Problem 7 - 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. $$ m=-6, \text { the point }(-1,0) $$

Step-by-Step Solution

Answer

Answer: The equation of the line is y = -6x - 6.

Step 1: Write down the given information

We are given the slope of the line: $$ m = -6 $$ And we are also given a point on the line: $$ (-1, 0) $$

Step 2: Substitute the slope and the point into the slope-intercept formula

To find the equation of the line, we will replace m with its value and (-1, 0) with (x, y). Then we are going to solve for b. $$ 0 = -6 \cdot (-1) + b $$

Step 3: Solve for b

In this step, we are going to solve the equation we created in step 2 for the y-intercept b: $$ 0 = 6 + b $$ Subtract 6 from both sides: $$ b = -6 $$

Step 4: Write the equation of the line in slope-intercept form

Now, we have all the information needed to write the equation of the line in slope-intercept form, which is y = mx + b. Replace m and b with their values from the previous steps: $$ y = -6x - 6 $$ This is the equation of the line in slope-intercept form with the given slope and point.

Key Concepts

Equation of a Line

Mathematically, a line can be described with an equation. An equation represents the entire set of points that make up a line on a graph. A very common way to express a line is using the **slope-intercept form** of an equation. In **slope-intercept form**, the equation is structured as: \( y = mx + b \), where:

• \( y \) stands for the value on the vertical axis (y-axis).
• \( x \) stands for the value on the horizontal axis (x-axis).
• \( m \) is the slope of the line, representing its steepness.
• \( b \) is the y-intercept, which is where the line crosses the y-axis.

This form is helpful because once you know the slope and y-intercept, you can easily write down the equation of the line, determine its steepness, and find other points on the line. When given a point and a slope, like in our exercise, you can substitute these into the formula to find the complete equation of the line.

Slope

The **slope** of a line indicates how steep the line is. In a graph, the slope tells you how much the line rises or falls as you move along the x-axis. Mathematically, the slope \( m \) can be calculated as the ratio of the change in the vertical direction \( \Delta y \) to the change in the horizontal direction \( \Delta x \). It's often described by the formula:\[ m = \frac{\Delta y}{\Delta x}\]

• If the slope is positive, the line rises as it moves from left to right.
• If the slope is negative, like \( -6 \) in our example, the line falls as you go from left to right.
• A slope of zero means the line is flat or horizontal.

In your homework problem, you were given a slope of \( -6 \). This means for every one unit you move to the right, the line drops by 6 units. Understanding the slope helps in predicting the direction of the line.

Y-Intercept

The **y-intercept** is a critical component in forming the equation of a line in the slope-intercept form. This is the point where the line crosses the y-axis of the graph, which means the value of \( x \) is zero at this point. The y-intercept is represented by \( b \) in the equation \( y = mx + b \). In a visual representation:

• The y-intercept shows exactly where the line begins when drawing it from the y-axis.
• It provides a starting point for the line, before it slopes up or down.

In our exercise, after substituting the point (-1, 0) and the slope into the slope-intercept form, we calculated that the y-intercept \( b \) is \( -6 \). Therefore, it tells us that this line crosses the y-axis at the point (0, -6). This information, combined with the slope, completes the equation of the line.