Problem 8 - Elementary Algebra

Question

Find the equation of each line given the following information. Use the slope- intercept form as the final form of the equation. $$ m=-1, \text { the point }(-5,-5) $$

Step-by-Step Solution

Answer

Answer: The equation of the line is y = -x - 10.

Step 1: Identify the slope and point on the line

We are given the slope, m = -1, and a point on the line, (-5, -5).

Step 2: Use the slope-intercept form, y = mx + b, and input the values

Now, we can use the given point (-5, -5) and the slope -1 to find the value of b. Plug the values for x, y, and m into the equation and solve for b: $$ -5 = -1(-5) + b $$

Step 3: Solve for b

Solve the equation to find the value of b: $$ -5 = 5 + b \Rightarrow b = -10 $$

Step 4: Write the final equation

Now that we have the values for m and b, we can write the equation of the line in slope-intercept form: $$ y = -1x - 10 $$ So, the equation of the line is: $$ y = -x - 10 $$

Key Concepts

Equation of a Line

When we talk about the equation of a line, we're referring to a mathematical expression that describes all the points lying on that specific line. The most common form used for this equation, especially in algebra, is the slope-intercept form, which is expressed as \( y = mx + b \). Here, \( y \) and \( x \) are the coordinates of any point on the line, \( m \) is the slope, and \( b \) is the y-intercept.
Understanding this form allows us to see the line's slope and where it crosses the y-axis. This makes plotting the line on a graph straightforward, and it's easy to identify and graph the line using just these two pieces of information.

Slope

The slope of a line, represented by \( m \) in the equation, is a measure of how steep the line is. It tells us how much the line goes up or down as we move from one point to another along the x-axis. In the formula for slope-intercept form, \( m \) shows us the rise over the run, which is the vertical change divided by the horizontal change between two points.
In our example, the slope \( m \) is \(-1\), meaning for each unit we move to the right along the x-axis, the line goes down by one unit.

• A positive slope means the line is going upward from left to right.
• A negative slope, like in our case, means the line is going downward.
• A zero slope indicates a completely horizontal line.
• An undefined slope is the result of a vertical line.

Recognizing the slope helps us understand the line's direction and inclination.

Using a Point to Find Y-Intercept

Once the slope is known, the next step in forming the line's equation is to find the y-intercept, represented by \( b \). This is where the line crosses the y-axis. To find \( b \), you can use a known point on the line. Suppose the slope \( m \) and a point \((x, y)\) are given—plug these values into the equation \( y = mx + b \).
In our solution, the point \((-5, -5)\) is used with the slope \( m = -1 \). Substituting these values into the equation allows us to solve for \( b \): \[-5 = -1(-5) + b \]By solving, we found that \( b = -10 \).

• The y-intercept \( b \) helps us locate where the line reaches the y-axis, which is crucial for graphing the line accurately.
• In the complete equation of the line, it provides the fixed vertical starting point from which the line continues with its slope.

This process demonstrates the combination of slope and y-intercept to represent a line elegantly on a Cartesian plane.