Problem 17 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(--4,\) passing through \((-4,0)\)

Step-by-Step Solution

Answer

The point-slope form of the line is \(y = -4x -16\) and the slope-intercept form is the same, \(y = -4x -16\).

Step 1: Find the Point-Slope Form

By inserting the known values into the point-slope formula, the equation of the line can be obtained. Here, the slope \(m\) is -4 and the point \((x_1, y_1)\) is (-4,0). Therefore, the equation becomes: \(y - 0 = -4(x - (-4))\). This simplifies as \(y = -4(x + 4)\). So, the point-slope form of the line is \(y = -4x -16\).

Step 2: Convert to the Slope-Intercept Form

The slope-intercept form of a line is given by: \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. For this line, the equation is already in this form, since we have isolated \(y\) on one side of the equation in the previous step.

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a very useful way to write the equation of a line when you know the slope and one point on the line. The general formula for the point-slope form is:\[ y - y_1 = m(x - x_1) \]where \(m\) represents the slope of the line, and \( (x_1, y_1) \) is a specific point on the line. In our exercise, the given point is \(-4, 0\), and the slope is \(-4\).
To substitute these values into the formula:

• m = -4
• (x₁, y₁) = (-4, 0)

After substituting, the formula becomes:\[ y - 0 = -4(x - (-4)) \]This equation simplifies to:\[ y = -4(x + 4) \]This expression is the point-slope form of the equation of the line. It highlights how the line changes or "slopes" from the provided point. Understanding and using the point-slope form is incredibly helpful for those tricky math problems where you're provided with any one point and the slope of the line.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express an equation of a line. The structure of this form is:\[ y = mx + b \]Here, \(m\) stands for the slope of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
In our scenario, we initially found the equation in point-slope form:\[ y = -4(x + 4) \]By distributing the \(-4\) and simplifying, we can also see this equation in slope-intercept form:\[ y = -4x - 16 \]This tells us that the slope is \(-4\) and the y-intercept is \(-16\). Every time \(x\) increases by 1, \(y\) will decrease by 4. Understanding this form is particularly useful because it easily shows both how a change in one variable affects another, and where the line crosses the y-axis. This can quickly help in graphing the equation or comparing it to other lines.

Slope of a Line

The slope of a line is an important concept that characterizes how steep a line is and the direction it moves. Slope is typically denoted by the letter \(m\) in equations and is calculated as "rise over run":\[ m = \frac{\text{change in } y}{\text{change in } x} \]Basically, it's the amount that "y" increases for every increase of one unit in "x." A positive slope indicates that the line is moving upwards as you go from left to right, while a negative slope suggests the line goes downwards.
For example, our slope \(-4\) means:

• For every step right on the x-axis, the line goes 4 units down on the y-axis.
• A slope of 0 would mean a flat horizontal line.
• An undefined slope corresponds to a vertical line.

Knowing the slope is crucial in identifying the incline or decline of a line and setting up its equation in various forms like the point-slope or slope-intercept form. It reveals a lot about the nature of the line without needing to plot it.