Problem 20 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-1,\) passing through \(\left(-4,-\frac{1}{4}\right)\)

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y + \frac{1}{4} = -1(x + 4)\) and in slope-intercept form it is \(y = -x - \frac{17}{4}\).

Step 1: Write the Point-Slope Form

To start, use the point-slope form equation and plug in the given values. The slope is -1 and the point is \(-4,-\frac{1}{4}\). Substitute these values into the point-slope form, which gives: \(y - (-\frac{1}{4}) = -1(x - (-4))\).

Step 2: Simplify the Point-Slope Form

Next step is to simplify the equation. This results in \(y + \frac{1}{4} = -1(x + 4)\).

Step 3: Write the Slope-Intercept Form

Now, convert the point-slope form to the slope-intercept form. To do this, distribute the -1 in the equation from step 2 over \(x + 4\), and then subtract \(\frac{1}{4}\) from both sides. This yields the slope-intercept form: \(y = -x - 4 - \frac{1}{4}\).

Step 4: Simplify the Slope-Intercept Form

Finally, simplify the slope-intercept equation from step 3. Subtracting \(\frac{1}{4}\) from 4 gives \(-\frac{17}{4}\), so the equation becomes \(y = -x - \frac{17}{4}\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing the equation of a line. It is one of the most commonly used forms in algebra. The equation is written as:
\[ y = mx + b \]
Here:

• \(y\) is the dependent variable.
• \(m\) is the slope of the line.
• \(x\) is the independent variable.
• \(b\) is the y-intercept, which is where the line crosses the y-axis.

This form is very useful because it immediately reveals the slope and the y-intercept of the line.
In the problem we solved, we converted from a point-slope form to a slope-intercept form. This helps us easily graph the line by starting at the y-intercept and using the slope to plot other points.

Slope

The slope of a line measures its steepness and direction. It's commonly represented by the letter \(m\). In the slope-intercept form, slope is a crucial component.
The slope of a line is calculated as the rise over run; this means the vertical change divided by the horizontal change between two points on the line. Mathematically, the slope is represented as:
\[ m = \frac{\text{change in } y}{\text{change in } x} \]
For the given exercise, the slope is \(-1\), indicating that for every unit increase in \(x\), \(y\) decreases by 1 unit. The negative sign shows that the line is descending.

Equation of a Line

The equation of a line represents all the combinations of \(x\) and \(y\) that make the line. It can be expressed in different forms, but they all describe the same line.
Two common forms are:

• Point-Slope Form: \( y - y_1 = m(x - x_1) \)
• Slope-Intercept Form: \( y = mx + b \)

In point-slope form, \((x_1, y_1)\) is a specific point on the line, and \(m\) is the slope. This form is very useful when you have one point on the line and the slope.
In our exercise, we started with a point-slope form using the given point \((-4, -\frac{1}{4})\) and slope \(-1\). Then we converted it to the slope-intercept form, which makes graphing the line straightforward.