An equation has no solution when you simplify it and get a contradiction or an absurdity.
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One way to determine if an equation has no solution is to simplify it and check for a contradiction or absurdity. For example, if we have the equation 2x + 5 = 2x + 7, we can subtract 2x from both sides to get 5 = 7, which is clearly a contradiction. This tells us that the equation has no solution.
According to the famous mathematician Srinivasa Ramanujan, “An equation means nothing to me unless it expresses a thought of God.” In other words, there is beauty and meaning in equations, but only if they are solvable and make sense.
Here are some interesting facts about equations and solutions:
- In mathematics, a solution to an equation is a value of the variable(s) that make the equation true.
- Some equations have infinitely many solutions, such as x = x or 2x = 2(x+1).
- Other equations have exactly one solution, such as x + 2 = 5 or 3x – 1 = 8.
- However, there are also equations that have no solution, such as x + 1 = x + 2 or 2x = x + 1.
- Equations can be solved using various algebraic and numerical methods, such as factoring, substitution, or graphing.
- In real-life applications, equations are used to model and solve problems in areas such as physics, engineering, and finance.
Here is a table summarizing the number of solutions that different types of equations can have:
|Type of equation||Example||Number of solutions|
|One-step||x + 2 = 5||1|
|Multi-step||3x – 1 = 8||1|
|Linear||2x + 3y = 7||1|
|Quadratic||x^2 + 2x – 3 = 0||2|
|Cubic||x^3 – 4x^2 + x + 6 = 0||3|
|No solution||x + 1 = x + 2||0|
|Infinite solutions||x = x or 2x = 2(x+1)||Infinite|
By understanding the nature of equations and solutions, we can better appreciate the importance and beauty of mathematics in our everyday lives.
A visual response to the word “How do you know if an equation has no solution?”
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.
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Equations with no solutions If a linear equation has the same variable term but different constant values on opposite sides of the equation, it has no solutions.
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 = …
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Considering this, What is an example of a no solution? 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.
Beside this, How do you know if an equation has infinite or no solutions?
In reply to that: 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.
Subsequently, What does it mean when an equation has no solution? The reply will be: Sometimes equations have no solution. This means that no matter what value is plugged in for the variable, you will ALWAYS get a contradiction.
Regarding this, Which equation has no solution 4x 2 =- 6?
Response to this: ⇒ |4x – 2| = -6 has a negative value -6. |3x – 6| = -5 etc will not have solutions for x. Thus, the equation |4x – 2| = -6 has no solution.
Herein, When does an equation have no solution? As a response to this: This type of equation is called a dependent pair of linear equations in two variables. If we plot the graph of this equation, the lines will coincide. Case 2. If (a 1 /a 2) = (b 1 /b 2) ≠ (c 1 /c 2 ), then there will be no solution. This type of equation is called an inconsistent pair of linear equations.
What is a system of equation with no solution?
When a system of linear equations has no solution, the lines are parallel to each other and will therefore never intersect. Every method you try will result in the elimination of both variables and simplifying it will leave you with a statement that is not true.
Thereof, Which system of equations has exactly one solution? Response: An independent system of equations has exactly one solution (x,y) . An inconsistent system has no solution, and a dependent system has an infinite number of solutions. Besides, how many solutions does this system have? Systems of linear equations can only have 0, 1, or an infinite number of solutions. These two lines cannot intersect twice.
When does a system have no solution? Answer will be: If (a 1 /a 2) = (b 1 /b 2) ≠ (c 1 /c 2 ), then there will be no solution. This type of system of equations is called an inconsistent pair of linear equations. If we plot the graph, the lines will be parallel and system of equations have no solution. Example Find the value of x and y -4x + 10y = 6 2x – 5y = 3 Solution: Given -4x + 10y = 6 … (i)