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Chapter 3: Problem 4

Determine the slope and \(y\) -intercept. $$ -x+y=0 $$

Short Answer

Expert verified
The slope is 1, and the y-intercept is 0.

Step by step solution

01

Rearranging Equation

The given equation is \(-x + y = 0\). To find the slope and \(y\)-intercept, first rearrange the equation into the slope-intercept form, \(y = mx + b\). So, add \(x\) to both sides to get \(y = x\).
02

Identifying the Slope

In the equation \(y = x\), compare it to the slope-intercept form \(y = mx + b\). Here, \(m\) is the slope. Therefore, the slope \(m\) is equal to 1.
03

Identifying the Y-Intercept

Next, in the equation \(y = x\), compare again with the slope-intercept form \(y = mx + b\). Here, \(b\) represents the \(y\)-intercept. Therefore, the \(y\)-intercept \(b\) is equal to 0.

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Key Concepts

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

Understanding the Slope
The slope is a fundamental concept in algebra that indicates how steep a line is on a graph. It is represented by the letter \(m\). The formula for finding the slope from two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
In our equation \(y = x\), which we derived from the original form, the slope \(m\) is 1. This means for each increase of 1 in the \(x\)-value, the \(y\)-value also increases by 1.
This can be visualized as a line rising at a 45-degree angle on the graph.
Understanding the slope helps in predicting how changes in \(x\) affect \(y\), making it an essential tool in graph analysis.
Grasping the Y-Intercept
The y-intercept is the point where a line crosses the y-axis. It's represented by the letter \(b\) in the equation \(y = mx + b\). This value shows the \(y\)-coordinate when \(x\) is zero.
  • In our rearranged equation \(y = x\), the y-intercept \(b\) is 0.
This means the line passes through the origin point \((0, 0)\) on the graph.
Identifying the y-intercept is crucial for graphing because it gives you a starting point.
Once you know where the line starts, it's easier to use the slope to determine its direction.
Algebra and Slope-Intercept Form
Algebra involves using variables, symbols, and numbers to express relationships. A common form used in algebra for linear equations is the slope-intercept form: \(y = mx + b\). This form clearly presents the slope \(m\) and y-intercept \(b\), making it easy to graph linear equations.
Converting an equation to this form simplifies identifying key attributes like:
  • Slope \(m\)
  • Y-intercept \(b\)
For example, with \(-x + y = 0\), adding \(x\) to both sides results in \(y = x\).
This adjustment aligns the equation with slope-intercept form and highlights the slope as 1 and y-intercept as 0.
Understanding this form is foundational in algebra, as it aids in efficiently solving and graphing linear equations.

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Most popular questions from this chapter

Determine the slope, given two points. (0,-5) and (-6,3)

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