Problem 28 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. \(x\) -intercept \(-8 ; \quad y\) intercept 6

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{3}{4}x + 6 \).

Step 1: Understanding the Intercepts

The line intersects the x-axis at \((-8, 0)\) and the y-axis at \((0, 6)\). These intercepts provide two points that will help us determine the slope and eventually, the equation of the line.

Step 2: Calculating the Slope

The slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the points \((-8, 0)\) and \((0, 6)\) into the formula gives:\[ m = \frac{6 - 0}{0 - (-8)} = \frac{6}{8} = \frac{3}{4} \].

Step 3: Formulating the Equation

With the slope known, we use the point-slope form of a line to find the equation. The formula is:\[ y - y_1 = m(x - x_1) \]Using the point \((0, 6)\) and the slope \(m = \frac{3}{4}\), we substitute into the formula: \[ y - 6 = \frac{3}{4}(x - 0) \].Simplifying:\[ y = \frac{3}{4}x + 6 \].

Key Concepts

Intercepts

Intercepts are the points at which a line crosses the x-axis and y-axis. These are important because they give us specific points on the line, making it easier to formulate equations. When a line crosses the x-axis, the y-coordinate is zero, which is why the x-intercept of this problem is \((-8, 0)\). Conversely, the y-intercept occurs when the x-coordinate is zero, giving us \((0, 6)\) for the y-intercept in this exercise.

Intercepts serve as reliable anchors in determining the path of a line across the coordinate plane. Knowing these points provides a direct way to visualize where the line splits the axes into different regions, packaging impactful information with simplicity. This information is critical when we move to the next step: calculating the slope.

Slope Calculation

The slope of a line is a measure of its steepness and direction. It can be calculated using any two points on the line. These points are plugged into the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Let's take a closer look at this process:

• Identify two points on the line. Here, these are \((-8, 0)\) and \((0, 6)\).
• Subtract the y-values (6 - 0) and the x-values (0 - (-8)).
• Divide the difference in the y-values by the difference in the x-values to find the slope.

Following these steps, we get a slope \(m = \frac{3}{4}\). A positive slope like this indicates the line tilts upwards as it moves from left to right.

Having a solid grasp of how to calculate the slope will set you up perfectly for writing the equation in the point-slope form, which is our next focus.

Point-Slope Form

The point-slope form is a powerful tool for deriving the equation of a line when you know the slope and one point on the line. The formula looks like this: \[ y - y_1 = m(x - x_1) \]. Let's break it down:

• \(y_1\) and \(x_1\) represent the y and x coordinates of a known point on the line.
• \(m\) is the slope of the line, which we've calculated as \(\frac{3}{4}\).

By substituting \(m = \frac{3}{4}\) and the point \((0, 6)\) into the form, we have:\[ y - 6 = \frac{3}{4}(x - 0) \].

Simplifying this gives us the final equation: \[ y = \frac{3}{4}x + 6 \]. This form is beneficial because it straightforwardly shows how one quantity changes with respect to another, providing a simple path from numeric data to a visual line on a graph.