Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the line passing through the points. \((4,3),(-4,-4)\)

Step-by-Step Solution

Answer

The equation of the line passing through the points (4,3) and (-4,-4) is \(y = 0.875x - 0.5\).

Step 1: Calculate the Slope

First of all, calculate the slope (m) of the line using the formula \((y2-y1)/(x2-x1)\), substituting the coordinates of the given points into the formula. So, \(m = (-4-3)/(-4-4) = -7/-8 = 0.875\).

Step 2: Find the Y-Intercept

Now utilize one of the given points and slope to find the y-intercept. Let's use point (4,3). Substitute these values into the equation. You'll get: \(3 = 0.875 * 4 + b\). Solving this equation for b yields \(b = -0.5\).

Step 3: Write the Equation of the Line

Finally, having the slope (m) and y-intercept (b), it's possible to write the equation of the line: \(y = 0.875x - 0.5\).

Key Concepts

Slope

The slope of a line is a measure of how "steep" or "tilted" the line is on the coordinate plane. To find the slope between two points \(x_1, y_1\) and \(x_2, y_2\), you can use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}.\]This formula gives you a ratio that describes how much the line rises or falls vertically for every unit it moves horizontally. In simpler terms, if the slope is positive, the line ascends as it moves from left to right; if negative, it descends. A zero slope means the line is perfectly horizontal, and if the slope is undefined, the line is vertical.
Let's see how this applies to our example: \(4,3\) and \(-4,-4\). Plugging these into our formula, we have: \[m = \frac{-4 - 3}{-4 - 4} = \frac{-7}{-8} = 0.875.\]This number, 0.875, tells us the line rises 0.875 units for every 1 unit it moves to the right.
Understanding slope helps you visualize the line and predict how changes in the x-values affect the y-values.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. It's a crucial component of the line's equation as it provides the starting point for the line on the graph. This point has coordinates \(0, b\), where \b\ is the y-intercept value.
To find the y-intercept, you can use the slope-intercept form of the equation of a line: \[y = mx + b.\]Here, \(m\) is the slope, and \(b\) is the y-intercept. If the slope is known, and one point on the line is known, the y-intercept can be calculated by substituting the point's coordinates and the slope into the equation and solving for \(b\).
In the given example, using the slope 0.875 and point (4,3), plug these into the equation:

• \(3 = 0.875\times4 + b\)

Solving for \(b\), you'll rearrange this to: \[b = 3 - 0.875\times4 = -0.5\]Therefore, the y-intercept, \(b\), is -0.5. This means the line intersects the y-axis at \(-0.5\). Understanding the y-intercept helps to pinpoint the exact position of the line on the graph.

Linear Equation

A linear equation is a fundamental concept in algebra that forms a straight line when graphed on a coordinate plane. It is typically written in the format \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
This format is known as the slope-intercept form and is advantageous because it immediately shows the rate of change (slope) and the starting value (y-intercept).

• The "\(y = mx + b\)" equation clearly outlines how the value of \(y\) depends on \(x\).
• The slope \(m\) demonstrates how much \(y\) changes for every change in \(x\).
• The y-intercept \(b\) denotes where the line crosses the y-axis, indicating where the line starts if continued backwards to \(x = 0\).

In our example, the line's equation, \(y = 0.875x - 0.5\), tells us that for every unit increase in \(x\), \(y\) increases by 0.875. Also, when \(x\) is zero, \(y\) is -0.5.
Understanding the linear equation allows us to explore how different modifications to \(x\) or \(b\) impact the overall line on a graph. This concept is essential for solving problems related to lines and patterns in mathematics.