91Ó°ÊÓ

Isolating variables is an essential skill in algebra. It means rearranging an equation so that you have the variable you're solving for by itself on one side of the equation. In our exercise, we start with the equation: \( 5x - 3y = 12 \). Our goal is to solve for y. To isolate y, we need to get rid of the term with x on the same side as y. By subtracting \(5x\) from both sides, we remove the \(5x\) term next to y. This is the principle of inverse operations. Whatever you do to one side of the equation, you must do to the other. After this step, our equation looks like: \( -3y = 12 - 5x \). This process helps simplify the problem and bring y to one side, preparing us for the next steps required to solve for y.

Once you've isolated y, it's time to solve for it. This means getting y by itself, with no coefficients or other terms. In our isolated equation: \( -3y = 12 - 5x \), y is still multiplied by -3. To solve for y, we need to eliminate this coefficient. We do this by dividing both sides of the equation by -3. Using inverse operations again, our equation now becomes: \[ y = \frac{12 - 5x}{-3} \]. This step has y on its own, but we can further simplify our expression to make it clearer. The key takeaway is that solving for a variable often involves isolating it first and then performing arithmetic operations to fully unlock what y equals.

Simplifying fractions makes expressions easier to understand and work with. After solving for y, we have the fraction: \[ y = \frac{12 - 5x}{-3} \]. To simplify, we divide each term in the numerator by the denominator separately. This looks like: \[ y = \frac{12}{-3} - \frac{5x}{-3} \]. Doing the division, we find: \[ y = -4 + \frac{5x}{3} \]. For a more standard format, we can rearrange it to: \[ y = \frac{5x}{3} - 4 \]. Simplifying fractions helps us arrive at a cleaner and more practical solution. It ensures each term is clear and expressed in its simplest form, crucial for understanding and further calculations.