Problem 32 - College Algebra

Question

For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible. Passes through (1,5) and (4,11)

Step-by-Step Solution

Answer

The linear equation is \(y = 2x + 3\).

Step 1: Identify Points for Equation

We are given two points, Point 1: \((1, 5)\) Point 2: \((4, 11)\). These will help us find the slope of the line.

Step 2: Calculate the Slope

The formula to calculate the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute the given points: \[m = \frac{11 - 5}{4 - 1} = \frac{6}{3} = 2\]Thus, the slope \(m\) is 2.

Step 3: Use Point-Slope Form

The point-slope form of a linear equation is: \[y - y_1 = m(x - x_1)\]Using Point 1 \((1, 5)\) and the slope \(m = 2\), substitute these values into the formula:\[y - 5 = 2(x - 1)\]

Step 4: Simplify to Slope-Intercept Form

Distribute and simplify the equation from step 3:\[y - 5 = 2x - 2\]Add 5 to both sides:\[y = 2x + 3\]This is the equation in slope-intercept form, \(y = mx + b\).

Key Concepts

Slope

The slope of a line is a measure of how steep the line is. Imagine you're going up or down a hill—the steeper the hill, the greater the slope. The slope is represented by the letter \( m \) in linear equations. It is calculated using the rise over run formula. This means you need to look at how much the line goes up (or down) for a certain amount of horizontal movement along the x-axis.

In mathematics, the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated with the formula:

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

This formula tells us the change in \( y \) (vertical change) divided by the change in \( x \) (horizontal change). The slope can take different values:

• A positive slope means the line goes upwards as you move from left to right.
• A negative slope means the line goes downwards as you move from left to right.
• A zero slope means the line is perfectly horizontal, and an undefined slope means the line is perfectly vertical.

In our problem, the slope was found to be 2, indicating a fairly steep rise as you move from left to right.

Point-Slope Form

The point-slope form is a linear equation that is especially useful when you know the slope of a line as well as one point that the line passes through. This form makes it straightforward to create a formula for a line.

The general structure of the point-slope form is:

• \( y - y_1 = m(x - x_1) \)

Here, \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope. In this formula:

• \( y_1 \) is the y-coordinate of the known point.
• \( x_1 \) is the x-coordinate of the known point.
• \( x \) and \( y \) will represent any other point on the line.

In the provided exercise, we used the point \((1, 5)\) and a slope of 2. When plugged into the point-slope form, the equation becomes \( y - 5 = 2(x - 1) \). This is a crucial intermediary step in deriving other forms of linear equations.

Slope-Intercept Form

The slope-intercept form serves as one of the most intuitive ways to represent a linear equation. It's great for quickly identifying the slope and the y-intercept of a line, which is where the line crosses the y-axis.

The formula for the slope-intercept form is:

• \( y = mx + b \)

In this equation, \( m \) is the slope, and \( b \) is the y-intercept. You can think of the y-intercept \( b \) as the point where the line "intercepts" or meets the y-axis.

• \( x \) represents the variable, and \( y \) is the dependent variable.
• Here, you instantly know the slope just by looking at \( m \), and the starting point on the y-axis is \( b \).

In our example, we simplified the point-slope form \( y - 5 = 2(x - 1) \) to the slope-intercept form \( y = 2x + 3 \). This tells us that the slope is 2, and the line intersects the y-axis at 3.