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Magazine Chapter 3: Problem 74
Graph each equation. $$ -5 x=10+5 y $$
Short Answer
Expert verified
Graph the line by plotting \( (0, -2) \) and \( (1, -3) \), then draw the line through these points.
Step by step solution
01
Rearrange the Equation
We start by rearranging the given equation to the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Our given equation is \( -5x = 10 + 5y \). First, we isolate \( y \) by subtracting 10 from both sides: \( -5x - 10 = 5y \). Then, divide every term by 5 to solve for \( y \): \( y = -x - 2 \).
02
Identify the Slope and Y-Intercept
Now that we have the equation in slope-intercept form: \( y = -x - 2 \), we can identify the slope \( m = -1 \) and the y-intercept \( b = -2 \). This will help us graph the equation.
03
Plot the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. Our y-intercept is -2, so we plot the point \( (0, -2) \) on the graph.
04
Use the Slope to Find Another Point
The slope \( m = -1 \) means that for every 1 unit we move to the right (positive x-direction), we move 1 unit down (negative y-direction). Starting from \( (0, -2) \), move right 1 unit to \( x = 1 \) and down 1 unit to \( y = -3 \). This gives us the point \( (1, -3) \). Plot this point on the graph.
05
Draw the Line
Using a ruler, draw a straight line through the points \( (0, -2) \) and \( (1, -3) \). Extend the line across the graph, ensuring it passes through these points, which represent the solutions to the equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
When dealing with linear equations, the slope-intercept form is your straightforward ticket to quickly graphing a line. This form is expressed as \( y = mx + b \). Here, \( m \) represents the slope of the line, indicating how steep it is, while \( b \) tells us where the line intersects the y-axis.
To convert an equation into this form, you want to isolate \( y \) on one side of the equation. Take the equation from the exercise: \( -5x = 10 + 5y \). The goal is to manipulate this equation so it looks like \( y = mx + b \).
With \( y \) neatly isolated, you've turned your equation into the slope-intercept form, making it easy to identify the line's slope and y-intercept.
To convert an equation into this form, you want to isolate \( y \) on one side of the equation. Take the equation from the exercise: \( -5x = 10 + 5y \). The goal is to manipulate this equation so it looks like \( y = mx + b \).
- First, get rid of any terms on the same side as \( y \) that aren't part of \( y \) itself. In this case, subtracting 10 gives \( -5x - 10 = 5y \).
- Then, divide everything by the coefficient of \( y \), which is 5 in this case, to get \( y \) by itself: \( y = -x - 2 \).
With \( y \) neatly isolated, you've turned your equation into the slope-intercept form, making it easy to identify the line's slope and y-intercept.
Slope Calculation
The slope of a line, noted as \( m \) in the slope-intercept form \( y = mx + b \), shows how the line tilts. It is a measure of rise over run.
For every unit the line moves horizontally, the slope tells you how much the line moves vertically.
For every unit the line moves horizontally, the slope tells you how much the line moves vertically.
- If \( m = 1 \), the line rises one unit for every unit it runs.
- If \( m = -1 \), like in our exercise equation \( y = -x - 2 \), the line falls one unit for every unit it runs.
- A positive slope means the line goes upwards from left to right, while a negative slope means it goes downwards.
Y-Intercept Identification
The y-intercept is another crucial detail for graphing a line in the slope-intercept form of \( y = mx + b \). The y-intercept, noted as \( b \), tells you where the line crosses the y-axis, directly giving one of the points needed to draw the line.
In our slope-intercept equation \( y = -x - 2 \), \( b = -2 \). Thus, the line intercepts the y-axis at the point \( (0, -2) \).
In our slope-intercept equation \( y = -x - 2 \), \( b = -2 \). Thus, the line intercepts the y-axis at the point \( (0, -2) \).
- This means when \( x = 0 \), \( y \) hits \(-2\).
- Plotting the point \( (0, -2) \) on the graph gives you a solid foundation for drawing the line.
