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Building a table of values is a great way to visualize the relationship between two variables in an equation. In this exercise, we're working with the linear equation \(y = 4x - 6\). The goal is to find five pairs of \(x\) and \(y\) values based on a starting \(x\) value and a specific interval. Let's break down the steps.

Start with the given starting \(x\) value, which is \(-8\). Next, determine the interval between each \(x\) value, which in this case is \(3\). This means you'll add \(3\) to the starting \(x\) value to find the next \(x\) value, and so on for the rest.

Let's use the interval to find the next four \(x\) values:

• Starting \(x\): \(-8\)
• \(x + 3 = -5\)
• \(-5 + 3 = -2\)
• \(-2 + 3 = 1\)
• \(1 + 3 = 4\)

So, our \(x\) values are \(-8, -5, -2, 1, 4\). Next, we’ll calculate corresponding \(y\) values based on these \(x\) values.

Solving linear equations means finding the values of the variable(s) that make the equation true. In \(y = 4x - 6\), we need to substitute the given \(x\) values into the equation to find the corresponding \(y\) values. Let’s do the math step-by-step for each \(x\):

• For \(x = -8\): \( y = 4(-8) - 6 = -38 \)
• For \(x = -5\): \( y = 4(-5) - 6 = -26 \)
• For \(x = -2\): \( y = 4(-2) - 6 = -14 \)
• For \(x = 1\): \( y = 4(1) - 6 = -2 \)
• For \(x = 4\): \( y = 4(4) - 6 = 10 \)

As we solve it, we see how each y-value is directly affected by changing the x-value. This shows the linear relationship between the two variables in the equation.

Intervals are useful in algebra for finding a series of values systematically. By knowing the interval (difference) between consecutive \(x\) values, you can easily generate a sequence of points. Let's understand this with our interval of \(3\) and the starting \(x\) value of \(-8\).

• Start with \(-8\)
• Add \(3\) to get the next \(x\)-value: \(-8 + 3 = -5\)
• Keep adding \(3\) to each subsequent \(x\)-value: \(-5 + 3 = -2, -2 + 3 = 1, 1 + 3 = 4\)

Interpreting intervals helps in various algebra problems, not just linear equations. It’s a systematic way to progress through values and verify how changes in one variable affect another.

It also helps in graphing equations, as you get clearly defined points that can be plotted to visualize the relationship. Keeping these techniques in mind, you'll be able to handle algebraic tasks more efficiently, ensuring accuracy and a better understanding of the relationships between variables.