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Chapter 1: Problem 32

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. \(\frac{x+1}{2}+\frac{y+1}{2}=1\)

Short Answer

Expert verified
Slope \( m = -1 \); y-intercept \( (0, 0) \).

Step by step solution

01

Simplify the Equation

Start by eliminating the fractions in the equation \( \frac{x+1}{2} + \frac{y+1}{2} = 1 \) by multiplying the entire equation by 2. This gives us the simplified equation: \( (x+1) + (y+1) = 2 \).
02

Solve for y in terms of x

Rearrange and simplify the equation \( x + y + 2 = 2 \) to isolate \( y \). Subtract 2 from both sides to get \( x + y = 0 \). Now subtract \( x \) from both sides to get \( y = -x \).
03

Identify the Slope and y-intercept

The equation \( y = -x \) is now in slope-intercept form, \( y = mx + b \), where \( m = -1 \) and \( b = 0 \). Thus, the slope \( m \) is -1, and the y-intercept is \( (0, 0) \).
04

Draw the Graph

To draw the graph, plot the y-intercept at the origin (0, 0) on a coordinate plane. From this point, use the slope \(-1\) to determine another point: for a 1-unit increase in \( x \), \( y \) decreases by 1 unit. Plot additional points like (1, -1) and (-1, 1), and draw a straight line through these points to extend it infinitely in both directions.

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Key Concepts

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

Graphing Linear Equations
Graphing linear equations can be a straightforward and visual way to understand mathematical relationships. Linear equations are those which can be represented as a straight line, and every equation can be expressed in different forms. One of the most common forms used for graphing is the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
To graph a linear equation:
  • First, identify the y-intercept from the equation, which provides a starting point on the y-axis (when \( x=0 \)).
  • Next, determine the slope \( m \); it tells you how steep the line will be and in which direction it will go.
  • Use the slope to find another point starting from the y-intercept. Repeat this step to plot multiple points along the line.
  • Finally, draw a line through the points to extend the line in both directions on the graph.
By following these steps, you can easily graph any linear equation and understand the relationship between the variables involved.
Slope of a Line
The slope of a line is a measure of how steep the line is and indicates the direction of the line. In mathematical terms, the slope is represented by \( m \) in the slope-intercept form of a line, \( y = mx + b \). It is calculated as the change in the y-value divided by the change in the x-value between two points on a line, often expressed as \( \frac{\Delta y}{\Delta x} \).
The slope tells us:
  • If the slope \( m \) is positive, the line rises to the right.
  • If \( m \) is negative, the line falls to the right.
  • If \( m \) is zero, the line is horizontal, indicating no change in y with respect to x.
  • If the slope is undefined, the line is vertical, indicating an infinite change in y with zero change in x.
Understanding slope helps in predicting how one variable will change with respect to another, which is essential in different fields such as physics, economics, and everyday problem solving.
Y-Intercept Calculation
The y-intercept of a line is where the line crosses the y-axis of a graph. In the slope-intercept form of a linear equation \( y = mx + b \), the y-intercept is represented by \( b \). This is the point \( (0, b) \) on the graph and is crucial in graphing linear equations. It indicates the value of \( y \) when \( x \) is zero.
To calculate the y-intercept:
  • Rearrange the linear equation into the form \( y = mx + b \) if it's not already presented in that structure.
  • Find the constant term \( b \) in the equation; this is the y-intercept value.
  • Verify by substituting \( x = 0 \) into the equation and solving for \( y \); this should yield the y-intercept value.
Knowing how to calculate the y-intercept is essential for plotting linear equations on a graph. It provides a solid starting point for sketching the line and checking the accuracy of your graph.

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