Problem 41 - Elementary Algebra

Question

For the following problems, write the equation of the line using the given information in slope-intercept form. $$ (1,6),(-1,-6) $$

Step-by-Step Solution

Answer

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

Step 1: Find the slope (m)

Using the formula for finding the slope between two points, (y2 - y1) / (x2 - x1), we can find the slope (m) of the line by plugging in the given points (1, 6) and (-1, -6): $$ m = \frac{-6 - 6}{-1 - 1} = \frac{-12}{-2} = 6 $$ The slope of the line is 6.

Step 2: Find the y-intercept (b)

Using one of the given points (either (1, 6) or (-1, -6)) and the slope we found in the previous step, we can find the y-intercept (b) by plug the point and the slope into the slope-intercept equation and solve for b. Let's use the point (1, 6): $$ 6 = 6(1) + b $$ Subtract 6 from both sides to solve for b: $$ b = 6 - 6 = 0 $$ The y-intercept (b) is 0.

Step 3: Write the equation of the line in slope-intercept form

Now that we have the slope (m = 6) and the y-intercept (b = 0), we can write the equation of the line in slope-intercept form, y = mx + b: $$ y = 6x + 0 $$ Simplify the equation by removing the "+ 0": $$ y = 6x $$ The equation of the line in slope-intercept form is y = 6x.

Key Concepts

Understanding Slope Calculation

The slope of a line is a measure of its steepness. It tells us how much the line rises or falls as it moves from left to right. To calculate the slope (\(m\)), you can use the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of two points on the line.
When using this formula, ensure:

• Subtract the y-coordinates in the same order as the x-coordinates (i.e., \(y_2 - y_1\) and \(x_2 - x_1\)).
• A positive slope means the line rises, and a negative slope means it falls.
• A slope of zero means the line is horizontal, while an undefined slope means the line is vertical.

For the given points, (1, 6) and (-1, -6), the calculation was \(m = \frac{-6 - 6}{-1 - 1} = 6\).
This means the line rises steeply as it moves to the right.

What is the Y-Intercept?

The y-intercept of a line is the point where the line crosses the y-axis. It is represented by the variable \(b\) in the slope-intercept form equation \(y = mx + b\).
To find the y-intercept:

• Use the slope and one of the points.
• Substitute these values into the equation, \(y = mx + b\).
• Solve for \(b\).

In this exercise, using the point (1, 6) and the slope 6, you set up the equation \(6 = 6(1) + b\). After solving, you find \(b = 0\).
This means the line crosses the y-axis at the origin, or point (0, 0).

Connecting It All: Linear Equations

Linear equations are used to describe straight lines and are written in the slope-intercept form \(y = mx + b\). Here:

• \(m\) is the slope, indicating the line's direction and steepness.
• \(b\) is the y-intercept, showing where the line crosses the y-axis.

The foundation of linear relationships in math, this formula lets us easily graph and interpret lines.
From our previous solutions, the equation \(y = 6x\) indicates a line with a slope of 6 and y-intercept 0. This means for every one unit increase in \(x\), \(y\) increases by 6. As \(b\) is 0, the line passes through the origin.
Understanding and using linear equations is crucial for grasping the basics of algebra and applications in real-life contexts.