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Chapter 3: Problem 45

Express in slope-intercept form and identify the slope and y-intercept. $$ 12 x-4 y=8 $$

Short Answer

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

Step by step solution

01

Write the Original Equation

Start with the given equation: \[ 12x - 4y = 8 \]
02

Solve for y

We want to express the equation in slope-intercept form, which is \( y = mx + b \). First, isolate \( y \) by moving the term with \( x \) to the other side. Subtract \( 12x \) from both sides: \[ -4y = -12x + 8 \]
03

Simplify to Get y by Itself

Divide each term by \(-4\) to solve for \( y \): \[ y = \frac{12}{4}x - \frac{8}{4} \] Simplify the fractions: \[ y = 3x - 2 \]
04

Identify the Slope and y-Intercept

In the slope-intercept form \( y = mx + b \), the slope \( m \) is the coefficient of \( x \) and \( b \) is the y-intercept. Here, \( m = 3 \) and \( b = -2 \).

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Key Concepts

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

Linear Equations
A linear equation is a type of mathematical statement that creates a straight line when graphed on a coordinate plane. Generally, it looks like this:
  • It can be written in the form \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants.
  • The graph of a linear equation will always be a straight line.
  • It describes a constant relationship between two variables, typically \( x \) and \( y \).
Knowing this, we can manipulate the equation to find specific forms, like the slope-intercept form, to learn more about its characteristics. Understanding linear equations is crucial because they model many real-world situations and help us predict and understand trends.
Solving for y
Solving for \( y \) is a common task, especially when converting an equation into the slope-intercept form. This type of transformation involves isolating \( y \) on one side of the equation.
Here's how you can approach it:
  • Start by rearranging the equation to get all terms containing \( y \) on one side. This often involves adding or subtracting terms.
  • Once the \( y \)-terms are isolated, divide them by the coefficient of \( y \) to solve for a single \( y \).
  • This will result in \( y = \, \text{function of } x \), which is the slope-intercept form.
In the exercise,
  1. We had \( 12x - 4y = 8 \).
  2. Subtract 12x from both sides to get \( -4y = -12x + 8 \).
  3. Then, divide each part by \(-4\) to solve for \( y \) giving us \( y = 3x - 2 \).
By solving for \( y \), we can directly see the relationship between \( x \) and \( y \), making it easier to graph and understand.
Slope
The slope is a key aspect of a linear equation in the slope-intercept form, usually represented as "\( m \)" in \( y = mx + b \). It indicates how steep a line is, which means:
  • The slope describes how much \( y \) changes when you change \( x \) by one unit.
  • A positive slope means the line rises as \( x \) increases, while a negative slope means it falls.
  • If the slope is zero, the line is horizontal, indicating no change in \( y \) regardless of \( x \).
In our problem, after solving for \( y \), the slope is found to be \( 3 \). This tells us that for each increase of 1 in \( x \), \( y \) increases by 3. Knowing the slope helps in predicting how changes in one variable affect the other.
y-Intercept
The y-intercept, represented as "\( b \)" in the slope-intercept equation \( y = mx + b \), is the point where the line crosses the y-axis. It provides essential information:
  • The y-intercept is the value of \( y \) when \( x \) is zero, giving us a starting point on the graph.
  • In the equation \( y = 3x - 2 \), the y-intercept is \(-2\).
  • This intercept tells us that when there are no changes in \( x \), the value of \( y \) is \(-2\).
Understanding the y-intercept is important as it helps in quickly sketching the graph of the equation. It marks the exact location where the line will cross the y-axis, giving a clearer idea of how the line sits on the coordinate plane.

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