- A linear equation \(ax + b = cx + d\) has infinitely many solutions if it simplifies to an identity (a statement that is true for all \(x\)).
- All provided equations are of the form \(f(x) = f(x)\), which means the left-hand side is identical to the right-hand side.
- For each option, subtracting the linear term simplifies the equation to the true numerical statement \(-37 = -37\).
- Therefore, all options A, B, C, and D are identities and possess infinitely many solutions.
- Final answer: \(\boxed{A, B, C, D}\).
Explanation
- Analyzing the Condition for Infinitely Many Solutions
A linear equation in one variable, generally written as \(ax + b = cx + d,\) has infinitely many solutions if the two sides of the equation are identical. This means the equation represents an identity, such as \(k = k,\) which is true for all real values of \(x\). If we subtract the terms involving \(x\) from both sides and obtain a true numerical statement, the original equation is satisfied by every real number.
- Evaluating Each Equation
We examine each equation provided:
- Option A: \(73x - 37 = 73x - 37\). Subtracting \(73x\) from both sides leaves \(-37 = -37\), which is a true statement for all values of \(x\).
- Option B: \(37x - 37 = 37x - 37\). Subtracting \(37x\) from both sides leaves \(-37 = -37\), which is a true statement for all values of \(x\).
- Option C: \(x - 37 = x - 37\). Subtracting \(x\) from both sides leaves \(-37 = -37\), which is a true statement for all values of \(x\).
- Option D: \(74x - 37 = 74x - 37\). Subtracting \(74x\) from both sides leaves \(-37 = -37\), which is a true statement for all values of \(x\).
- Conclusion
Since every equation simplifies to the identity \(-37 = -37\), all of them are true for any real number \(x\). Therefore, every given equation has infinitely many solutions. Thus, options A, B, C, and D are all correct. The final answer is \(\boxed{A, B, C, D}\).
Examples
In real-world applications, identifying equations with infinitely many solutions is like recognizing when two different descriptions actually define the exact same constraint. For example, if two recipes for a lemonade mixture lead to the same ratio of ingredients regardless of the volume produced, they describe the same outcome. In engineering or finance, recognizing such identities simplifies models, allowing you to cancel out redundant variables and focus on the core relationship between parameters.