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Chapter 4: Problem 40

Solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}2(2 x+3 y)=0 \\ 7 x=3(2 y+3)+2\end{array}\right.$$

Short Answer

Expert verified
There are infinitely many solutions, represented by the set \(\{(x, y) \mid y = -\frac{2x}{3}, x \in \mathbb{R}\}\).

Step by step solution

01

Simplify each equation

Start with the given system of equations: \(2(2x+3y) = 0\) and \(7x = 3(2y+3) + 2 \). Simplifying these, we get the following system of equations: \n \[ \begin{align*} 4x + 6y &= 0, \\ 7x &= 6y + 11. \end{align*} \]
02

Eliminate one variable

Now, we need to eliminate one variable by adding the two equations together. Let's multiply the first equation by 7 and the second by 4 and add them together, we get 28x = 28x. This is always a true statement.
03

Determine the number of solutions

Since we got a statement that is always true, it means that there are an infinite number of solutions, with \(y = -\frac{2x}{3}\).
04

Express the solution in set notation

We can write the solution as \(\{ (x, y) \mid y = -\frac{2x}{3}, x \in \mathbb{R} \}\).

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Key Concepts

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

Addition Method
The Addition Method, also known as the Elimination Method, is a technique used to solve systems of linear equations by eliminating one of the variables. The main goal is to combine the equations in such a way that one of the variables cancels out, allowing you to solve for the remaining variable. This is achieved by:
  • Aligning the equations.
  • Choosing a variable to eliminate.
  • Multiplying the equations, if necessary, to make the coefficients of one variable equal but opposite.
  • Adding or subtracting the equations to eliminate the chosen variable.
Once a variable is eliminated, you can solve for the other variable. It's efficient because it reduces the number of equations and variables step by step. This method works best when the coefficients of the chosen variables can easily be equalized.
Infinite Solutions
Infinite Solutions occur in a system of equations when the equations describe the same line. This means every solution to one equation is a solution to the other, resulting in an infinite number of solutions.
  • After simplifying the system, check if the resulting equations are dependent (multiples of each other).
  • Finding a true identity like "0 = 0" when trying to eliminate a variable often indicates infinite solutions.
In such cases, you can express the relationship between the variables in terms of each other, like in this exercise where we found that the relationship was expressed as \(y = -\frac{2x}{3}\). This tells us that any \(x\) value can be paired with a corresponding \(y\) value that satisfies this relationship.
Set Notation
Set Notation is a formal way to express the solutions of a system of equations, especially when there are infinitely many solutions. It uses a set-builder format which typically looks like this: \[ \{ (x, y) \mid \text{constraint} \} \]In this format:
  • The curly brackets \(\{ \}\) signify a set.
  • The ordered pair \((x, y)\) shows a typical element of this set.
  • The vertical bar \(\mid\) is read as "such that."
  • The constraint describes the relation between \(x\) and \(y\).
For our case, the set notation \( \{ (x, y) \mid y = -\frac{2x}{3}, x \in \mathbb{R} \} \) describes all the possible solutions, where \(x\) is any real number, and \(y\) is determined through the equation \(y = -\frac{2x}{3}\). This gives a concise and precise expression of the infinite solutions.
Equation Simplification
Equation Simplification is a crucial first step when solving any system of equations. It involves rewriting the equations in a simpler form without changing their solutions. This makes them easier to handle, especially when performing operations like addition or subtraction to eliminate variables.
  • Distribute constants into parenthetical expressions.
  • Combine like terms to make the equations as straightforward as possible.
  • Streamline coefficients by factoring or simplifying unnecessary complex terms.
For the given problem, the original complicated equations were simplified to \(4x + 6y = 0\) and \(7x = 6y + 11\), which made it simpler to apply the addition method. Simplifying eliminates potential errors and provides clarity, making mathematical problem-solving more manageable.

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Most popular questions from this chapter

Use the four-step strategy to solve each problem. Use \(x, y,\) and \(z\) to represent unknown quantities. Then translate from the verbal conditions of the problem to a system of three equations in three variables. A person invested \(\$ 17,000\) for one year, part at \(10 \%,\) part at \(12 \%,\) and the remainder at \(15 \% .\) The total annual income from these investments was \(\$ 2110 .\) The amount of money invested at \(12 \%\) was \(\$ 1000\) less than the amounts invested at \(10 \%\) and \(15 \%\) combined. Find the amount invested at each rate.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I'm solving a three-variable system in which one of the given equations has a missing term, so it will not be necessary to use any of the original equations twice when I reduce the system to two equations in two variables.

When a small plane flies with the wind, it can travel 800 miles in 5 hours. When the plane flies in the opposite direction, against the wind, it takes 8 hours to fly the same distance. Find the rate of the plane in still air and the rate of the wind.

Write a system of equations modeling the given conditions. Then solve the system by the addition method and find the two numbers. Five times a first number increased by a second number is 14. The difference between four times the first number and the second number is \(4 .\) Find the numbers.

F. For thousands of years, gold has been considered one of Earth's most precious metals. One hundred percent pure gold is 24 -karat gold, which is too soft to be made into jewelry. In the United States, most gold jewelry is 14 -karat gold, approximately \(58 \%\) gold. If 18 -karat gold is \(75 \%\) gold and 12 -karat gold is \(50 \%\) gold, how much of each should be used to make a 14 -karat gold bracelet weighing 300 grams?
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