Problem 66 - Elementary Algebra

Question

Find the coordinates of two points on the given line, and then use those coordinates to find the slope of the line. $$y=6 x$$

Step-by-Step Solution

Answer

Two points are (0, 0) and (1, 6); slope is 6.

Step 1: Choose Values for x

To find coordinates on the line, we start by selecting values for \( x \). Let's choose \( x = 0 \) and \( x = 1 \).

Step 2: Calculate y for x = 0

Substitute \( x = 0 \) into the equation \( y = 6x \). This gives \( y = 6(0) = 0 \). So, one point is \((0, 0)\).

Step 3: Calculate y for x = 1

Substitute \( x = 1 \) into the equation \( y = 6x \). This gives \( y = 6(1) = 6 \). So, another point is \((1, 6)\).

Step 4: Apply the Formula for Slope

The slope \( m \) is calculated using the formula \( m = (y_2 - y_1) / (x_2 - x_1) \). Using the points \((0, 0)\) and \((1, 6)\), we find the slope: \( m = (6 - 0) / (1 - 0) = 6 \).

Step 5: Conclusion

The coordinates of the two points on the line are \((0, 0)\) and \((1, 6)\). The slope of the line is 6.

Key Concepts

Coordinates of Points on a Line

To identify points that lie on a given line, such as the line defined by the equation \( y = 6x \), we need to find pairs of \( x \) and \( y \) values that satisfy the equation. These pairs are known as coordinates, represented as \((x, y)\). The process starts with selecting any values for \( x \), since the line is defined in terms of \( x \), and then substituting these into the equation to find corresponding \( y \) values. For example:

• If we choose \( x = 0 \), substituting into the equation gives \( y = 6(0) = 0 \), resulting in the coordinate \((0,0)\).
• Choosing \( x = 1 \), substituting gives \( y = 6(1) = 6 \), resulting in the coordinate \((1,6)\).

This method of choosing different \( x \) values helps us find multiple points, all of which lie on the line described by the equation. These points can then be plotted on a graph to visualize the line.

Equation of a Line

The equation of a line provides a mathematical description of the relationship between the \( x \) and \( y \) coordinates on a two-dimensional plane. In the equation \( y = 6x \), you'll notice it's in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The term \( 6x \) indicates the slope is 6, and since there's no \( + b \) term, \( b = 0 \), meaning the line intercepts the y-axis at 0.

This equation tells us several things:

• The line passes through the origin point \((0, 0)\).
• For every unit of increase in \( x \), \( y \) increases by 6 units, indicating a steep incline.
• The line extends infinitely in both directions along its path defined by this linear relationship between \( x \) and \( y \).

Calculating Slope

Slope is a key concept in understanding lines and their behavior. It measures the steepness or incline of a line and is calculated as the ratio of the change in \( y \) to the change in \( x \) between two points on the line. Given two points \((x_1, y_1)\) and \((x_2, y_2)\), the formula for calculating slope \( m \) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

• Using the points \((0, 0)\) and \((1, 6)\), substitute into the formula to get \( m = \frac{6 - 0}{1 - 0} = 6 \).
• This tells us the slope of the line is 6, consistent with the coefficient of \( x \) in the line's equation \( y = 6x \).
• A slope of 6 means for every one unit increase in \( x \), \( y \) increases by 6 units.

Understanding slope is fundamental for analyzing the direction and steepness of lines, which is essential when plotting graphs or interpreting linear relationships.