Problem 73 - Introductory and Intermediate Algebra for College Students 4th

Question

Explain what it means to solve a formula for a variable.

Step-by-Step Solution

Answer

Solving a formula for a variable means getting that variable alone on one side of the equation. This is done by performing operations that won't change the balance of the equation. In this case, we successfully isolated \(x\) by dividing both sides of the equation by \(a\), hence \(x = \frac{b}{a}\).

Step 1: Understanding the Problem

Firstly, to solve a formula for a variable means to manipulate the formula so that the variable is on one side of the equation and everything else is on the other side. Essentially, the aim here is to make the formula illustrate how to calculate the value of the variable in terms of the other variables or constants. Let's consider the formula \(ax = b\), and we are to solve for \(x\).

Step 2: Isolate the Variable

To get \(x\) alone, we need to eliminate \(a\) from the left side of the equation. We do this by using fundamental operations that keep the equation balanced. Since \(a\) is being multiplied by \(x\), we can get rid of \(a\) by doing the opposite, that is, dividing both sides of the equation by \(a\). This gives us \(x = \frac{b}{a}\).

Step 3: Interpreting the Result

Now, \(x\) has been isolated on one side of the equation. This means we have solved the equation for \(x\), and \(x\) can now be calculated if the values of \(b\) and \(a\) are known.