Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write an equation of the line passing through the two points. Use the slope- intercept form, if possible. If not possible, explain why. $$(3,4),(10,6)$$

Step-by-Step Solution

Answer

The equation of the line passing through the points (3,4) and (10,6) is \(y = (2/7)x + 20/7\).

Step 1: Calculate the Slope

First, we calculate the slope (m) using the formula: \(m = (y_2 - y_1)/(x_2 - x_1)\). The two points given are (3, 4) and (10, 6), so \(m = (6 - 4)/(10 - 3) = 2/7\).

Step 2: Find the Y-Intercept

Now, use the slope and one of the points to find the y-intercept (b). Rearrange the slope-intercept equation to solve for b: \(b = y - mx\). Using point (3, 4) and the slope 2/7: \(b = 4 - (2/7) * 3 = 20/7\).

Step 3: Write the Equation

Now that we have both the slope (m) and the y-intercept (b), we can write the equation in slope-intercept form: \(y = mx + b\). Plug in our values to get: \(y = (2/7)x + 20/7\).

Key Concepts

Equation of a Line

The equation of a line is generally written in two primary forms: the slope-intercept form and the point-slope form. In this exercise, we focus on the slope-intercept form, which is a common way to express linear equations due to its simplicity.

This form is expressed as:

• \( y = mx + b \)

Here, \( m \) represents the slope of the line, while \( b \) represents the y-intercept, the point where the line crosses the y-axis.

The slope-intercept form provides a convenient way to graph a line or to understand the relationship between the variables \( x \) and \( y \). You can easily identify how steep a line is and where it starts on the y-axis by just looking at these parameters.

In any situation where you're given points or a graph, translating it into a linear equation helps predict and calculate future or unknown values.

Slope Calculation

The slope of a line is a measure of its steepness and direction. It is simply a ratio that compares the change in the y-values to the change in the x-values between two points on the line.

The formula for calculating slope \( m \) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This expresses how much \( y \) changes for a unit change in \( x \).

In the problem, using the points (3,4) and (10,6):

• Change in y: \( 6 - 4 = 2 \)
• Change in x: \( 10 - 3 = 7 \)

Thus, the slope is \( \frac{2}{7} \), indicating that for every 7 units the line moves horizontally, it moves 2 units vertically.

A positive slope means the line goes upwards from left to right, while a negative slope means it goes downwards. A slope of zero implies a horizontal line, and an undefined slope (when \( x_2 = x_1 \)) suggests a vertical line.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. At this point, the value of \( x \) is zero, making it a straightforward check on where the line begins in vertical terms.

To find the y-intercept \( b \), we rearrange the line equation in slope-intercept form:

• \( b = y - mx \)

Using one of the given points (3,4) in our original problem, we substitute in the slope \( \frac{2}{7} \):
\( b = 4 - \frac{2}{7} \times 3 \)
Calculating this gives us \( b = \frac{20}{7} \).

This means the line crosses the y-axis at \( \frac{20}{7} \), or approximately 2.86. The y-intercept is vital for establishing the vertical starting point of the line when graphing and helps to complete the equation of the line. This term provides insight into the initial value of a dependent variable when all other influences (x-values) are nullified.