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Chapter 1: Problem 27

For each equation, find the slope \(m\) and \(y\) intercept \((0, b)\) (when they exist) and draw the graph. \(x-y=0\)

Short Answer

Expert verified
The slope \(m\) is 1 and the y-intercept \(b\) is 0.

Step by step solution

01

Rewrite the Equation

Rearrange the given equation \(x - y = 0\) into the standard form of a linear equation, which is \(y = mx + b\). Start by isolating \(y\) on one side of the equation.\[\begin{align*} x - y & = 0 \ -y & = -x \ y & = x \end{align*}\]Now the equation is in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
02

Identify the Slope and Y-intercept

Once the equation is in the form \(y = mx + b\), identify the slope \(m\) and y-intercept \(b\).\[y = x \implies y = 1x + 0\]Here, the slope \(m\) is 1, and the y-intercept \(b\) is 0, which means the y-intercept point is \((0, 0)\).
03

Draw the Graph

To draw the graph of the equation \(y = x\), plot the y-intercept \((0, 0)\) on a coordinate plane. Then, use the slope \(m = 1\), which means that for every unit increase in \(x\), \(y\) also increases by 1 unit. From the point \((0, 0)\), move up 1 unit and right 1 unit to plot a second point, \((1, 1)\). Connect these points with a straight line extending in both directions.

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Key Concepts

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

Graphing Linear Equations
Graphing linear equations is a fundamental skill in mathematics that helps visualize relationships between two variables. To graph an equation like \( y = x \), begin by understanding the format: a straight line equation typically presented as \( y = mx + b \). Here, the graphing process involves plotting points based on these identified components:
\
    \
  • First, pinpoint the y-intercept on the graph, which is where the line crosses the y-axis.
    • \
  • Next, use the slope, or rate of change, to plot additional points. This involves moving upwards (or downwards) and sideways on the graph from the y-intercept, depending on the slope value.
    • \
  • Finally, draw a straight line through the plotted points to extend the line across the grid.
    • \
Ensuring the line is straight is key, as any curve or bend would suggest a mistake since linear equations should form straight lines on a graph.
Finding Slope
The slope of a line describes its steepness and direction. It's represented as \( m \) in the equation \( y = mx + b \). Finding the slope involves understanding its role:

  • The slope is calculated as the rise over run, or the change in \( y \) values divided by the change in \( x \) values between two points.
  • For the equation \( y = x \), the slope is 1. This implies that for every increase of 1 in \( x \), \( y \) also increases by the same amount, resulting in a line that rises at a 45-degree angle.
  • Positive slopes mean the line ascends as you move right, while negative slopes mean it descends.
The slope is a key component in determining the angle and direction of the line on a graph. Understanding it is essential for predicting how changes in one variable affect another.
Y-Intercept in Linear Equations
The y-intercept is an integral part of linear equations, denoted by \( b \) in \( y = mx + b \). It represents the point where the line crosses the y-axis, serving as a starting point from which the slope extends the line across the graph. To find and interpret the y-intercept:

    \
  • Identify the y-intercept by setting \( x \) to 0 in the equation. This simplifies to \( y = b \).
    • \
  • In the given example \( y = x \), the y-intercept \( b \) is 0, indicating that the line passes directly through the origin (0,0).
    • \
  • The y-intercept provides a reference point for plotting the line, ensuring you start at the correct location on the graph.
    • \
  • It's crucial for understanding how the line interacts with the y-axis, and zero intercept means the line starts directly at the origin.
    • \
The y-intercept is essential for constructing accurate linear graphs and for quickly understanding the initial value of \( y \) when \( x \) is zero.

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