Problem 37 - Introductory Algebra for College Students

Question

In Exercises \(1-44,\) 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. $$\left\\{\begin{array}{l} x=5-3 y \\ 2 x+6 y=10 \end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations has infinite solutions, which can be expressed in set notation as\[\{(x,y)| x = 5 - 3y, y\in\mathbb{R}\}\]

Step 1: Write the system in standard form

Transform the given system of equations to standard form:\[\begin{array}{l} x+3y=5\\ 2x+6y=10 \end{array}\]

Step 2: Make the coefficients of x in both equations equal

We can already see that the coefficients of x in both equations are already equal. If they were not, we would need to multiply one of the equations by a constant to make these coefficients equal.

Step 3: Subtract the equations

Subtract the first equation from the second equation to eliminate x:\[\begin{array}{l} (2x+6y)-(x+3y)=10 - 5\\ x + 3y = 5\end{array}\] Here, we can see that both equations are identical. This suggests that there may be an infinite number of solutions.

Step 4: Check your work

A quick check verifies this. Substituting from first equation to second equation, we get exactly the same equation, hence, the system have infinite solution.

Step 5: Write the solution in set notation

If the system has an infinite number of solutions, these solutions can be expressed in set notation as:\[\{(x,y)| x = 5 - 3y, y\in\mathbb{R}\}\] meaning that for each real y, the value of x will be calculated from the equation x = 5 - 3y

Key Concepts

Systems of Linear Equations

Linear equations are at the heart of algebra and represent relationships where variables change in a consistent manner. When we have two or more linear equations with the same variables, we call this a system of linear equations. These systems can be solved using various methods, one such being the addition (or elimination) method.

The addition method involves manipulating the equations such that adding them together eliminates one variable, allowing us to solve for the other. This is particularly useful when the coefficients of one variable are opposites.

• If the result is a true statement (like 0=0), the system has infinitely many solutions.
• If the result is a contradiction (like 0=5), the system has no solution.
• Otherwise, the result will give a unique solution for the variables involved.

Standard Form of Linear Equations

A linear equation can be expressed in various forms, and one of the most common is the standard form. This form is expressed as Ax + By = C, where A, B, and C are integers, and A should be non-negative. In the context of the addition method, having the system in standard form is quite advantageous. It makes it easier to manipulate the equations to get like terms lined up - a crucial step for cancellation.

Converting a given linear equation to standard form often requires simple algebraic manipulations, such as moving terms from one side of the equation to the other and ensuring the x and y terms are on the same side.

Set Notation

Set notation is a concise and standardized way to express collections of objects, which in the context of algebra, typically refers to numbers or solutions. When writing solutions to a system of linear equations, using set notation can effectively express multiple or even infinite possible answers.

For example, \(\{ (x,y)| x = 5 - 3y, y \in \mathbb{R}\}\) is read as \'the set of all pairs (x, y) such that x equals 5 minus 3 times y, where y is any real number\'. It\'s a straightforward way to encapsulate the relationship between x and y without listing out every single possible pair.

Infinite Solutions to Systems

When a system of linear equations yields an identity after applying the addition method (or any method of solution), it indicates that the equations are not distinct but rather are different representations of the same line. This results in an infinite number of solutions, as every point on the shared line is a solution to the system.

In our exercise, this occurs when the manipulation resulted in equivalent equations. Representing this infinite set of solutions in set notation is very useful as it describes all possible solutions in a compact form. It tells us that any point that satisfies the equation x = 5 - 3y is part of the solution set, covering infinitely many possibilities where y is a real number.