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Chapter 3: Problem 33

Find an equation of the line that passes through the two given points. Write the equation in slope-intercept form, if possible. See Example 2. Passes through \((5,5)\) and \((7,5)\)

Short Answer

Expert verified
The equation of the line is \( y = 5 \).

Step by step solution

01

Identify the Slope Formula

The slope formula is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points the line passes through.
02

Substitute Point Values

Substitute the coordinates of the given points into the slope formula: \((x_1, y_1) = (5, 5)\) and \((x_2, y_2) = (7, 5)\). Substitute these values into the formula: \[ m = \frac{5 - 5}{7 - 5} = \frac{0}{2} = 0 \].
03

Write the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by \( y = mx + b \). Since the slope \( m = 0 \), the equation becomes \( y = 0 \cdot x + b \), simplifying to \( y = b \).
04

Find the y-intercept

Since the line passes through \((5, 5)\), substitute \( x = 5 \) and \( y = 5 \) into \( y = b \) to find \( b \). This gives \( 5 = b \).
05

Write the Final Equation

Substitute \( b = 5 \) back into the equation \( y = b \). Therefore, the equation of the line is \( y = 5 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation of a Line
When you hear someone talking about the "equation of a line", they are referring to a mathematical expression that describes a line's path on a plane. It shows the relationship between the x-coordinates and y-coordinates of any point on that line. A common form of a line's equation is the slope-intercept form, which is written as \( y = mx + b \).
Here:
  • \( y \) represents the dependent variable, usually plotted on the vertical axis.
  • \( x \) is the independent variable, typically on the horizontal axis.
  • \( m \) denotes the slope of the line, representing how steep the line is or the rate of change.
  • \( b \) is the y-intercept, indicating where the line crosses the y-axis.
In our problem, since the slope \( m \) is zero, the equation simplifies to \( y = b \). Specifically, it means any point on this line has the same y-coordinate, forming a horizontal line.
Slope Formula
The slope formula is a straightforward method for calculating the steepness or incline of a line. It's given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula essentially tells you how much y changes for a certain change in x as you move along the line. Here is what you need to know about its parts:
  • \( m \) represents the slope.
  • \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two distinct points on the line.
  • The numerator \( y_2 - y_1 \) shows the vertical change, or rise.
  • The denominator \( x_2 - x_1 \) reflects the horizontal change, or run.
In our exercise, after substituting the points \( (5, 5) \) and \( (7, 5) \) into this formula, we obtain \( m = 0 \). This indicates no vertical change, meaning the line is perfectly horizontal.
Linear Equations
Linear equations are a pivotal concept in algebra where the relationship between variables is expressed as a straight line when graphed. These equations are first-degree, meaning the highest power of the variable (like x or y) is one. The most familiar form of a linear equation is the slope-intercept form \( y = mx + b \). Here are key features of linear equations:
  • They produce straight lines when plotted on a graph.
  • Each solution to the equation corresponds to a point on the graph.
  • They can model real-life situations with constant rates of change.
  • The coefficients in the equation affect the line's slope and position on the graph.
In simpler terms, a linear equation defines a straight path, and knowing its equation helps us predict or find any point along this path. For our specific example, the line represented by \( y = 5 \) is horizontal, demonstrating a unique case of linear equations where only the y-value changes.

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Most popular questions from this chapter

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