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Chapter 1: Problem 12

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

Short Answer

Expert verified
Slope: \(-3\), y-intercept: \(2\).

Step by step solution

01

Identify the slope-intercept form

The given equation is already in the slope-intercept form, which is written as: $$y = mx + b$$.Here, $$m$$ is the slope and $$b$$ is the y-intercept.
02

Determine the slope

In the equation $$y = -3x + 2$$, the coefficient of $$x$$ is $$-3$$. Therefore, the slope, $$m$$, is \(-3\).
03

Determine the y-intercept

In the equation $$y = -3x + 2$$, the constant term is $$2$$. Therefore, the y-intercept, $$b$$, is \(2\).
04

Interpret the slope and y-intercept

The slope (\(-3\)) indicates that for every unit increase in $$x$$, the value of $$y$$ decreases by 3 units. The y-intercept (\(2\)) is the point where the line intersects the y-axis.

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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 an equation that represents a straight line when graphed on a coordinate plane. The general form of a linear equation is given by
\[ y = mx + b \]
where:
  • y represents the dependent variable
  • x represents the independent variable
  • m is the slope of the line
  • b is the y-intercept
Linear equations can be used to model and solve real-world problems where there is a constant rate of change between variables. These equations are simple yet powerful tools in mathematics and are foundational for more advanced concepts.
slope
The slope of a line, denoted by m, measures the steepness and direction of the line. In the slope-intercept form equation, \( y = mx + b \), the slope is the coefficient of the $$x$$ term. For example, in the equation \( y = -3x + 2 \), the slope \( m \) is \(-3\).

The slope tells us how much \( y \) changes for a unit change in \( x \). If the slope is positive, the line rises as \( x \) increases. If the slope is negative, the line falls as \( x \) increases.
Specifically:
  • If m = -3, this means that for every 1 unit increase in \( x \), the value of \( y \) decreases by \( 3 \) units.
  • m = 0 leads to a horizontal line, indicating no change in \( y \) as \( x \) changes.
  • An undefined slope results from a vertical line where the change in \( x \) is zero.
y-intercept
The y-intercept is the point where a line crosses the y-axis of a graph. It is represented by the variable \( b \) in the slope-intercept form \( y = mx + b \).

For the equation \( y = -3x + 2 \), the \( b \) value is \( 2 \). This means that the line intersects the y-axis at the point \( (0, 2) \).

The y-intercept provides a starting point for graphing a linear equation. To find it:
  • Set \( x \) to zero in the equation.
  • Solve for \( y \).
For example, in our equation, setting \( x = 0 \) gives \( y = 2 \).
This indicates the location on the graph where the line meets the y-axis.
graphing
Graphing a linear equation involves plotting points on a coordinate plane and drawing a line through them. Here's a step-by-step guide using the equation \( y = -3x + 2 \):

  1. Identify the y-intercept \( (0, 2) \) and plot it on the graph.
  2. Use the slope to find another point. Since the slope \( m \) is \( -3 \), move 1 unit to the right (positive direction of \( x \)) and 3 units down (negative direction of \( y \)). This gives the point \( (1, -1) \).
  3. Plot the second point \( (1, -1) \) on the graph.
  4. Draw a straight line through the points. Extend the line across the graph.

Graphing helps visualize the relationship between \( x \) and \( y \). By interpreting this graph, we can easily see how changes in \( x \) affect \( y \) and better understand the linear relationship.

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