Solve for y, show all your work:
$15 x+3 y=30$
The given equation is \(15x + 3y = 30\). This is a linear equation in two variables, \(x\) and \(y\). Our objective is to solve for \(y\), which means we need to isolate the variable \(y\) on one side of the equation. To do this, we will apply the properties of equality, such as the subtraction property and the division property, to systematically move terms involving \(x\) and constants to the other side.
To start isolating \(y\), we first remove the term \(15x\) from the left side. We do this by subtracting \(15x\) from both sides of the equation:
\[3y = 30 - 15x\]
Next, we need to isolate \(y\) by removing its coefficient, which is 3. We divide both sides of the equation by 3:
\[y = \frac{30 - 15x}{3}\]
To simplify this fraction, we divide each term in the numerator by 3:
\[y = \frac{30}{3} - \frac{15x}{3}\]
\[y = 10 - 5x\]
Thus, we have successfully expressed \(y\) in terms of \(x\). The final simplified linear equation is:
\[y = -5x + 10\]
This shows the linear relationship between \(y\) and \(x\), where the slope is \(-5\) and the \(y\)-intercept is \(10\), so the final solution is \(\boxed{y = -5x + 10}\).
In real life, linear equations like this are often used to model costs and budgets. For example, suppose you have a budget of \(\$30\) to spend on two types of items, where Item A costs \(\$15\) and Item B costs \(\$3\). If \(x\) is the quantity of Item A and \(y\) is the quantity of Item B, the equation \(15x + 3y = 30\) models the relationship between the quantities you can afford. Solving for \(y = 10 - 5x\) helps you easily determine how many of Item B you can buy for any given number of Item A you choose to purchase.