How many solutions does this equation have?
[tex]12.9 f=-5.3 f+18.2 f[/tex]
* no solution
* one solution
* infinitely many solutions
Answer (2)
Combine like terms: − 5.3 f + 18.2 f = 12.9 f .
Rewrite the equation: 12.9 f = 12.9 f .
Subtract 12.9 f from both sides: 0 = 0 .
Since the equation is always true, there are infinitely many solutions: in f ini t e l y man yso l u t i o n s .
Explanation
Understanding the Problem We are given the equation 12.9 f = − 5.3 f + 18.2 f . Our goal is to determine how many solutions this equation has. The options are no solution, one solution, or infinitely many solutions.
Simplifying the Equation First, let's simplify the right side of the equation by combining the terms with f . We have − 5.3 f + 18.2 f = ( − 5.3 + 18.2 ) f . Calculating − 5.3 + 18.2 , we get 12.9 . Therefore, the equation becomes 12.9 f = 12.9 f .
Isolating the Variable Now, let's subtract 12.9 f from both sides of the equation. This gives us 12.9 f − 12.9 f = 12.9 f − 12.9 f , which simplifies to 0 = 0 .
Determining the Number of Solutions Since the equation 0 = 0 is always true, regardless of the value of f , this means that any value of f will satisfy the original equation. Therefore, the equation has infinitely many solutions.
Examples
Consider a scenario where you're trying to balance a budget. If both sides of your equation always balance out, no matter how much you spend or earn, you have infinite possibilities for your income and expenses. This is similar to having infinitely many solutions to an equation, where any value you choose for the variable will satisfy the equation. Understanding such equations helps in modeling situations where multiple outcomes are possible and valid.
The equation 12.9 f = − 5.3 f + 18.2 f simplifies to 0 = 0 , indicating that it holds true for all values of f . As a result, there are infinitely many solutions to this equation. Thus, the chosen option is infinitely many solutions.
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