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Linear equations are fundamental in algebra and they represent straight lines when graphed on a coordinate plane. The general form of a linear equation is: \[Ax + By = C\], where A, B, and C are constants.

Understanding this basic structure is key because it forms the basis for converting equations into different forms, like the slope-intercept form.

The slope-intercept form is particularly useful because it immediately gives you the slope and y-intercept of the line. This form is written as \[y = mx + b\], where m is the slope (the rate of change) and b is the y-intercept (the point where the line crosses the y-axis).

In this given exercise, we started with the linear equation \[10x - 7y = 70\] and our goal was to convert it to the slope-intercept form.

Solving for y means isolating y on one side of the equation. This lets you express the equation in terms of y.

Here's how you solve for y step-by-step:
1. Start with the given equation: \[10x - 7y = 70\].
2. Move the x term to the other side by subtracting 10x from both sides: \[-7y = -10x + 70\].
3. Divide each term by -7 to isolate y: \[y = \frac{-10}{-7}x + \frac{70}{-7}\].

When you solve for y, you are essentially changing the equation’s format to make it easier to understand and work with.

This rearrangement also lays the groundwork for graphing the equation, as now you have it in a form where the slope (m) and y-intercept (b) are clearly visible.

Simplifying fractions is the process of making a fraction as simple as possible. This usually means dividing the numerator and the denominator by their greatest common divisor (GCD).

When we solved for y, we came across the fractions \[\frac{-10}{-7}\] and \[\frac{70}{-7}\]. Let's simplify these:

• \[\frac{-10}{-7}\] simplifies to \[\frac{10}{7}\] because the negative signs cancel each other out.
• \[\frac{70}{-7}\] simplifies to -10 because \[70 \div -7 = -10\].

Therefore, after simplifying, the equation in slope-intercept form becomes: \[y = \frac{10}{7}x - 10\].

Simplifying fractions makes equations cleaner and easier to read. It is a crucial step that ensures your final answer is in its simplest and most understandable form.