Problem 33 Write the equation in slope-inte... [FREE SOLUTION]

Chapter 0: Problem 33

Write the equation in slope-intercept form. Identify the slope and the $y$ -intercept. $$x+3 y=6$$

Short Answer

Expert verified

The slope-intercept form is $y = -\frac{1}{3}x + 2$; slope = $-1/3$, $y$-intercept = 2.

Step by step solution

Identify the standard form

The given equation is in standard form, which is written as $Ax + By = C$. In this exercise, $A = 1$, $B = 3$, and $C = 6$.

Solve for $y$

To convert the equation to the slope-intercept form $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept, isolate $y$ on one side of the equation. Start by subtracting $x$ from both sides: \[ 3y = -x + 6 \]

Divide by the coefficient of $y$

Divide every term by 3 to solve for $y$: \[ y = -\frac{1}{3}x + 2 \] This is the slope-intercept form of the equation.

Identify the slope and $y$-intercept

From the equation $y = -\frac{1}{3}x + 2$, we identify the slope $m$ as $-\frac{1}{3}$ and the $y$-intercept $b$ as 2.

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Key Concepts

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

Standard Form

The standard form of a linear equation is one of the most straightforward ways to represent a straight line. It is expressed as $Ax + By = C$, where $A$, $B$, and $C$ are integers, and $A$ should typically be a positive number. This form of writing reminds us that linear equations can be simplified into a neat and organized format. In a classroom context, it is often the starting point for various conversions, paving the way for easier comparisons between multiple equations.

Key aspects of standard form include:

Both $A$ and $B$ typically are not fractions or decimals.
This form is useful for determining the x- and y-intercepts of the graph directly, which are key points where the line crosses the axes.

Keep in mind that when given a linear equation in standard form, you usually have the freedom to convert it into other forms to suit the problem at hand.

Equation Conversion

Converting equations from standard form to slope-intercept form is a common task in algebra. The slope-intercept form is represented by $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. This form tells you instantly how the line behaves and where it crosses the y-axis, making it great for graphing.

To convert from standard to slope-intercept form, the goal is to solve for $y$. Here鈥檚 a simple process to follow:

Start by isolating the $y$ term by moving $x$ and constant terms to the opposite side of the equation. For example, subtract $x$ from both sides if needed.
Divide every term by the coefficient in front of $y$ to solve for $y$.

This conversion is valuable because it simplifies graphing and analyzing the line's properties, like slope and y-intercept, in numerical and graphical forms.

Identifying Slope

The slope of a line indicates how steep it is and the direction in which it inclines. In the equation format $y = mx + b$, the letter $m$ represents the slope. The slope can be a fraction, whole number, or decimal, and it tells us how much $y$ changes for a change in $x$.

Here is how you can identify and interpret the slope:

The slope of $-\frac{1}{3}$ means that for every 3 units you move horizontally to the right on the x-axis, the line moves down 1 unit vertically.
A positive slope means the line inclines upward as you move from left to right, while a negative slope means it declines.

This measure is essential for understanding the rate of change between variables represented in the equation.

Identifying y-intercept

The y-intercept is the point where the line crosses the y-axis on a graph. In the slope-intercept form, $y = mx + b$, this is represented by $b$. The y-intercept is a critical value because it marks where the dependent variable $y$ is located when $x = 0$.

Understanding the y-intercept involves:

Recognizing that it provides a starting point for constructing the line graphically.
The y-intercept value is 2 in our equation, meaning the line crosses the y-axis at the point (0, 2).

The y-intercept supplies valuable information about the initial condition or baseline of the linear relationship being modeled.

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91影视

Write the equation in slope-intercept form. Identify the slope and the \(y\) -intercept. $$x+3 y=6$$

Short Answer

Step by step solution

Identify the standard form

Solve for \(y\)

Divide by the coefficient of \(y\)

Identify the slope and \(y\)-intercept

Key Concepts

Standard Form

Equation Conversion

Identifying Slope

Identifying y-intercept

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