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Magazine Chapter 3: Problem 66
(a) find three solutions of the equation. (b) graph the equation. \(y=4 x-6\)
Short Answer
Expert verified
Solutions: (-1, -10), (0, -6), (1, -2). Graph: A straight line through these points.
Step by step solution
01
- Choose values for x
Select three different values for the variable x to use in the equation. It's helpful to choose simple numbers such as -1, 0, and 1.
02
- Calculate corresponding y values
Substitute each chosen value of x into the equation to find the corresponding y values. For example, if x = -1, 0, and 1, calculate y for each of these x values by plugging them into the equation y = 4x - 6.
03
- Solve for y when x = -1
When x = -1, substitute it into the equation: \[ y = 4(-1) - 6 \]\[ y = -4 - 6 \]\[ y = -10 \]So, one solution is (-1, -10).
04
- Solve for y when x = 0
When x = 0, substitute it into the equation: \[ y = 4(0) - 6 \]\[ y = 0 - 6 \]\[ y = -6 \]So, another solution is (0, -6).
05
- Solve for y when x = 1
When x = 1, substitute it into the equation: \[ y = 4(1) - 6 \]\[ y = 4 - 6 \]\[ y = -2 \]So, another solution is (1, -2).
06
- Plot the solutions on a graph
Plot the three points (-1, -10), (0, -6), and (1, -2) on a coordinate plane. These points should form a straight line because the equation is linear.
07
- Draw the line
Draw a straight line through the points (-1, -10), (0, -6), and (1, -2) to graph the equation y = 4x - 6.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Functions
A linear function is a type of function that forms a straight line when graphed on a coordinate plane.
The general form of a linear function is: during the text use,meaning if you want to mention then only use the math tag \( y = mx + b \),
where \( m \) represents the slope and \( b \) represents the y-intercept. It's called linear because it creates a line.
This simplicity is why it's often one of the first types of functions you'll encounter in algebra. Understanding linear functions helps build a strong foundation for more complex math concepts.
For instance, the equation \( y = 4x - 6 \) has a slope ( m ) of 4 and a y-intercept ( b ) of -6.
The general form of a linear function is: during the text use,meaning if you want to mention then only use the math tag \( y = mx + b \),
where \( m \) represents the slope and \( b \) represents the y-intercept. It's called linear because it creates a line.
This simplicity is why it's often one of the first types of functions you'll encounter in algebra. Understanding linear functions helps build a strong foundation for more complex math concepts.
For instance, the equation \( y = 4x - 6 \) has a slope ( m ) of 4 and a y-intercept ( b ) of -6.
Graphing Equations
Graphing linear equations involves plotting points on a coordinate plane and drawing a line through them.
You start by selecting values for x and solving for y. These pairs ( x, y ) form your points.
To graph \( y = 4x - 6 \), let's work through the solution from the exercise:
This line represents the equation \( y = 4x - 6 \).
You start by selecting values for x and solving for y. These pairs ( x, y ) form your points.
To graph \( y = 4x - 6 \), let's work through the solution from the exercise:
- Step 1: Choose x values, such as -1, 0, and 1.
- Step 2: Calculate corresponding y values:
- For x = -1: y = 4(-1) - 6 = -10
- For x = 0: y = 4(0) - 6 = -6
- For x = 1: y = 4(1) - 6 = -2
This line represents the equation \( y = 4x - 6 \).
Solving for y in Linear Equations
Solving for y in linear equations means isolating y on one side of the equation.
This helps to understand the relationship between x and y. Let's go through solving \( y = 4x - 6 \):
Mastering this technique is crucial for graphing linear functions and other algebraic operations.
This helps to understand the relationship between x and y. Let's go through solving \( y = 4x - 6 \):
- Step 1: For x = -1, substitute in the equation: \( y = 4(-1) - 6 = -10 \).
- Step 2: For x = 0, substitute in the equation: \( y = 4(0) - 6 = -6 \).
- Step 3: For x = 1, substitute in the equation: \( y = 4(1) - 6 = -2 \).
Mastering this technique is crucial for graphing linear functions and other algebraic operations.
Coordinate Plane
The coordinate plane is a two-dimensional surface used for plotting points.
It contains two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).
Where they intersect is called the origin (0,0).
Each point on the plane is identified by a pair of numerical coordinates ( x, y ).
In the exercise, we worked with points (-1, -10), (0, -6), and (1, -2).
These coordinates tell you exactly where to locate the points on the plane.
It contains two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).
Where they intersect is called the origin (0,0).
Each point on the plane is identified by a pair of numerical coordinates ( x, y ).
In the exercise, we worked with points (-1, -10), (0, -6), and (1, -2).
These coordinates tell you exactly where to locate the points on the plane.
- x-coordinates tell you how far to move left (negative) or right (positive) from the origin.
- y-coordinates tell you how far to move up (positive) or down (negative) from the origin.
