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Chapter 11: Problem 106

In the following exercises, graph each equation. $$ 2 x+y=6 $$

Short Answer

Expert verified
Rewrite the equation as \( y = -2x + 6 \), plot the points (0, 6) and (1, 4), and draw the line.

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 isolating \( y \) in the original equation: \[ 2x + y = 6 \] Subtract \( 2x \) from both sides: \[ y = -2x + 6 \]
02

Identify the slope and y-intercept

From the equation \( y = -2x + 6 \), identify the slope \( m \) as \( -2 \) and the y-intercept \( b \) as \( 6 \).
03

Plot the y-intercept

On a graph, locate the y-intercept (0, 6) and plot this point.
04

Use the slope to find another point

The slope is \( -2 \), which means for every 1 unit increase in \( x \), \( y \) decreases by 2 units. From the y-intercept (0, 6), move 1 unit to the right to \( x = 1 \) and 2 units down to \( y = 4 \). Plot this point (1, 4).
05

Draw the line

Draw a straight line through the points (0, 6) and (1, 4). This line represents the graph of the equation \( y = -2x + 6 \).

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Key Concepts

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

slope-intercept form
The slope-intercept form is a way of writing linear equations. It describes how changes in one variable affect another. The general form is \ y = mx + b \, where \( m \) is the slope and \( b \) is the y-intercept. This representation helps us quickly understand the behavior of linear relationships. By converting an equation to slope-intercept form, we can easily identify the slope and y-intercept, making it simpler to graph the equation. In our example, the equation \ 2x + y = 6 \ was rewritten as \ y = -2x + 6 \.
slope
The slope of a line describes its steepness and direction. It shows how much \( y \) changes for a change in \( x \). Mathematically, it's represented as \( m \) in the slope-intercept form \ y = mx + b \. The value of \( m \) can be positive, negative, or zero. In our example, the slope \( m = -2 \) tells us that for every unit increase in \( x \), \( y \) decreases by 2 units. This negative slope indicates a downward slant from left to right. Understanding the slope helps us predict and plot additional points on the graph.
y-intercept
The y-intercept is the point where the line crosses the y-axis. It gives us a starting point for graphing the equation. In the general form \ y = mx + b \ , the y-intercept is represented by \( b \). To find this point, set \( x \) to zero and solve for \( y \). For our equation \ y = -2x + 6 \, the y-intercept \( b = 6 \) means the line crosses the y-axis at the point (0, 6). Plotting this point on the graph provides a reference to start plotting the line.
plotting points
Plotting points involves marking points on a graph to represent the equation visually. We start with the y-intercept, as it's a known point. For example, the point (0, 6) from our equation. Next, we use the slope to find additional points. The slope \( m = -2 \) means for every 1 unit we move right (positive direction for x), we move down 2 units (negative direction for y). From (0, 6), moving 1 unit right and 2 units down, we get another point (1, 4). By connecting these points with a straight line, we graph the equation \ y = -2x + 6 \.

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Most popular questions from this chapter

In the following exercises, graph the vertical and horizontal lines. $$ x=-2 $$

In the following exercises, graph the line given a point and the slope. $$ (2,-2) ; m=\frac{5}{2} $$

Do you prefer to graph the equation \(4 x+y=-4\) by plotting points or intercepts? Why?

In the following exercises, graph each pair of equations in the same rectangular coordinate system. $$ y=5 x \text { and } y=5 $$

What does the sign of the slope tell you about a line?
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