Solution Sets and Linear Equations

• e(ho0n3
In summary, the conversation discusses proving that two linear equations with the same solution set are actually the same equation. It is stated that this is true only if the equations are the same line. One person expresses skepticism and confusion, but it is explained that this is a valid concept and the conversation ends with understanding and agreement.
e(ho0n3
Problem
Prove that, where a, b, c, d, e are real numbers and a <> 0, if ax + by = c has the same solution set as ax + dy = e, then they are the same equation.

Given Solution
If a <> 0 then solution set of the first equation is {(x,y) | x = (c - by)/a}. Taking y = 0 gives the solution (c/a, 0), and since the second equation is supposed to have the same solution set, substituting into it gives a(c/a) + d(0) = e, so c = e. Then taking y = 1 in x = (c - by)/a gives a((c - b)/a) + d = e, which gives b = d. Hence they are the same equation.

My Thoughts
I don't buy into the solution above because it assumes that one of the members of the solution set has y = 0 and that another has y = 1. And anyways, you can take any two-variable two-equation linear system (where the equations aren't equal) and solve to get the solution set. Is this problem bogus or what?

No. The equations define straight lines. They are the same solution set iff they are the same line, which is iff they are "the same equation"

I don't buy into the solution above because it assumes that one of the members of the solution set has y = 0 and that another has y = 1.

So what you're saying is that if a != 0, then it's possible that ax + b = c (or ax = c) are not solvable (for x)?

OK. I see where I was confused. This makes sense now. Cheers.

1. What is a solution set in linear equations?

A solution set in linear equations is the set of values that satisfy the equation, making it true. In other words, it is the set of all possible solutions that make the equation balanced.

2. How do you find the solution set of a linear equation?

To find the solution set of a linear equation, you need to solve for the variable in the equation. This can be done by using algebraic methods such as addition, subtraction, multiplication, and division to isolate the variable. The resulting value or values will be the solution set.

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

Yes, a linear equation can have more than one solution. In fact, there are three types of solution sets for linear equations: no solution, one solution, or infinitely many solutions. This depends on the number of variables and equations in the system.

4. What is the difference between a solution set and a solution?

A solution set is a set of all possible solutions that make the equation true, while a solution is a specific value or values that satisfy the equation. In other words, a solution set is a collection of solutions, while a solution is a single value or set of values.

5. How are solution sets used in real-world applications?

Solution sets are used in many real-world applications, such as in engineering, physics, and economics. They are used to solve problems and make predictions by finding the values that satisfy a given equation or system of equations. For example, they can be used to determine the optimal production level for a company or the best route for a delivery truck.

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