Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible. Passes through (2,4) and (4,10)

Step-by-Step Solution

Answer

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

Step 1: Identify the Slope

To find the slope of the line that passes through the points (2,4) and (4,10), we use the formula for slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the coordinates: \( m = \frac{10 - 4}{4 - 2} = \frac{6}{2} = 3 \). So, the slope \( m \) is 3.

Step 2: Use the Point-Slope Formula

With the slope \( m = 3 \) and using the point (2,4), apply the point-slope formula: \( y - y_1 = m(x - x_1) \). Substituting in the values: \( y - 4 = 3(x - 2) \).

Step 3: Simplify to Slope-Intercept Form

Simplify the equation from the point-slope form to the slope-intercept form \( y = mx + b \). Start by distributing the 3: \( y - 4 = 3x - 6 \). Then solve for \( y \) by adding 4 to both sides: \( y = 3x - 2 \).

Step 4: Verify the Equation

Use the original points to verify that they satisfy the equation \( y = 3x - 2 \). For point (2,4), substitute 2 for \( x \) and check if \( y = 4 \): \( y = 3(2) - 2 = 6 - 2 = 4 \). For point (4,10), substitute 4 for \( x \): \( y = 3(4) - 2 = 12 - 2 = 10 \). Both points satisfy the equation.

Key Concepts

Slope Calculation

Before we jump into linear equations, understanding how to calculate the slope is crucial. The slope, denoted as \( m \), measures the steepness and the direction of a line. Think of it as describing how fast and in which way a line climbs or falls between two points. Let's dive into the steps on how to calculate it. Using two points on a line, such as \((2,4)\) and \((4,10)\), you can determine the slope with the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of the two points. It's a simple subtraction and division process.

• First, subtract the \( y \)-values: \( 10 - 4 = 6 \)
• Next, subtract the \( x \)-values: \( 4 - 2 = 2 \)
• Now, divide these results: \( \frac{6}{2} = 3 \)

The slope \( m \) comes out to be 3, indicating that for every unit increase in \( x \), \( y \) increases by 3. A positive slope like this tells us the line moves upwards as we move from left to right on the graph.

Point-Slope Form

Once you have the slope, the next step is to express the linear equation using the point-slope form. This form is especially handy when you know a point on the line and the slope. The formula you'll use here is: \[ y - y_1 = m(x - x_1) \] Consider we have a point, say \((2,4)\), and the slope \( m = 3 \). Substituting these values, you get:\[ y - 4 = 3(x - 2) \] This is the point-slope form. It's a preliminary version of your linear equation.

• It clearly shows one point \((x_1, y_1)\) the line goes through.
• It uses the slope to describe the line's inclination.

This form is a bridge that connects a specific point on a line with its slope, allowing us to set a foundation to find the more common slope-intercept form.

Slope-Intercept Form

To make the linear equation more user-friendly, we simplify the point-slope form to the slope-intercept form. This involves a bit of algebra to isolate \( y \) on one side of the equation. The slope-intercept form is signified by: \[ y = mx + b \] where \( m \) is the slope, and \( b \) is the y-intercept. Let's transform our equation from the point-slope form we derived earlier:\[ y - 4 = 3(x - 2) \] First, distribute the \( 3 \): \[ y - 4 = 3x - 6 \]Then, solve for \( y \) by adding 4 to both sides:\[ y = 3x - 2 \] This is your slope-intercept form.

• It offers an immediate view of the slope \( m = 3 \).
• It shows the y-intercept, \( b = -2 \), where the line crosses the y-axis.

This clean form can easily be interpreted and graphed, providing a straightforward formula to see how \( y \) changes with \( x \). It's the go-to format for plotting straight lines.