Problem 46 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. Passing through \((1990,4)\) and parallel to the line passing through \((1980,3)\) and \((2000,8)\)

Step-by-Step Solution

Answer

The equation is \( y = \frac{1}{4}x - 493.5 \).

Step 1: Finding the Slope of the Reference Line

To find the slope-intercept form of the line, we first need to determine the slope. Since our line is parallel to the line passing through \((1980,3)\) and \((2000,8)\), we calculate the slope using the formula for slope: \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \). Plugging in the given points: \( m = \frac{{8 - 3}}{{2000 - 1980}} = \frac{5}{20} = \frac{1}{4} \). The slope of our line is \(\frac{1}{4}\).

Step 2: Writing the Slope-Intercept Equation

With the slope of the line known, we can write the equation in slope-intercept form \( y = mx + b \). The slope \( m = \frac{1}{4} \), so we have \( y = \frac{1}{4}x + b \).

Step 3: Solving for the Y-Intercept

To find the y-intercept \( b \), substitute the given point \((1990, 4)\) into the equation \( y = \frac{1}{4}x + b \). This gives us \( 4 = \frac{1}{4}(1990) + b \). Simplify \( \frac{1}{4} \times 1990 = 497.5 \). So the equation becomes \( 4 = 497.5 + b \). Solve for \( b \) by subtracting 497.5 from both sides: \( b = 4 - 497.5 = -493.5 \).

Step 4: Finalizing the Equation

Now that we have the slope \( \frac{1}{4} \) and the y-intercept \( b = -493.5 \), the equation of the line in slope-intercept form is \( y = \frac{1}{4}x - 493.5 \).

Key Concepts

Slope Calculation

Calculating the slope is the first step in understanding how steep or flat a line is. The slope is often represented by the letter \( m \). When finding the slope between two points

• The formula used is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
• Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

For example, suppose we have two points

• \((1980, 3)\) and \((2000, 8)\).

We substitute these values into the formula:

• \( m = \frac{8 - 3}{2000 - 1980} = \frac{5}{20} = \frac{1}{4} \).

This slope tells us the line rises 1 unit for every 4 units it runs horizontally.

Y-Intercept

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

• We use a known point on the line and the slope.
• Substitute these into the equation to solve for \( b \).

In our example with the point \((1990, 4)\) and slope \( \frac{1}{4} \):

• \( 4 = \frac{1}{4} \times 1990 + b \).
• This simplifies to ‎ ‎ ‎ \( 4 = 497.5 + b ‎ ‎ ‎\), solving for \( b \)
• we get \( ‎ b = -493.5 \).

Thus, the y-intercept of the line is \(-493.5\).

Parallel Lines

Parallel lines are lines that have the same slope but different y-intercepts. They run alongside each other and never intersect. Since they share the same slope:

• In our exercise, the original line through points \((1980, 3)\) and \((2000, 8)\) and our line pass through different y-intercept points.
• Both have a slope of \( \frac{1}{4} \).

This consistency in slope shows that they are parallel but their different y-intercepts mean they are separate lines. It's crucial to maintain the slope consistency to ensure the lines remain parallel.

Linear Equation

A linear equation in slope-intercept form is written as \( y = mx + b \), where \( m \) denotes the slope and \( b \) the y-intercept. It represents a straight line on a graph. In our example:

• The equation came out as \( y = \frac{1}{4}x - 493.5 \).
• This shows a linear relationship between \( x \) and \( y \).

The slope \( \frac{1}{4} \) indicates a mild incline as \( x \) increases, while the y-intercept \(-493.5\) tells us where the line crosses the y-axis. Understanding linear equations helps in visualizing how changes in \( x \) affect \( y \).