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Chapter 0: Problem 22

Find the slope and \(y\) -intercept. $$2 x-y+3=0$$

Short Answer

Expert verified
The slope is 2 and the \(y\)-intercept is 3.

Step by step solution

01

Rewrite the equation in slope-intercept form

The slope-intercept form of a linear equation is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. Start by rewriting the given equation: First, add \(y\) to both sides to isolate terms involving \(y\) on one side: \[2x + 3 = y\]Then, rewrite it as: \[y = 2x + 3\]
02

Identify the slope

In the equation \(y = 2x + 3\), identify the coefficient of \(x\). This coefficient is the slope \(m\). Thus, the slope \(m = 2\).
03

Identify the \(y\)-intercept

In the equation \(y = 2x + 3\), identify the constant term. This constant term is the \(y\)-intercept \(b\). Thus, the \(y\)-intercept \(b = 3\).

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Key Concepts

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

linear equations
Linear equations are mathematical expressions that describe a straight line when plotted on a graph.
They generally have the form:
ewline \[Ax + By + C = 0\] where:
- A, B, and C are constants.
- x and y are variables.
Linear equations showcase a unique relationship between x and y.
The solutions to linear equations are pairs of x and y values that make the equation true.
For example, the original exercise provided the linear equation: \[2x - y + 3 = 0\]which will form a straight line when plotted.
slope-intercept form
The slope-intercept form of a linear equation is one of the easiest ways to write linear equations.
It is written as:
ewlinell \[y = mx + b\] where:
- y = dependent variable.
- m = slope of the line (rate at which y changes with respect to x).
- x = independent variable.
- b = y-intercept (value of y when x is 0).
To convert a linear equation into slope-intercept form, isolate y on one side.
From the example, the equation \[2x - y + 3 = 0\] was rearranged to \[y = 2x + 3\]
graphing linear equations
Graphing linear equations is a straightforward process.
Here's what you need:
1. Start with the slope-intercept form of the equation (\(y = mx + b \)).
2. Identify the slope (m). This tells you how steep the line is, and the direction it goes (up or down as you move along the x-axis).
3. Determine the y-intercept (b). This tells you where the line crosses the y-axis.
ewlinell In the provided example:
- The slope (m) is 2. This means for every 1 unit increase in x, y increases by 2 units.
- The y-intercept (b) is 3. This means the line crosses the y-axis at (0, 3).
Now, you can plot these points and draw your line to represent the equation \[y = 2x + 3\].

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