Problem 34 - 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 (-2,8) and (4,6)

Step-by-Step Solution

Answer

The linear equation is \( y = -\frac{1}{3}x + \frac{22}{3} \).

Step 1: Identify the Slope Formula

To find the equation of a line given two points, we start by calculating the slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are the given points.

Step 2: Apply the Slope Formula

Substitute the points \((-2,8)\) and \((4,6)\) into the slope formula: \( m = \frac{6 - 8}{4 - (-2)} = \frac{-2}{6} = -\frac{1}{3} \). Thus, the slope \( m = -\frac{1}{3} \).

Step 3: Point-Slope Form Equation

Use the point-slope form \( y - y_1 = m(x - x_1) \) to construct the equation. Choose one of the points, say \((-2, 8)\), and substitute: \( y - 8 = -\frac{1}{3}(x - (-2)) \).

Step 4: Simplify the Equation

Simplify the equation from Step 3: \( y - 8 = -\frac{1}{3}(x + 2) \). This simplifies to \( y - 8 = -\frac{1}{3}x - \frac{2}{3} \).

Step 5: Solve for y

Add 8 to both sides to solve for \( y \): \( y = -\frac{1}{3}x - \frac{2}{3} + 8 \). Write 8 as \( \frac{24}{3} \). Combine terms to get: \( y = -\frac{1}{3}x + \frac{22}{3} \).

Step 6: Confirm the Line Equation

The final equation of the line is \( y = -\frac{1}{3}x + \frac{22}{3} \). This equation is in slope-intercept form \( y = mx + b \) with \( m = -\frac{1}{3} \) and \( b = \frac{22}{3} \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way to express the equation of a straight line. It's particularly useful because it easily shows the slope of the line and its y-intercept, where the line crosses the y-axis. The general format for this form is \( y = mx + b \), where:

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

In our example, after simplifying, we arrived at the line equation \( y = -\frac{1}{3}x + \frac{22}{3} \). Here, the slope \( m = -\frac{1}{3} \) shows the steepness and direction (downwards as it is negative). The y-intercept \( b = \frac{22}{3} \) tells us that the line crosses the y-axis at \( y = \frac{22}{3} \). Understanding this form helps recognize how changes in \( m \) and \( b \) affect the line's position and orientation.

Point-Slope Form

The point-slope form is another method for writing the equation of a line. This form is useful when you know the slope of the line and one of its points. The formula for the point-slope form is \( y - y_1 = m(x - x_1) \), where:

• \( (x_1, y_1) \) is a known point on the line.
• \( m \) is the slope.

In our step-by-step solution, we used this form with the point \((-2, 8)\) and the calculated slope \( m = -\frac{1}{3} \). This led to the equation \( y - 8 = -\frac{1}{3}(x - (-2)) \). Point-slope form is versatile because it can be easily manipulated into slope-intercept form or standard form, making it a valuable tool in algebra.

Slope Formula

The slope formula is fundamental in finding the equation of a line, especially when two points are known. This formula is a way to calculate the slope \( m \), which is the ratio of the change in y-coordinates to the change in x-coordinates between two points on the line. The formula is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Where:

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

In the original exercise, we utilized the slope formula with the points \((-2, 8)\) and \((4, 6)\). Substituting these points gives us \( m = \frac{6 - 8}{4 - (-2)} = -\frac{1}{3} \). This negative value indicates the line slopes downwards as it moves from left to right. Calculating the slope correctly is crucial because it determines the angle and direction of the line in any equation form.