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Chapter 3: Problem 44

If a system has an infinite number of solutions, use set-builder notation to write the solution set. If a system has no solution, state this. Solve using the elimination method. $$ \begin{aligned} &8 s+12 t=16\\\ &6 s+9 t=12 \end{aligned} $$

Short Answer

Expert verified
The solution set is \( \{(s, t)\, | \,2s + 3t = 4\} \).

Step by step solution

01

- Write the system of equations

The system of equations is given as: \[\begin{aligned} 8s + 12t &= 16\6s + 9t &= 12 \end{aligned} \]
02

- Simplify the second equation

Divide all terms in the second equation by 3 to simplify it: \[\frac{6s}{3} + \frac{9t}{3} = \frac{12}{3} \implies 2s + 3t = 4\]
03

- Simplify the first equation

Divide all terms in the first equation by 4 to simplify it: \[\frac{8s}{4} + \frac{12t}{4} = \frac{16}{4} \implies 2s + 3t = 4\]
04

- Compare the simplified equations

Now both equations are: \[\begin{aligned} 2s + 3t &= 4\2s + 3t &= 4 \end{aligned} \] Since these are the same equation, the system has an infinite number of solutions.
05

- Write the solution set using set-builder notation

Since the system has infinite solutions, the solution set can be written as: \[\{(s, t)\, | \,2s + 3t = 4\}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

elimination method
The elimination method is a powerful technique for solving systems of linear equations. This method involves combining the equations to eliminate one of the variables, thereby reducing the system to a simpler form. To apply the elimination method:

  • First, align the equations vertically by their variables and constants.
  • Next, manipulate the equations through addition, subtraction, multiplication, or division to cancel one of the variables.
If the equations can be simplified such that one variable cancels out, you'll be left with a single equation in one variable. Solving this equation gives the value of the remaining variable. You can then substitute this value back into one of the original equations to find the value of the other variable. The elimination method is especially useful when the coefficients of the variables are easily manipulated.
infinite solutions
Sometimes, a system of equations might not have a unique solution. When you simplify the equations and end up with the same equation repeated, it indicates that there are infinite solutions. Infinite solutions mean that any number of ordered pairs \(s, t\) can satisfy the system of equations. For instance, in our example, both simplified equations turned out to be the same: \[2s + 3t = 4.\] Since these are essentially the same line on a graph, any point on the line is a solution, leading to an infinite number of possible solutions. Such systems are known as 'consistent and dependent,' meaning the equations represent the same linear relationship.
set-builder notation
Set-builder notation is a concise way of expressing a set of solutions. It specifies the property or condition that the elements of the set must satisfy. In our example with infinite solutions, set-builder notation helps us to neatly define the entire solution set. For the system \[ \{(s, t) | 2s + 3t = 4\}\], this reads as: 'the set of all \(s, t\) such that \ 2s + 3t = 4.\' This notation is particularly useful in expressing complex solution sets without having to list every possible solution, which is impractical when there are infinite solutions.`
simplification
Simplification is a vital step in solving equations, as it makes the problem easier to handle. In the given system, simplification involved dividing each term of both equations by a common factor to reduce the coefficients. The first equation \[8s + 12t = 16\] simplifies to \[2s + 3t = 4\] when each term is divided by 4. Similarly, the second equation \[6s + 9t = 12\] simplifies to \[2s + 3t = 4\] when each term is divided by 3. Simplifying equations makes it easier to identify relationships and solutions. It's an essential part of mathematical problem-solving that ensures the equations are in their simplest, most manageable form before further analysis or solution.

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Most popular questions from this chapter

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