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Chapter 15: Problem 29

Find the slope and the \(y\) -intercept of each equation. \(y=2 x-5\)

Short Answer

Expert verified
The slope is 2 and the y-intercept is -5.

Step by step solution

01

Identify the Slope-Intercept Form

The given equation is in the slope-intercept form which is generally written as \[y = mx + b\]. Here, \(m\) represents the slope and \(b\) represents the y-intercept.
02

Locate the Slope

In the equation \(y = 2x - 5\), compare it to the slope-intercept form. The coefficient of \(x\) is \(2\). Therefore, the slope \(m\) is \(2\).
03

Locate the Y-Intercept

Identify the constant term in the equation. In \(y = 2x - 5\), the constant term is \(-5\). Therefore, the \(y\)-intercept \(b\) is \(-5\).

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Key Concepts

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

slope
The slope of a line measures how steep it is. Imagine a hill; the slope tells you how quickly you go up or down as you move from left to right. It can be a positive or negative number. The formula for slope (often represented by the letter 'm') in the slope-intercept form is straightforward:
The slope tells you the ratio of the change in y (the vertical change) to the change in x (the horizontal change).
To find the slope in a linear equation like the one given:
Step 1: Write the equation in the form \(y = mx + b\).
Step 2: Locate the coefficient of \(x\). This is your slope.
Let's take the example from the exercise:
  • The equation is \(y = 2x - 5\).
  • Here, the coefficient of \(x\) is 2.
  • So, the slope \(m\) = 2.
The slope helps you understand how the line behaves. For instance, a slope of 2 means that for every unit you move to the right, the line goes up by 2 units.
y-intercept
The y-intercept is the point where the line crosses the y-axis. It's the value of y when x is 0.
In the slope-intercept form \(y = mx + b\), the y-intercept is represented by the 'b' term.
Here鈥檚 how to find it:
Step 1: Make sure the equation is in slope-intercept form: \(y = mx + b\).
Step 2: Identify the constant term (the part without \(x\)):
  • In the example \(y = 2x - 5\), the constant term is \(-5\).
  • Therefore, the y-intercept \(b\) = -5.
This means the graph of the line will cross the y-axis at -5. The y-intercept offers a starting point for graphing the line. Just plot this point on the y-axis, and use the slope to determine the direction and steepness of the line.
linear equations
Linear equations are equations of straight lines. They鈥檙e called 'linear' because the graphed equation produces a line.
The general form of a linear equation is \(y = mx + b\), where:
  • 'y' is the dependent variable
  • 'x' is the independent variable
  • 'm' is the slope
  • 'b' is the y-intercept


To understand linear equations better: Think of 'x' as an input and 'y' as an output. For any given x, the equation tells you what y will be. The slope-intercept form is particularly useful because:
  • You can easily find the slope, which tells you how steep the line is.
  • You can quickly identify the y-intercept, which tells you where the line crosses the y-axis.
To sum up, linear equations are simple yet powerful tools in algebra. They describe a direct relationship between two variables, with a constant rate of change represented by the slope, and a starting point represented by the y-intercept.

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