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Magazine Chapter 3: Problem 34
Graph equation by hand. \(x+y=-5\)
Short Answer
Expert verified
Rewrite as y = -x - 5, plot (0, -5) and (1, -6), then draw the line through these points.
Step by step solution
01
- Rewrite the Equation
Rewrite the given equation in slope-intercept form (\text{y = mx + c}), where m is the slope and c is the y-intercept. Start with the equation:\[ x + y = -5 \]Subtract x from both sides to solve for y:\[ y = -x - 5 \]
02
- Identify the Slope and Y-Intercept
From the equation in slope-intercept form \( y = -x - 5 \), identify the slope (m) and the y-intercept (c). Here:\[ m = -1 \]\[ c = -5 \]
03
- Plot the Y-Intercept
On a coordinate plane, locate the y-intercept, which is the point where the line crosses the y-axis. In this case, plot the point (0, -5).
04
- Use the Slope to Find Another Point
From the y-intercept point (0, -5), use the slope of -1 to find another point on the line. Since the slope is -1, it means for every 1 unit increase in x, y decreases by 1 unit. Move 1 unit to the right along the x-axis to x = 1, then move 1 unit down along the y-axis to y = -6. Plot the point (1, -6).
05
- Draw the Line
Draw a straight line through the two points (0, -5) and (1, -6) on the graph. Extend the line in both directions, making sure it runs through both plotted points accurately.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
slope-intercept form
To graph a linear equation, it's helpful to start by rewriting it in the slope-intercept form. This form is written as \text{y = mx + c}. Here, m represents the slope, and c represents the y-intercept. For example, given the equation x + y = -5, we can rearrange it to be in the slope-intercept form:
1. Start with the equation:
\[ x + y = -5 \]
2. Subtract x from both sides:
\[ y = -x - 5 \]
Now, it's in the slope-intercept form, \text{y = mx + c}, with m = -1 and c = -5.
1. Start with the equation:
\[ x + y = -5 \]
2. Subtract x from both sides:
\[ y = -x - 5 \]
Now, it's in the slope-intercept form, \text{y = mx + c}, with m = -1 and c = -5.
plotting points
Plotting points on a graph is essential for visualizing linear equations. Once you have an equation in slope-intercept form, you can easily plot points:
(0, -5). To find another point, use the slope m = -1. Move 1 unit to the right on the x-axis, and 1 unit down on the y-axis, locating the point (1, -6). By plotting these points, you can draw the line representing the equation.
- First, plot the y-intercept. This is where the line crosses the y-axis.
- Next, use the slope to determine another point on the line.
(0, -5). To find another point, use the slope m = -1. Move 1 unit to the right on the x-axis, and 1 unit down on the y-axis, locating the point (1, -6). By plotting these points, you can draw the line representing the equation.
coordinate plane
A coordinate plane allows us to visually represent equations. It consists of two perpendicular lines: the vertical y-axis and the horizontal x-axis. Each point on the plane is defined by an (x, y) coordinate. The first number represents its position on the x-axis, and the second on the y-axis. To graph a line equation, you need:
equation y = -x - 5, we first locate the y-intercept at (0, -5) on the coordinate plane. Then using the slope, we find another point, such as (1, -6). By drawing a line through these points, we effectively represent the equation on the coordinate plane.
- A coordinate plane setup
- An equation in slope-intercept form
- The points plotted accurately according to the x and y values
equation y = -x - 5, we first locate the y-intercept at (0, -5) on the coordinate plane. Then using the slope, we find another point, such as (1, -6). By drawing a line through these points, we effectively represent the equation on the coordinate plane.
slope and y-intercept
Understanding the slope and y-intercept is key to graphing linear equations. The slope m describes how steep the line is and the direction it goes鈥攚hether it rises or falls as you move from left to right. A positive slope means the line rises, and a negative slope means it falls. The y-intercept c is straightforward; it tells you where the line crosses the y-axis.
For the equation y = -x - 5:
For the equation y = -x - 5:
- The slope m is -1, signifying the line falls at a 45掳 angle downward.
- The y-intercept c is -5, marking the point (0, -5) on the y-axis.
