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Chapter 3: Problem 28

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. $$ \begin{array}{r} {2 x+y=6} \\ {x-y=3} \end{array} $$

Short Answer

Expert verified
The solution set is \( \{ (3, 0) \} \).

Step by step solution

01

Write down the system of equations

The given system of equations can be written as: \[ 2x + y = 6 \] \[ x - y = 3 \]
02

Add the equations

To eliminate variable \( y \), add the two equations: \[ (2x + y) + (x - y) = 6 + 3 \] This simplifies to: \[ 3x = 9 \]
03

Solve for \( x \)

Solve the simplified equation for \( x \): \[ x = \frac{9}{3} \] \[ x = 3 \]
04

Substitute \( x \) back into one of the original equations

Use the value of \( x = 3 \) and substitute it back into the second original equation: \[ 3 - y = 3 \] Solve for \( y \): \[ y = 0 \]
05

Write the solution set

Since \( x = 3 \) and \( y = 0 \), the solution is \( (3, 0) \). The system has a unique solution, so it does not have an infinite number of solutions or no solution.

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Key Concepts

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

System of Equations
A system of equations is a collection of two or more equations with the same set of variables. In these equations, we seek values for the variables that satisfy all the equations simultaneously. For example, the given system of equations is:
\[ 2x + y = 6 \] \[ x - y = 3 \]
This means we are looking for values of both \( x \) and \( y \) that make both equations true at the same time. There are several methods to find these solutions, such as graphing, substitution, and elimination.
Set-Builder Notation
Set-builder notation is a mathematical notation used to describe a set by stating the properties that its members must satisfy. It's particularly useful in situations where you have an infinite number of solutions or when describing solutions in a concise manner.
The general form of set-builder notation looks like this:
\[ \text{Set} = \{ \text{x | property of x} \} \]
For instance, if a system of equations has infinite solutions, you could use set-builder notation to describe the set of all possible solutions. However, in cases like the given exercise where there's a unique solution, the notation simplifies, and we just state the solution pair directly.
Unique Solution
When solving a system of linear equations, it鈥檚 crucial to identify whether the solution is unique, infinite, or non-existent. A unique solution means that there is exactly one set of values for the variables that satisfies all the given equations.
In our exercise, after using the elimination method, we found that:
  • \( x = 3 \)
  • \( y = 0 \)
This means the system has precisely one solution in the form of the ordered pair \( (3, 0) \). This unique solution is the intersection point of the two lines represented by the equations in a Cartesian plane.

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Most popular questions from this chapter

Brushstroke Computers, Inc., is planning a new line of computers, each of which will sell for \(\$ 970 .\) The fixed costs in setting up production are \(\$ 1,235,580,\) and the variable costs for each computer are \(\$ 697 .\) a) What is the break-even point? b) The marketing department at Brushstroke is not sure that \(\$ 970\) is the best price. Their demand function for the new computers is given by \(D(p)=-304.5 p+374,580\) and their supply function is given by \(S(p)=788.7 p-\) \(576,504 .\) To the nearest dollar, what price \(p\) would result in equilibrium between supply and demand? c) If the computers are sold for the equilibrium price found in part \((b),\) what is the break-even point?

Reba works at a Starbucks "coffee shop where a 12 -oz cup of coffee costs 1.85 dollars, a 16 -oz cup costs 2.10 dollars, and a 20 -oz cup costs 2.45 dollars. During one busy period, Reba served 55 cups of coffee, emptying six \(144-\mathrm{oz}\) "brewers" while collecting a total of 115.80 dollars. How many cups of each size did Reba fill?

Small-Business Loans. Chelsea took out three loans for a total of 120,000 dollars to start an organic orchard. Her business-equipment loan was at an interest rate of \(7 \%,\) the small-business loan was at an interest rate of \(5 \%\), and her home-equity loan was at an interest rate of \(3.2 \% .\) The total simple interest due on the loans in one year was 5040 dollars. The annual simple interest on the home-equity loan was 1190 dollars more than the interest on the business equipment loan. How much did she borrow from each source?

Solve. $$ \begin{aligned} &\frac{x+2}{3}-\frac{y+4}{2}+\frac{z+1}{6}=0\\\ &\frac{x-4}{3}+\frac{y+1}{4}-\frac{z-2}{2}=-1\\\ &\frac{x+1}{2}+\frac{y}{2}+\frac{z-1}{4}=\frac{3}{4} \end{aligned} $$

The tens digit of a two-digit positive integer is 2 more than three times the units digit. If the digits are interchanged, the new number is 13 less than half the given number. Find the given integer. (Hint. Let \(x=\) the tens-place digit and \(y=\) the units-place digit; then \(10 x+y\) is the number.
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