30 million students worldwide
Create an account