Problem 74 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. $$ (4,4), \quad(5,1) $$

Step-by-Step Solution

Answer

Answer: The equation of the line is y = -3x + 16.

Step 1: Calculate the slope

First, we need to find the slope. The slope formula is given as: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Using the given points (4,4) and (5,1), let (x1, y1) = (4,4) and (x2, y2) = (5,1). Plugging the points into the slope formula: $$ m = \frac{1 - 4}{5 - 4} = -\frac{3}{1} = -3 $$ So, the slope of the line is -3.

Step 2: Determine the y-intercept

Now that we have the slope, we need to find the y-intercept. Using one of the given points (we will use (4,4)) and the slope we just found, plug it into the equation of a line (y = mx + b), and solve for b: $$ 4 = -3(4) + b $$ Simplify and solve for b: $$ 4 = -12 + b $$ Therefore: $$ b = 4 + 12 = 16 $$ The y-intercept is 16.

Step 3: Write the equation in slope-intercept form

Now that we have the slope (m=-3) and y-intercept (b=16), we can plug them into the slope-intercept form equation (y = mx + b): $$ y = -3x + 16 $$ The equation of the line in slope-intercept form is: $$ y = -3x + 16 $$

Key Concepts

Calculating Slope

Understanding how to calculate the slope is crucial when dealing with linear equations. The slope is a measure of how steep a line is and the direction it goes. When calculating the slope between two points on a coordinate plane, you can use the slope formula, which 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 simple terms, it shows how much the 'y' value (rise) changes for a unit change in the 'x' value (run).

A positive slope means the line is rising from left to right, whereas a negative slope means the line is falling. A slope of zero indicates a horizontal line, while an undefined or infinite slope (caused by a zero denominator) signifies a vertical line. Calculating the slope correctly is essential because it's used to determine the angle and direction of the line on a graph.

Finding y-intercept

The y-intercept is another fundamental concept in algebra, specifically when working with the equation of a line. It denotes the point where the line crosses the y-axis on a graph. To find the y-intercept, we use the formula for a line, \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. Once the slope is known, you can choose any point on the line to solve for \(b\).

To find the y-intercept, simply rearrange the equation to solve for \(b\): \[b = y - mx\]. Use the slope obtained from the slope formula and plug in the coordinates of either point into the equation, and solve for \(b\). The result will give you the exact point on the y-axis where the line intercepts, which is crucial for graphing the line or understanding its equation.

Equation of a Line

The equation of a line represents the relationship between the x and y coordinates of any point on the line. The most common form used is the slope-intercept form which is written as \(y = mx + b\). In this equation, \(m\) stands for the slope of the line, and \(b\) represents the y-intercept, which is the point where the line crosses the y-axis. This form is highly popular because it provides a straightforward method to graph a line when the slope and y-intercept are known.

To graph a line using the slope-intercept form, you start by plotting the y-intercept on the y-axis. From there, you use the slope to determine the direction and steepness of the line. If the slope is positive, the line will slant upwards as you move to the right, and if it's negative, it will slope downwards. The absolute value of the slope indicates the steepness of the line. The steeper the line, the greater the absolute value of the slope.

Slope Formula

The slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) is fundamental for understanding linear relationships in algebra. It's based on the concept of 'rise over run', which means for every horizontal movement along the x-axis ('run'), there is a corresponding vertical change along the y-axis ('rise'). The formula allows you to calculate the exact slope of a straight line when you know the coordinates of any two points on the line.

Example of Calculating Slope

Let’s say we have two points \((x_1, y_1)\) and \((x_2, y_2)\), such as (4, 4) and (5, 1). By plugging these values into the formula, \(m = \frac{1 - 4}{5 - 4} = -3\), we find that the slope is -3. This indicates that for every one unit the line moves to the right (positive direction along the x-axis), it drops by three units (negative direction along the y-axis). Grasping the usage of the slope formula is a crucial step in various applications in math and the physical sciences.