Problem 16 - 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. $$\left\\{\begin{array}{l} 5 x-4 y=19 \\ 3 x+2 y=7 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution is \(\{ (3, -1) \}\)

Step 1: Multiply Equations Suitably To Cancel Out A Variable

We want to eliminate one of the variables by adding the two equations. We can multiply the first equation by 2 and the second equation by 4 to make the coefficients of y the same in both equations, so they cancel out when we add them.\nSo now the equations are: \n\(10x - 8y = 38\) \nand \(12x + 8y = 28\)

Step 2: Add the equations

We add the two resulting equations from step 1 to get \(22x = 66\). Simplifying that gives us the solution for x which is \(x=3\)

Step 3: Substitute x Value To Solve for y

Now we can substitute x = 3 into the second original equation to find the value of y: \n\(3*3 + 2y = 7\) which simplifies to \(9 + 2y = 7\). Solving for y, we get \(2y = -2\) and thus \(y = -1\)

Step 4: State Solution In Set Notation

The solution to the system of equations is the set \(\{ (3, -1) \}\)

Key Concepts

Solving Systems of Equations

Solving systems of equations is a fundamental concept in algebra. It involves finding the values of variables that satisfy multiple equations simultaneously. In many practical scenarios, systems of equations are used to describe relationships between different quantities. For instance, they can represent the intersection points of lines on a graph or the balance of chemical equations.

When tackling a system of equations, one must determine whether a single, multiple, or no solutions exist. Techniques for solving these systems include graphing, substitution, and the addition or elimination method. Each method has its advantages depending on the type of system and its complexity. Graphical solutions provide a visual insight, while algebraic methods offer precise solutions.

In the given exercise, the goal is to use the addition method to solve the system. Identifying the right approach greatly simplifies the problem and helps achieve the solution efficiently.

Elimination Method

The elimination method is a strategic approach to solve systems of linear equations. It focuses on eliminating one variable by adding or subtracting the equations, transforming the system into one that is easier to solve.

Here's how the method works:

• Configure the equations so that adding or subtracting them will cancel out one variable.
• Multiply the equations by suitable numbers to equate the coefficients of one of the variables. This makes it possible to eliminate the variable when the equations are added together.
• Add or subtract the equations to remove one variable, thus simplifying the system to a single-variable equation.
• Solve the resulting equation to find the value of one variable, and then substitute back into one of the original equations to find the other variable.

In the problem provided, we transformed the equations so that the coefficients of the variable \( y \) are opposites, facilitating its elimination. Once \( y \) is gone, the problem reduces to finding \( x \) from a simple equation. This technique highlights the power of strategically manipulating equations to unlock their solutions.

Set Notation

Set notation is a precise way to express the solution to a system of equations. It presents solutions clearly and succinctly, showing all the possible solutions if there are multiple or announcing they are nonexistent if the case demands.

In mathematics, a set is a collection of distinct elements, and using set notation helps communicate these elements effectively. For example, if a system of equations leads to a solution, it can be written in set notation as \( \{ (x, y) \} \), where \( x \) and \( y \) represent the solution coordinates.

• Use curly braces \( \{ \} \) to denote sets.
• Represent each solution as an ordered pair \( (x, y) \).
• For systems with no single solution, express this with an empty set \( \emptyset \) or with wording specifying infinitely many solutions.

This method was used in the exercise to present the solution \( \{ (3, -1) \} \), which is the point where the lines represented by the equations intersect. Set notation is not only mathematically accurate but also helps convey solutions in an easily understandable format.