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Magazine Chapter 3: Problem 41
Graph each pair of linear equations on the same set of axes. Discuss how the graphs are similar and how they are different. See Example 6. $$ y=5 x ; y=5 x+4 $$
Short Answer
Expert verified
The lines are parallel with the same slope but different y-intercepts.
Step by step solution
01
Identify the Equations
We have two linear equations to graph: \( y = 5x \) and \( y = 5x + 4 \). The first equation has no y-intercept term other than zero, while the second equation has a y-intercept of 4.
02
Determine the Y-Intercepts
For the equation \( y = 5x \), the y-intercept is \(0\) (when \(x = 0, y = 0\)). For the equation \( y = 5x + 4 \), the y-intercept is \(4\) (when \(x = 0, y = 4\)).
03
Calculate the Slopes
Both equations have the same slope of \(5\). This is determined by the coefficient of \(x\) in both equations.
04
Graph the First Equation
Start at the y-intercept of zero on the y-axis for the equation \( y = 5x \). From this point, use the slope \(5\) to find another point, for example, move up 5 units and go 1 unit to the right. Connect these points with a straight line.
05
Graph the Second Equation
Start at the y-intercept of 4 on the y-axis for the equation \( y = 5x + 4 \). From this point, use the slope \(5\) to find another point, move up 5 units and go 1 unit to the right. Connect these points with a straight line.
06
Compare the Graphs
Both graphs are parallel because they share the same slope (\(5\)). However, they differ in their y-intercepts: \(0\) for \( y = 5x \) and \(4\) for \( y = 5x + 4 \). The graph of \( y = 5x + 4 \) is vertically shifted 4 units up from \( y = 5x \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Equations
Linear equations are mathematical expressions that create straight lines when graphed on a coordinate plane. They usually take the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
These equations are called 'linear' because they represent lines. The key components of linear equations are consistently related to their graphical representations:
A linear equation like \( y = 5x \) simplifies to just having a slope of 5 and a y-intercept of 0. On the other hand, \( y = 5x + 4 \) still keeps the slope at 5 but shifts the y-intercept to 4, showing the versatility of linear equations in describing lines through different starting points.
These equations are called 'linear' because they represent lines. The key components of linear equations are consistently related to their graphical representations:
- The slope \( m \), showing the direction and steepness of the line.
- The y-intercept \( b \), indicating the point where the line crosses the y-axis.
A linear equation like \( y = 5x \) simplifies to just having a slope of 5 and a y-intercept of 0. On the other hand, \( y = 5x + 4 \) still keeps the slope at 5 but shifts the y-intercept to 4, showing the versatility of linear equations in describing lines through different starting points.
Slope
The slope of a line is a measure of its steepness. It is one of the key elements in a linear equation, represented by the letter \( m \) in the standard equation form \( y = mx + b \).
The slope is calculated as the "rise over run," meaning the vertical change divided by the horizontal change between two points on a line.
For the equations \( y = 5x \) and \( y = 5x + 4 \), both have a slope of 5. This indicates that for every unit you move to the right along the x-axis, the line rises 5 units. The identical slopes mean these lines are parallel, as they have the same steepness and direction.
The slope is calculated as the "rise over run," meaning the vertical change divided by the horizontal change between two points on a line.
For the equations \( y = 5x \) and \( y = 5x + 4 \), both have a slope of 5. This indicates that for every unit you move to the right along the x-axis, the line rises 5 units. The identical slopes mean these lines are parallel, as they have the same steepness and direction.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis on a graph. It is represented by the \( b \) value in the equation \( y = mx + b \).
Understanding the y-intercept is crucial, as it gives the starting point of the line when \( x = 0 \). This makes it straightforward to plot the first point of a line.
The difference in y-intercepts between these two lines helps us visualize that \( y = 5x + 4 \) is simply \( y = 5x \) shifted upwards by 4 units.
Understanding the y-intercept is crucial, as it gives the starting point of the line when \( x = 0 \). This makes it straightforward to plot the first point of a line.
- For the equation \( y = 5x \), the y-intercept is 0, meaning the line crosses the origin.
- For \( y = 5x + 4 \), the y-intercept is 4, meaning the line meets the y-axis 4 units above the origin.
The difference in y-intercepts between these two lines helps us visualize that \( y = 5x + 4 \) is simply \( y = 5x \) shifted upwards by 4 units.
Parallel Lines
Parallel lines are lines in the same plane that never intersect. They have the same slope but different y-intercepts. This makes them equidistant from each other at all points along their lengths.
In the case of \( y = 5x \) and \( y = 5x + 4 \), both lines are parallel because they share the same slope of 5. Even though their y-intercepts differ, this only affects their starting points along the y-axis and not the distance or angles between them.
This distinction is a simplified way to understand that parallel lines, while they look different depending on their y-intercepts, will always move in the same direction indefinitely.
In the case of \( y = 5x \) and \( y = 5x + 4 \), both lines are parallel because they share the same slope of 5. Even though their y-intercepts differ, this only affects their starting points along the y-axis and not the distance or angles between them.
This distinction is a simplified way to understand that parallel lines, while they look different depending on their y-intercepts, will always move in the same direction indefinitely.
