Problem 37
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. \(\left\\{\begin{array}{c}x+3 y-2 \\ 3 x+9 y-6\end{array}\right.\)
Step-by-Step Solution
Verified Answer
This system of equations has infinitely many solutions, and the solution set is represented by: \(\{(x, y) ∈ ℝ² | x + 3y = 2\}\)
1Step 1: Simplify the second equation
The second equation can be simplified by noting that all its terms are divisible by 3. Thus, divide each term by 3 to get a simplified second equation: \(x + 3y - 2\)
2Step 2: Compare the two equations
Now the first equation and the simplified second equation are as follows: \(x + 3y - 2\) and \(x + 3y - 2\). They are the same equation, depicted differently, thus the two equations are dependent. Therefore, every solution to one equation is also a solution to the other equation.
3Step 3: Identify solution
Since both equations are the same, there are an infinite number of solutions. This is because any real number pair \((x, y)\) that satisfies the equation, including solutions when X and Y are zero, will solve this system. In set notation, the solution set is all real number pairs \((x, y)\) such that they fit the equation. More formally, the solution set is \(\{(x, y) ∈ ℝ² | x + 3y = 2\}\)
Key Concepts
Dependent SystemsInfinite SolutionsSet NotationSimplifying Equations
Dependent Systems
When we talk about dependent systems in the context of equations, we're referring to a situation where two or more equations are essentially the same. In other words, they overlap entirely when graphed, sharing every point on the line or curve they form.
This occurs because one equation can be derived from another through multiplication or addition of constants. For instance, if you simplify two allegedly different equations and end up with identical equations, it means you have a dependent system.
In the exercise, by simplifying the second equation, we see both equations become \( x + 3y - 2 = 0 \). This indicates dependency as every point that makes one equation true also makes the other true.
This occurs because one equation can be derived from another through multiplication or addition of constants. For instance, if you simplify two allegedly different equations and end up with identical equations, it means you have a dependent system.
In the exercise, by simplifying the second equation, we see both equations become \( x + 3y - 2 = 0 \). This indicates dependency as every point that makes one equation true also makes the other true.
Infinite Solutions
An infinite solutions scenario arises when an entire line or plane is shared by all equations in a system. This means there isn't just one single solution or a few; rather, there are countless solutions.
For a system of linear equations, this typically happens when the equations are dependent. Since they represent the same line, any point on this line is a valid solution.
In the exercise example, because both equations reduce to the same line, every point \((x, y)\) lying on \(x + 3y = 2\) is a solution. These are infinite because there are unlimited points along a line that extends forever in both directions.
For a system of linear equations, this typically happens when the equations are dependent. Since they represent the same line, any point on this line is a valid solution.
In the exercise example, because both equations reduce to the same line, every point \((x, y)\) lying on \(x + 3y = 2\) is a solution. These are infinite because there are unlimited points along a line that extends forever in both directions.
Set Notation
Set notation is a mathematical way to show a collection of elements or numbers. It is a precise way of expressing the entire solution set of equations.
When a system of equations has infinite solutions, we often use set notation to succinctly describe all possible solutions.
In the given example, once simplified, the solution set can be expressed as \( \{(x, y) ∈ ℝ² \mid x + 3y = 2\} \). This means that every pair of \((x, y)\) in the real number plane \(ℝ²\), if plugged into this equation, satisfies it and thus is part of the solution set.
When a system of equations has infinite solutions, we often use set notation to succinctly describe all possible solutions.
In the given example, once simplified, the solution set can be expressed as \( \{(x, y) ∈ ℝ² \mid x + 3y = 2\} \). This means that every pair of \((x, y)\) in the real number plane \(ℝ²\), if plugged into this equation, satisfies it and thus is part of the solution set.
- \(\{...\}\) denotes a set
- \((x, y) ∈ ℝ²\) indicates pairs of real numbers \(x\) and \(y\)
- \(\mid\) "such that" specifies a condition follows next
Simplifying Equations
Simplifying equations involves reducing them to their simplest and most understandable form. It makes equations easier to solve or compare by removing unnecessary complexity.
In the exercise, simplifying the second equation involved dividing each term by 3, leading it to be identical to the first equation \(x + 3y - 2 = 0\).
To simplify:
In the exercise, simplifying the second equation involved dividing each term by 3, leading it to be identical to the first equation \(x + 3y - 2 = 0\).
To simplify:
- Look for the greatest common factor (GCF) to divide through the equation
- Combine like terms if they’re present
- Simplify both sides as much as possible to reveal relationships among variables
Other exercises in this chapter
Problem 37
In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$-2 \leq x
View solution Problem 37
Solve each system by the method of your choice. $$\left\\{\begin{array}{l} x^{2}+(y-2)^{2}-4 \\ x^{2}-2 y-0 \end{array}\right.$$
View solution Problem 38
Exercises 37-39 will help you prepare for the material covered in the first section of the next chapter. Solve the system: $$\left\\{\begin{aligned}w-x+2 y-2 z
View solution Problem 38
Write the partial fraction decomposition of each rational expression. $$\frac{x^{2}+2 x+3}{\left(x^{2}+4\right)^{2}}$$
View solution