91Ó°ÊÓ

At its core, the process of isolating a variable is about rewriting equations to make one variable the subject. This technique is foundational in solving algebraic problems, such as graphing linear equations. Consider the equation from our exercise, \(x + 4y = 48\). The goal is to express \(y\) in terms of \(x\), so we can easily compute the value of \(y\) for any given \(x\).

To do this, we perform operations that 'undo' whatever is being done to \(y\). Since \(y\) is being multiplied by 4 and \(x\) is added to it, we reverse these steps—subtract \(x\) from both sides, followed by dividing everything by 4. This gives us \(y = 12 - 0.25x\), which is a clearer way to see how changing \(x\) will affect \(y\). It's critical to remember to apply the same operation to both sides of the equation to maintain balance.

In learning to isolate variables, it's vital to master operations with numbers and understand the 'balance' concept of equations—what we do to one side, we must do to the other. This ensures that students can manipulate equations to extract needed information efficiently.

When graphing equations, a table of values is a strategic tool that enumerates several pairs of \(x\) and \(y\) that satisfy the equation. Our exercise involves creating such a table for the equation \(y = 12 - 0.25x\). To begin, choose a range of \(x\) values. It often helps to start with 0, as it simplifies calculations for the corresponding \(y\).

Here's how you might construct this table:

• Start with \(x = 0\) and substitute that into the equation to find the corresponding \(y\) value.
• Next, select another \(x\) value, like 8, and again compute \(y\).
• Repeat this process for as many points as you feel are necessary to accurately depict the line on a graph—commonly, at least three points are used.

As you can see, the table of values translates the abstract equation into concrete points that you can visualize and work with when graphing.

The final step to bring the equation \(y = 12 - 0.25x\) to a visual form is plotting the points on a Cartesian plane. Using the table of values, we mark the specific \(x, y\) coordinates on the graph. For the given points (0,12), (8,10), and (16,8), plot each point where the horizontal (\(x\)) and vertical (\(y\)) coordinates intersect.

It's like playing connect-the-dots. You start by locating the \(x\)-coordinate along the horizontal axis, then move vertically to meet the \(y\)-coordinate. Once all points are in place, you'll draw a straight line through them, extending it across the graph. This line represents all the possible solutions to the equation—a complete picture of the relationship between \(x\) and \(y\). It's essential for students to practice this skill to confidently translate algebraic concepts into graphical representations.