Certainty of being able to solve linear equation

In summary: This equation is not very helpful in determining a unique solution. In fact, any values of a and b that satisfy the equation are solutions. Therefore, the system is undertermined and has infinitely many solutions.In summary, the conversation discusses the concept of undertermined systems and infinitely many solutions in linear equations. It is commonly understood that a system with two variables and one equation is undertermined and has infinitely many solutions. However, this may not always be the case as seen in the example provided, where there is one unique solution after comparing coefficients. This is because the solution is restricted to a specific set of values, such as integers. In conclusion, the number of solutions in a linear equation depends on the type of equation and
  • #1
Mr Davis 97
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I have the linear equation ##a + (2a - b) \sqrt{2} = 4 + \sqrt{2}##. I commonly hear that for linear equations, if we have two variables and one equation, then the system is undertermined and there are infinitely many solutions. However, how does that jive with this example, where there is one unique solution after comparing coefficients?
 
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  • #2
Mr Davis 97 said:
I have the linear equation ##a + (2a - b) \sqrt{2} = 4 + \sqrt{2}##. I commonly hear that for linear equations, if we have two variables and one equation, then the system is undertermined and there are infinitely many solutions. However, how does that jive with this example, where there is one unique solution after comparing coefficients?
I may be missing something here, but what "unique solution" are you referring to? This is a linear expression so it should have an infinite amount of solutions, as there are no discontinuites in its graph.
 
  • #3
Mr Davis 97 said:
I have the linear equation ##a + (2a - b) \sqrt{2} = 4 + \sqrt{2}##. I commonly hear that for linear equations, if we have two variables and one equation, then the system is undertermined and there are infinitely many solutions. However, how does that jive with this example, where there is one unique solution after comparing coefficients?
It seems to me that you are restricting your solution to the integers. Solving your equation with respect to a gives [itex]a=\frac{(b+1)\sqrt{2} + 4}{1+2\sqrt{2}} [/itex], which gives one value of a for each value of b you insert in the formula.
 
  • #4
Mr Davis 97 said:
I have the linear equation ##a + (2a - b) \sqrt{2} = 4 + \sqrt{2}##. I commonly hear that for linear equations, if we have two variables and one equation, then the system is undertermined and there are infinitely many solutions. However, how does that jive with this example, where there is one unique solution after comparing coefficients?
It depends on what type of equation you're working with.
If there are no conditions on a and b, then there are infinitely many solutions, one of which is ##a=\sqrt 2, b = 0##.
If a and b must be integers, then the coefficient of ##\sqrt 2## must be the same on both sides of the equation, which yields the unique solution a = 4 and b = 7.
 
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  • #5
Mr Davis 97 said:
if we have two variables and one equation, then the system is undertermined and there are infinitely many solutions.

Consider the single equation in two variables: ##a + b = a + b + 1##
 

1. What is a linear equation?

A linear equation is an algebraic equation that involves two variables and can be written in the form of y = mx + b, where m is the slope of the line and b is the y-intercept. In simpler terms, it is an equation that forms a straight line when graphed.

2. How do you solve a linear equation?

To solve a linear equation, you need to isolate the variable on one side of the equation. This is done by using inverse operations, such as addition, subtraction, multiplication, and division, to cancel out any constants or coefficients attached to the variable. Once the variable is isolated, you can solve for its value.

3. What is the importance of the certainty of being able to solve a linear equation?

The certainty of being able to solve a linear equation is crucial because it ensures that the equation has a unique solution. This means that there is only one possible value for the variable that will make the equation true. Without this certainty, the equation may have no solution or an infinite number of solutions.

4. How do you know if a linear equation is solvable?

A linear equation is solvable if it meets two criteria: there must be only one variable in the equation, and the variable must be raised to the first power. If these two conditions are met, then the equation can be solved using the methods mentioned in the answer to question 2.

5. Can a linear equation have more than one solution?

No, a linear equation can only have one solution. This is because a linear equation represents a straight line on a graph, and a line can only intersect the x-axis at one point. If you solve a linear equation and get a value for the variable that makes the equation true, that is the only solution.

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