If solving an equation leads to a contradiction (such as 1 = 0), then there are no solutions. Similarly, if the equation involves taking the square root of a negative number, there are no real solutions.
So let us dig a little deeper
One way to tell if an equation has no solutions is if solving the equation leads to a contradiction. This means that the equation is impossible to satisfy, and therefore there are no solutions. For example, solving the equation 1 = 0 leads to a contradiction because it is not possible for the number 1 to be equal to the number 0.
Another way to tell if an equation has no solutions is if the equation involves taking the square root of a negative number. The square root of a negative number does not have a real solution, so the equation has no real solutions. For example, the equation x^2 = -1 has no real solutions because the square of any real number is always a positive number or zero.
In mathematical terms, if an equation has no solutions, it is said to be inconsistent or contradictory. This means that there is no value of the variable that will make the equation true. On the other hand, if an equation has one or more solutions, it is said to be consistent.
According to the famous mathematician, Jean le Rond d’Alembert, “An equation is a bond between two quantities, arrived at by equating them to each other, and has the advantage of eliminating any arbitrary elements that may be present in each of them.”
Here’s a table summarizing the different ways to tell if an equation has no solution:
| Method | Example | Conclusion |
|---|---|---|
| Leads to a | Equation has no | |
| contradiction | 1 = 0 | solution |
| Square root of a | x^2 = -1 | Equation has no real |
| negative number | solutions |
In summary, an equation has no solutions if it leads to a contradiction or if it involves taking the square root of a negative number. It is important to identify whether an equation has solutions or not, as this can affect the validity of any solutions presented.
There are several ways to resolve your query
Some equations have no solutions. In these equations, there is no value for the variable that makes the equation true. You can tell that an equation has no solutions if you try to solve the equation and get a false statement.
In an equation with no solution, move the variable to one side will result in the variable adding to zero on both sides. You are then left with constants only on both sides. If the sides are not equal, the equation has no solution. If the constants are equal, then the equation has a solution of all real numbers.
When finding how many solutions an equation has you need to look at the constants and coefficients. The coefficients are the numbers alongside the variables. The constants are the numbers alone with no variables. If the coefficients are the same on both sides then the sides will not equal, therefore no solutions will occur.
Solve the system – if you solve the system and get a nonsense equation (such as 0 = 1), then there is no solution. Look at the graph – if the two lines are parallel (they never touch), then there is no solution to the system. Look at the slope and y-intercept – solve both equations for y to get slope-intercept form, y = mx + b.
If you cancel out all of the x terms via addition or subtraction, and you get something along the lines of 1 = 2, then you have no solution.
11x + 4 = 11x + 7 Subtract 11x from both sides
4 = 7 No value for x will satisfy this equation.
If you cancel out all the x terms via addition or subtraction and you get something along the lines of 1 = 1, then you have infinite solutions.
2(x + 1) = 2x + 2 Expand the left side using the distributive property
2x + 2 = 2x + 2 Subtract 2x from both sides
2 = 2 Every value for x will satisfy this equation
If you can’t cancel out all the x terms with addition or subtraction, you probably have 1 solution.
5x + 2 = 3x + 100 Subtract 3x from both sides
2x + 2 = 100 Subtract 2 from both sides
2x = 98 Divide by 2 on both sides
x = 49 The only x value that satisfies this equation is 49
There are other cases where functions of x aren’t injective, meaning there’s more than one x value that satisfies the equation. Here’s what I mean.
x^2 = 4
x = …
Video response to “How can you tell if an equation has no solutions?”
The YouTube video “One Solution, No Solution, or Infinitely Many Solutions – Consistent & Inconsistent Systems” explains how to determine if a system of equations is consistent or inconsistent, dependent or independent, and contains one solution, no solution, or many solutions. By solving the system of equations, a single value for x and y indicates one solution, a contradiction shows no solution, and a statement like 0 = 0 or x = x means many solutions. The video also shows examples and uses the elimination method to obtain equations that indicate whether the system is consistent, dependent, or independent.
These topics will undoubtedly pique your attention
Consequently, How do you know if there is no solution or infinite solutions?
We can identify which case it is by looking at our results. If we end up with the same term on both sides of the equal sign, such as 4 = 4 or 4x = 4x, then we have infinite solutions. If we end up with different numbers on either side of the equal sign, as in 4 = 5, then we have no solutions.
Also Know, How can you tell when two equations in a system will have no solution?
Response to this: A system of two linear equations has no solution if the lines are parallel. Parallel lines on a coordinate plane have the same slope and different y-intercepts (see figure 3 for an example of this). If the lines look parallel, confirm it by checking that they have the same slope.
Regarding this, What does it mean when an equation has no solution?
Response to this: Sometimes equations have no solution. This means that no matter what value is plugged in for the variable, you will ALWAYS get a contradiction.
Thereof, What is an example of a no solution?
Answer: As an example, consider 3x + 5 = 3x – 5. This equation has no solution. There is no value that will ever satisfy this type of equation.
How do you determine if a system of equations has no solution?
Determine whether the following system of equations have no solution, infinitely many solution or unique solutions. x+2y = 3, 2x+4y = 15 So, the system of equations has no solution. Define a Linear equation. A Linear equation is an equation that has one or more variables having degree one. Give an example of a Linear equation in two variables.
What if a system of linear equations has infinitely many solutions?
Answer to this: Give the condition for a system of linear equations that has infinitely many solutions. If (a 1 /a 2) = (b 1 /b 2) = (c 1 /c 2 ), then there will be infinitely many solutions. Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!
Beside this, Is there a possible solution to a fraction with no solution? Response: Correct answer: no possible solution Explanation: Substituting 3 in for x, you will get 0 in the denominator of the fraction. It is not possible to have 0 be the denominator for a fraction so there is no possible solution to this equation. Report an Error Example Question #2 : How To Find Out When An Equation Has No Solution I. x= 0 II.
Likewise, How do you solve a quadratic equation with no real solution?
As an answer to this: In either of those cases, we are taking the square root of a negative, which gives us two complex solutions to the quadratic (that is, no real solution). For example, the quadratic equation x 2 + 4 = 0 has no real solution. In this case, a = 1, b = 0, and c = 4.
How do you know if an equation has no solution? Since the coefficients of x are both 4, but the constants are different, you know there are no solutions because if you took it to the end, you would get 2=0 which can never be true. Is there any real world application for making an equation with no solution? No, there can’t be, because it wouldn’t exist.
One may also ask, How do you know if a variable has a solution?
The answer is: This trick is based on simplifying and as soon as you see the same coefficients of the variable on both sides and any different numbers on the two sides, you know that there are no solutions.
Consequently, What if we don’t have to solve a problem entirely? So if at any point we might not have to solve this entirely if we somehow get something that’s nonsensical which will tells us there’s no solutions. Or we might have to go further and see if it’s one or infinite solutions, although it looks like one solution isn’t an option here, given how this question is phrased.
Why does math make no sense?
Response: You know, Math makes no sense, you can literally end up with answers like this: 8=3. or something confusing like that. So why does this work? If you have ended with an expression like 8 = 3, there is an error in your solution or, if you are working with a system of equations, then there is no solution that satisfies all the equations in the system.