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The slope-intercept form is a way of writing a linear equation. This form makes it easy to identify the slope and the y-intercept of the line. The general formula for the slope-intercept form is:
\[ y = mx + b \]
Here, \(m\) is the slope of the line, and \(b\) is the y-intercept.

Understanding slope and intercepts is crucial. The slope tells us how steep the line is. It shows how much \(y\) changes for a one-unit change in \(x\).
The y-intercept is the point where the line crosses the y-axis. This happens when \(x = 0\).
For example, in the equation \(y = 2x + 3\), the slope \(m\) is 2, meaning for each increase of 1 in \(x\), \(y\) increases by 2. The y-intercept \(b\) is 3, so the line crosses the y-axis at the point (0, 3).

Perpendicular lines are lines that intersect at a right angle (90 degrees). The slopes of perpendicular lines are special. If one line has a slope \(m\), the slope of a line perpendicular to it will be the negative reciprocal.

For instance, if the slope of a line is \(-1/4\), the slope of a perpendicular line would be \(4\). This is because the negative reciprocal of \(-1/4\) is found by flipping the fraction and changing the sign.

Let's consider our example problem:

• The line given was \(x + 4y = -5\).
• First, we rewrote it in slope-intercept form: \(y = -\frac{1}{4}x - \frac{5}{4}\).
• The slope is \(-\frac{1}{4}\).
• Therefore, the slope of the perpendicular line we need is \(4\).

Perpendicular slopes are a key concept in geometry and algebra, important for solving many types of problems.

The y-intercept is a fundamental concept in understanding lines. It is the point where the line crosses the y-axis. In the slope-intercept form \(y = mx + b\), the \(b\) value is the y-intercept.

To find the y-intercept of a line, set \(x = 0\) and solve for \(y\). For example, in the equation \(y = 4x - 1\), we can see that when \(x = 0\), \(y\) is \(-1\). Hence, the line crosses the y-axis at (0, -1).

Now back to our problem: the y-intercept was given as \( (0, -1) \).

• We took the known slope \(4\) and the y-intercept \(-1\) and used them in the slope-intercept form.
• This gave us the equation \( y = 4x - 1 \).

Understanding the y-intercept helps you quickly sketch graphs and grasp the relationship between variables.