Problem 31 - College Algebra

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: Find the Slope

To find the slope (m) of the line passing through two points (2, 4) and (4, 10), use the formula for the slope: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the given points into the formula:\[ m = \frac{10 - 4}{4 - 2} = \frac{6}{2} = 3 \]So, the slope is 3.

Step 2: Write the Equation in Point-Slope Form

Now we use the point-slope form of the equation of a line which is:\[ y - y_1 = m(x - x_1) \]Using the slope from Step 1 (m = 3) and one of the points, say (2, 4), plug these values into the form:\[ y - 4 = 3(x - 2) \]

Step 3: Convert to Slope-Intercept Form

Now, convert the equation from point-slope form to slope-intercept form (y = mx + b).Start by distributing the slope (3) on the right side:\[ y - 4 = 3x - 6 \]Then add 4 to both sides to solve for y:\[ y = 3x - 6 + 4 \]Simplify:\[ y = 3x - 2 \]This is the equation of the line in slope-intercept form.

Key Concepts

Slope Calculation

The slope of a line is a measure of how steep the line is. You calculate it by determining the change in the y-values divided by the change in the x-values between two points on the line. This is known as "rise over run." Here's how to do it:

• Identify two points: Find two points on the line, such as (2, 4) and (4, 10).
• Use the slope formula: The formula for the slope, denoted as \(m\), is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].
• Plug in the values: Substitute the coordinates of the points into the formula. Calculate: \[ m = \frac{10 - 4}{4 - 2} = \frac{6}{2} = 3 \].

The slope \(m = 3\) means for every 1 unit increase in x, y increases by 3 units. This linear relationship is essential for defining the line's direction and steepness in a coordinate plane.

Point-Slope Form

The point-slope form of a linear equation is useful when you know the slope of a line and one point on the line. It is represented by the equation:

• \(y - y_1 = m(x - x_1)\)
• Select a point: Choose one point from your two points, for example, (2, 4).
• Apply the slope: Use the slope calculated earlier, \(m = 3\).
• Fill in the formula: Substitute the slope and the chosen point into the point-slope form: \(y - 4 = 3(x - 2)\).

This form is particularly beneficial in understanding how a specific point and the slope relate to the entire equation. It's a straightforward method to transition from understanding the physical geometry of a line to writing an equation.

Slope-Intercept Form

The slope-intercept form of a linear equation, \(y = mx + b\), is one of the most intuitive forms for graphing. It directly shows both the slope and the y-intercept.

• Begin with the point-slope form: From the previous step, we have \(y - 4 = 3(x - 2)\).
• Distribute the slope: Expand by distributing the 3: \(y - 4 = 3x - 6\).
• Solve for y: Add 4 to both sides to isolate y: \(y = 3x - 6 + 4\).
• Simplify the equation: Combine terms for the final result: \(y = 3x - 2\).

This form is fantastic for quickly graphing a line since the slope (3) and y-intercept (-2) are clearly defined. You can easily see how changes in x affect y, making prediction and graph plotting a breeze.