Problem 40 - Elementary Algebra Within Reach

Question

In Exercises 39-42, find \(m\) and \(b\) such that \(y=m x+b\) is the equation of the line through the points. $$ (1,3),(4,9) $$

Step-by-Step Solution

Answer

The equation of the line that passes through the points (1,3) and (4,9) is \(y = 2x + 1\).

Step 1: Calculate the Slope (m)

The slope (\(m\)) of a line passing through two points (\(x_1, y_1\)) and (\(x_2, y_2\)) can be calculated using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the given points into this formula to get the slope: \(m = \frac{9 - 3}{4 - 1} = 2\)

Step 2: Calculate the Y-Intercept (b)

With the slope \(m\) and one of the points (e.g., \((1, 3)\)), the y-intercept \(b\) can be calculated using the formula: \(b = y - mx\). This results in \(b = 3 - 2*1 = 1\).

Step 3: Write the Equation of the Line

The equation of the line that passes through the points (1,3) and (4,9) in slope-intercept form is \(y = mx + b\). Substitute the calculated values of \(m\) and \(b\) into this equation to obtain the final answer: \(y = 2x + 1\).

Key Concepts

Slope

The slope of a line is a measure of its steepness. If you're familiar with skiing, you might think of it like the slope of a hill.
In math, we describe the slope as the "rise" over the "run," or how much a line goes up versus how far it goes horizontally.

• This is calculated as the change in the y-values divided by the change in the x-values between two points on the line.
• Using our formula, the slope \( m \) becomes \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
• For example, with the points \((1,3)\) and \((4,9)\), our slope is \( m = \frac{9 - 3}{4 - 1} = 2 \).

A positive slope means the line is increasing or going upwards as you move from left to right.
A negative slope means the line is decreasing or going downwards as you move from left to right.
In our example, the slope is positive, indicating an upward trend.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis.
This is an important feature of a line because it tells us where the line will hit the vertical axis when \( x = 0 \).

• The y-intercept is represented by the symbol \( b \) in the slope-intercept form \( y = mx + b \).
• To find \( b \), use the formula \( b = y - mx \) with a point on the line.
• Using our point \((1,3)\) and slope \( m = 2 \), we compute \( b = 3 - 2 \times 1 = 1 \).

Hence, the y-intercept, \( b = 1 \), tells us the line will cross the y-axis at the point \((0,1)\).
This gives us a starting point for drawing or interpreting the line.

Slope-Intercept Form

The slope-intercept form is an efficient way to express the equation of a line.
It highlights two major properties of the line: its slope and its y-intercept.

• The form is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
• This form allows us to quickly graph the line by first plotting the y-intercept and then using the slope to determine the direction and steepness of the line.
• From our calculations, the equation of our line is \( y = 2x + 1 \).

By substituting \( m = 2 \) and \( b = 1 \) into this form, we can visualize the line on a graph.
Start at the y-intercept \((0,1)\), and draw a line rising 2 units for every 1 unit it goes to the right.
This form makes it easy to understand and graph linear relationships.