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The slope-intercept form of a linear equation is an essential tool in algebra. It is written as: \(y = mx + b\), where

• \(m\) represents the slope of the line
• \(b\) represents the y-intercept, or the point where the line crosses the y-axis

To transform any linear equation into the slope-intercept form, you must solve for \(y\).
For example, given the equation \(5x - 6y = 4\), we can rewrite it as follows:

• Subtract \(5x\) from both sides to get: \(-6y = -5x + 4\)
• Divide everything by \(-6\) to isolate \(y\): \(y = \frac{5}{6}x - \frac{2}{3}\)

Now, you have the slope \(\frac{5}{6}\) and the y-intercept \(-\frac{2}{3}\). This line will graph as a straight line with these characteristics. Knowing how to find and work with slope-intercept form is crucial for understanding more complex algebraic concepts.

The point-slope form is another way to write the equation of a line. It's particularly useful when you have a point on the line and the slope but not the y-intercept.
The formula for point-slope form is: \(y - y_1 = m(x - x_1)\), where

• \(m\) is the slope
• \((x_1, y_1)\) is a specific point on the line

For instance, if we have a point \((3, -4)\) and a slope \(m = \frac{5}{6}\), we can plug these values into the point-slope formula: \(y + 4 = \frac{5}{6}(x - 3)\).
By distributing \(\frac{5}{6}\) and simplifying the equation, you can convert this into slope-intercept form if needed for easier graphing or interpretation:

• Distribute \(\frac{5}{6}\): \(y + 4 = \frac{5}{6}x - 2.5\)
• Subtract 4 from both sides: \(y = \frac{5}{6}x - 6.5\)

Both the point-slope and slope-intercept forms have their uses, depending on the specific requirements of the problem.

Parallel lines are lines in a plane that never intersect. They always have the same slope.
For example, the line \(5x - 6y = 4\) has a slope of \(\frac{5}{6}\). Any line parallel to this one must also have a slope of \(\frac{5}{6}\).
If we need a line parallel to this one that passes through a specific point, say \((3, -4)\), we can use the point-slope form to find it. Using the slope \(\frac{5}{6}\) and the point \(3, -4\), we get: \(y + 4 = \frac{5}{6}(x - 3)\).
This line will be parallel to \(5x - 6y = 4\) because it shares the same slope. Remember, whenever you come across the requirement for a parallel line, simply ensure the slopes are equal, and use the point-slope or slope-intercept forms to write the full equation. It's a beautiful way to see the consistency and patterns within linear algebra.