Problem 22 - 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. $$\left\\{\begin{array}{l} 9 x=25+y \\ 2 y=4-9 x \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is the set \( {(6, 29)} \)

Step 1: Rearrange the Equations

Rearrange the equations to align like terms, for each the coefficient of \(x\) in the first equation equals the negative of the coefficient of \(x\) in the second equation, so when the two equations are added, \(x\) can be eliminated. Here are the rearranged equations: \[9x - y = 25\] and \[-9x + 2y = 4\]

Step 2: Add the Equations to Eliminate One Variable

Add the two rearranged equations to eliminate \(x\): \[(9x - y) + (-9x + 2y) = 25 + 4\] This simplification gives: \(y = 29\)

Step 3: Substitute the Found Value Into One of the Initial Equations

Substitute \(y = 29\) into the first initial equation to find \(x\): \(9x = 25 + 29\). Solving this gives: \(x = 54/9 = 6\)

Step 4: Write the Solution in Set Notation

The coordinates (x, y) are the solution set. Using set notation to express the solution: \( {(6, 29)} \)

Key Concepts

System of Equations

Solving a system of equations involves finding values for the variables that make all the equations true at the same time. A system usually consists of two or more equations. In our example, we have two equations:

• 9x = 25 + y
• 2y = 4 - 9x

Each equation represents a line on a graph, and solving the system means finding the point at which these lines intersect. This point is the solution of the system, indicating the values of x and y that satisfy both equations. Understanding the setup of a system is crucial for choosing the right method to solve it.

Elimination Method

The elimination method, also known as the addition method, is an effective way to solve systems of equations. By strategically adding or subtracting equations, one can eliminate a variable, making it easier to solve for the remaining one.
In our system:

• First, we aligned the terms in each equation to 9x - y = 25 and -9x + 2y = 4.
• Next, we added these equations to get rid of the x variable:
• (9x - y) + (-9x + 2y) = 25 + 4 becomes y = 29, as the 9x and -9x terms cancel each other out.

This method is particularly useful when the coefficients of a variable are opposites, quickly simplifying the problem to a single-variable equation, which is much easier to solve.

Set Notation

Set notation is a way of expressing solutions in a simplified, mathematical format. It perfectly encapsulates the solution, especially when dealing with systems of equations.
Once we have determined the solution, we can write it concisely. For our system:

• The solution for x was found to be 6, and for y, it was 29.
• In set notation, the solution is expressed as a pair (x, y): \( \{(6, 29)\} \)

This notation not only indicates that these values simultaneously satisfy both equations but also presents the result in a clear, understandable format. It's especially useful in larger math problems to quickly convey exact pairs without long explanations.

Solve Equations

Solving equations is all about finding the unknown values that make the equation true. For the system at hand, after eliminating a variable using the elimination method, we solved for y first, getting y = 29.
Next, we substituted y back into one of the original equations to solve for x:

• Substituting y = 29 into 9x = 25 + y gives 9x = 54.
• Solving for x yields x = 54 / 9 = 6.

By following these steps, we find values for the variables that satisfy all the given conditions. It's crucial to substitute back into any initial equation to verify the accuracy of the results. Understanding these steps is vital for efficiently solving equations, especially when dealing with multiple variables in systems of equations.