Problem 33
Question
Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(\left\\{\begin{array}{l}{y=3 x-5} \\ {21 x-35=7 y}\end{array}\right.\)
Step-by-Step Solution
Verified Answer
The system of equations has infinitely many solutions, represented by the equation of the line y = 3x - 5. The solution set in set notation is {(x, y) | y = 3x - 5, x ∈ R }.
1Step 1: Identify the equations
The given system of equations is: \( y=3x-5\) (equation 1)\( 21x-35=7y\) (equation 2), which can be rewritten as \( y = 3x -5 \) (after dividing throughout by 7)
2Step 2: Compare the equations
Both equations in the system are identical when written in the slope-intercept form, implying they represent the same line in the graph.
3Step 3: Formulate the solution
Since both equations are representing the same line, there are infinitely many solutions. All points on the line y = 3x-5 are solutions to this system. In set notation, the solution set can be expressed as \( {(x, y) | y = 3x - 5, x ∈ R} \) , where \( R \) denotes the set of all real numbers.
Key Concepts
Method of Choice for Solving SystemsNo Solution SystemsInfinitely Many SolutionsSet NotationSlope-Intercept Form
Method of Choice for Solving Systems
When faced with a system of equations, it's crucial to decide on the most efficient method to find the solution. There are several techniques at your disposal, such as graphing, substitution, and elimination. Choosing the right method often depends on how the equations are presented and what the question is asking for.
In the given exercise, we are asked to use the method of our choice. Upon inspection, we see that the equations are in a format that lends itself easily to the slope-intercept form. Given this, one could argue that comparison is the most straightforward method, especially since this method allows us to quickly identify the relationship between the two equations, which is essential for determining if there are none, one, or infinitely many solutions.
In the given exercise, we are asked to use the method of our choice. Upon inspection, we see that the equations are in a format that lends itself easily to the slope-intercept form. Given this, one could argue that comparison is the most straightforward method, especially since this method allows us to quickly identify the relationship between the two equations, which is essential for determining if there are none, one, or infinitely many solutions.
No Solution Systems
Systems of equations that have no solution are called inconsistent systems. These occur when the equations represent parallel lines that never intersect. In the coordinate plane, this means that despite having the same slope, they have different y-intercepts.
To identify such a system, you'd look for equations that, when plotted, never touch each other. In algebraic terms, if you arrive at a false statement, such as 0 = 5 after using any method to solve the system, it indicates that there's no solution. No solution systems are significant because they notify us that the equations at hand cannot simultaneously satisfy the same value pair (x, y).
To identify such a system, you'd look for equations that, when plotted, never touch each other. In algebraic terms, if you arrive at a false statement, such as 0 = 5 after using any method to solve the system, it indicates that there's no solution. No solution systems are significant because they notify us that the equations at hand cannot simultaneously satisfy the same value pair (x, y).
Infinitely Many Solutions
On the other side of the coin, some systems have infinitely many solutions, indicating the lines coincide, or lay atop one another, comprising all the same points. Such systems are considered consistent and dependent.
Algebraically, if the two equations simplify to the same equation, or if you end up with a true statement like 0 = 0, then you're looking at a system with infinitely many solutions. As seen in the provided exercise where both equations represent the same line, every point on this line is a solution to the system.
Algebraically, if the two equations simplify to the same equation, or if you end up with a true statement like 0 = 0, then you're looking at a system with infinitely many solutions. As seen in the provided exercise where both equations represent the same line, every point on this line is a solution to the system.
Set Notation
Set notation is a standardized way of describing a set of numbers or solutions. It is compact and communicates information efficiently. For systems of equations, set notation can be used to define solution sets, whether they are single points, a range of points, or even no points at all.
For a system with infinitely many solutions, like the one in the exercise, you would denote the solution set as \( \{(x, y) | y = 3x - 5, x \in \mathbb{R}\} \), where \( \mathbb{R} \) stands for the set of all real numbers. This means that for every real number value of x, there is a corresponding y value that makes the equation true.
For a system with infinitely many solutions, like the one in the exercise, you would denote the solution set as \( \{(x, y) | y = 3x - 5, x \in \mathbb{R}\} \), where \( \mathbb{R} \) stands for the set of all real numbers. This means that for every real number value of x, there is a corresponding y value that makes the equation true.
Slope-Intercept Form
The slope-intercept form is one of the most significant tools in algebra for understanding linear equations and their graphs. This form is written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept—the point where the line crosses the y-axis.
By expressing both equations from the system in the slope-intercept form, as done in the exercise, it becomes easier to compare the equations and recognize their relationships. The slope determines the steepness and direction of the line, and if two lines have the same slope and y-intercept, like in our exercise, they are the same line and thus share all their points.
By expressing both equations from the system in the slope-intercept form, as done in the exercise, it becomes easier to compare the equations and recognize their relationships. The slope determines the steepness and direction of the line, and if two lines have the same slope and y-intercept, like in our exercise, they are the same line and thus share all their points.
Other exercises in this chapter
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