Problem 80 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. $$ (2,1), \quad(6,1) $$

Step-by-Step Solution

Answer

Question: Find the equation of a line in slope-intercept form, given two points on the line: (2,1) and (6,1). Answer: The equation of the line in slope-intercept form is y = 1.

Step 1: Find the slope

Given the points \((2,1)\) and \((6,1)\), we can find the slope (m) using the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Plugging in the coordinates: $$ m = \frac{1 - 1}{6 - 2} $$ Simplifying: $$ m = \frac{0}{4} $$ Therefore, the slope m is: $$ m = 0 $$

Step 2: Find the y-intercept

Now that we have the slope, we can use the slope-intercept form of a linear equation to find the y-intercept (b). The slope-intercept equation is: $$ y = mx + b $$ Plugging in one of the given points, say (2,1), and the slope, we get: $$ 1 = 0(2) + b $$ Simplifying the equation: $$ 1 = b $$ Thus, the y-intercept is 1.

Step 3: Write the equation of the line in slope-intercept form

Now that we have both the slope (m) and the y-intercept (b), we can plug them into the slope-intercept equation: $$ y = mx + b $$ Substitute m=0 and b=1, we get: $$ y = 0x + 1 $$ As a final step, simplify the equation: $$ y = 1 $$ So the equation of the line in slope-intercept form is: $$ y = 1 $$

Key Concepts

Linear Equations

Linear equations represent straight lines on a graph. In mathematics, these equations express a relationship where each input value (x) has exactly one output value (y).
For a standard understanding, a linear equation is usually arranged in the form: \[ y = mx + b \] where:

• \( y \) is the dependent variable.
• \( m \) represents the slope of the line.
• \( x \) is the independent variable.
• \( b \) is the y-intercept.

Linear equations are simple yet very fundamental in mathematics. They help depict situations with constant rates of change, making them applicable in numerous fields such as physics, economics, and even everyday life scenarios like calculating expenses or distances.
Understanding linear equations means being able to interpret how the slope and y-intercept define the line's direction and position.

y-intercept

The y-intercept is a critical component of the linear equation. It refers to the point where the line crosses the y-axis.
In the slope-intercept form, \( y = mx + b \), \( b \) represents the y-intercept. Essentially, it shows what the value of \( y \) is when \( x \) is zero. This intersection helps in graphing the line and understanding its position on the graph.Some key characteristics of the y-intercept:

• It is always in the form (0, \( b \)).
• The y-intercept is constant. It remains the same no matter where you look along the x-axis.
• In our example, using points \((2,1)\) and \((6,1)\), the y-intercept is: \[ b = 1 \]This means the line crosses the y-axis at \((0,1)\).

The y-intercept allows you to pinpoint the starting position of a line before any change occurrs from the slope.

Finding Slope

Finding the slope of a line is a crucial step in understanding linear equations. The slope indicates the line's steepness and direction, showing how much \( y \) changes for a given change in \( x \).
The slope \( m \) can be calculated given two points on the line, using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula finds the ratio of the vertical change (rise) to the horizontal change (run) between two points:

• When \( m \) is positive, the line inclines upward.
• When \( m \) is negative, the line declines downward.
• When \( m \) is zero, it means the line is horizontal, having no incline or decline, which was the case with points \((2,1)\) and \((6,1)\), showing a slope of 0.

Understanding how to find slope helps in graphing lines and comprehending how variables interact. Whether estimating trends in data or analyzing movement over time, calculating slope is an essential skill.