Question

matching list
explain how an equation with one solution is different from equations with no solution and equations that are true for every value of x.
one solution \t\t a. same variable, different constant
no solution
infinite solutions \t b. different variables
\t\t\t c. same variable, same constant

Explanation:

• For "One solution": When solving linear equations, if after simplifying, we have different variables (or the variable terms simplify to different variables), or more accurately, when the coefficients of the variable are such that we can solve for a unique value (e.g., in the case of different variables in a sense where the equation can be solved uniquely), but more precisely, when we have equations like \(ax + b = cx + d\) where \(a

eq c\) (different "effective" variables in terms of coefficients), we get one solution. But the option "b. Different variables" is the match here as it leads to a unique solution when solved.

• For "No solution": If we have the same variable but different constants after simplifying (e.g., \(2x + 3 = 2x + 5\), subtracting \(2x\) gives \(3 = 5\) which is false), so "a. Same variable, different constant" leads to no solution.
• For "Infinite solutions": If we have the same variable and same constant after simplifying (e.g., \(3x + 4 = 3x + 4\), subtracting \(3x\) and \(4\) gives \(0 = 0\) which is always true), so "c. Same variable, same constant" leads to infinite solutions.

Answer:

One solution - b. Different variables
No solution - a. Same variable, different constant
Infinite solutions - c. Same variable, same constant