Problem 18 - Elementary Algebra

Question

For the following problems, find the equation of the line using the information provided. Write the equation in slope-intercept form. passes through the points (5,2) and (2,1) .

Step-by-Step Solution

Answer

Answer: The equation of the line is y = (1/3)x + 1/3.

Step 1: Find the slope (m)

To find the slope, use the formula: m = (y2 - y1)/(x2 - x1). With points (5,2) and (2,1), we have x1 = 5, y1 = 2, x2 = 2, and y2 = 1. Plugging these values into the formula, we get: m = (1 - 2)/(2 - 5) = -1/(-3) = 1/3.

Step 2: Find the y-intercept (b)

Now that we have the slope (m = 1/3), we can find the y-intercept (b) by plugging in one of the points into the slope-intercept equation (y = mx + b). We will use point (5,2), so x = 5 and y = 2: 2 = (1/3)(5) + b.

Step 3: Solve for b

Continue solving the equation from step 2: 2 = (1/3)(5) + b 2 = 5/3 + b To isolate b, subtract 5/3 from both sides: 2 - 5/3 = b 1/3 = b

Step 4: Write the final equation in slope-intercept form

Now that we have the slope (m = 1/3) and the y-intercept (b = 1/3), we can write the equation in slope-intercept form: y = mx + b y = (1/3)x + 1/3.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a popular way to express the equation of a straight line. It is written as:

• \( y = mx + b \)

In this equation, \( m \) represents the slope of the line, while \( b \) is the y-intercept. This form is convenient because it directly provides both the slope and where the line crosses the y-axis. Thus, when given a line, putting it into slope-intercept form can quickly show us key properties of the line.
The structure of the slope-intercept form helps in easily graphing the line on a coordinate plane. You start by plotting the y-intercept and then use the slope to determine the next points. This method is very intuitive for both beginners and advanced learners. To gain confidence, you can practice converting different line equations into this form.

Finding Slope

When calculating the slope between two points, the formula you use is:

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

The slope \( m \) describes how steep a line is. In simple terms, it's the 'rise over run', or how much a line goes up (or down) for each unit it goes right. If the slope is positive, the line goes upwards; if it's negative, the line slopes downward. A zero slope means the line is perfectly horizontal.
For example, with points (5,2) and (2,1), you calculate:

• \( m = \frac{1 - 2}{2 - 5} = \frac{-1}{-3} = \frac{1}{3} \)

This means for every 3 units you move to the right along the x-axis, you go 1 unit up along the y-axis, creating a gentle slope upwards. Understanding slope helps you analyze and interpret the direction and steepness of a line, which is critical in many areas of math and real-world context.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis on a graph. In the equation \( y = mx + b \), the y-intercept is represented by \( b \). This point can be found by setting \( x \) to zero and solving the equation for \( y \).
For our line passing through points (5,2) and (2,1) with a slope of \( \frac{1}{3} \), we discovered \( b \) by plugging one of the points into the slope-intercept equation:

• \( 2 = \frac{1}{3}(5) + b \)
• Simplifying gives: \( 2 = \frac{5}{3} + b \)
• Subtract \( \frac{5}{3} \) from both sides to find: \( 1/3 = b \)

Thus, the y-intercept is \( \frac{1}{3} \), meaning the line cuts the y-axis a little above the origin. Knowing where the y-intercept is allows you to begin graphing or analyzing the line, since it's a fixed starting point. The y-intercept can provide context about what's being analyzed, such as an initial value or starting amount in a problem situation.