Question

solution
6 ria solves the equation 5 + 3r = 4 + 4r and gets r = r. she concludes that the equation has infinitely many solutions. what is the correct solution? what mistake did ria make?

Explanation:

Step1: Solve the equation

Start with the equation \(5 + 3r = 4 + 4r\). Subtract \(3r\) from both sides: \(5 + 3r - 3r = 4 + 4r - 3r\), which simplifies to \(5 = 4 + r\). Then subtract 4 from both sides: \(5 - 4 = 4 + r - 4\), so \(r = 1\).

Step2: Analyze Ria's mistake

Ria got \(r = r\), which is always true, but she misapplied the concept. An equation with infinitely many solutions has a form like \(a = a\) (e.g., \(2 = 2\)) after simplifying, but here, when solving correctly, we get a specific solution. She didn't properly isolate the variable and misinterpreted the result \(r = r\) as indicating infinitely many solutions, but in reality, when we solve the equation, we should get a specific value for \(r\) (or a contradiction for no solution).

Answer:

The correct solution is \(r = 1\). Ria's mistake was misinterpreting the result \(r = r\) as meaning the equation has infinitely many solutions; in reality, she should have continued solving to find the specific solution \(r = 1\) (or recognized that \(r = r\) here was a result of an incomplete solving process and that the equation has a unique solution).