Problem 44 - 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. $$\left\\{\begin{array}{l} \frac{x}{3}+y=3 \\ \frac{x}{2}-\frac{y}{4}=1 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \({(x, y)|x = 3, y = 2}\)

Step 1: Convert to More Manageable Form

First, multiply each equation by a factor that will eliminate the fraction. For the first equation, multiply both sides by 3 and for the second equation, multiply both sides by 4. This transforms the system to: \[\{\begin{array}{l} x + 3y = 9 \ 2x - y = 4\end{array}\]

Step 2: Addition Method

The addition method involves manipulating the equations so that adding them will eliminate one variable. In this case, multiply the second equation by 3 and add it with the first equation to eliminate y. This gives: \(2x\cdot3 - y\cdot3 + x + 3y = 4\cdot3 + 9\). Simplifying this leads to: \(7x = 21\). Therefore, \(x = 21/7 = 3\).

Step 3: Solve for the Other Variable

Substitute the value of x (which is 3) into the first equation. Hence, we get: \(3 + 3y = 9\). Solving for y gives \(y = 2\).

Step 4: Express in Set Notation

The solution in set notation is expressed as \({(x, y)|x = 3, y = 2}\)

Key Concepts

System of Linear Equations

When we talk about a system of linear equations, we're referring to a set of equations that all need to be true at the same time. Each equation in the system typically depicts a line on a graph, and the solution to the system is the point or points where these lines intersect.

Consider our original problem, which presents a system of two equations: one with the variable x divided by 3 and added to y, and another with x divided by 2 and y divided by 4, with certain constants on the other side. This represents two lines in a two-dimensional plane. Solving this system means finding the x and y values that satisfy both equations simultaneously, which is at the point where these two lines would cross on a graph.

Solving Equations Algebraically

The process of solving equations algebraically often involves steps that simplify the equations to make it easier to identify the solution. In the context of our system, algebraic manipulation is used to get rid of the fractions, and then rearrange and combine the equations to isolate one of the variables.

Specifically, we started by eliminating fractions to obtain whole number coefficients alongside our variables. Then, we used the addition method, which is a technique where we adjust the coefficients so that adding or subtracting equations will cancel out one of the variables, allowing us to solve for the other. After finding the value of x, this value is substituted back into one of the original equations to find y. This step-by-step approach reduces the complexity of the system and leads us directly to the solution.

Fraction Elimination in Equations

Dealing with fractions in equations can be daunting, so fraction elimination is a strategical step to streamline the solving process. By multiplying each term by the least common denominator, we can convert the fractional coefficients to whole numbers, which simplifies the algebra involved.

Let's reflect on the problem at hand. We multiplied the first equation by 3 and the second by 4, the denominators of the respective fractions, effectively 'clearing' the fractions from the equations. This operation is fundamental as it not only simplifies the equations, but also prepares them for the addition method by setting up compatible coefficients for elimination.

Set Notation Solutions

Finally, once we've solved for the variables, we use set notation to express the solution in a precise mathematical way. Set notation is a standardized method to describe collections of objects, which, in the context of algebra, often refers to the possible values of variables.

In this scenario, after finding the values x = 3 and y = 2, we can represent the solution set as \( \{ (x, y) | x = 3, y = 2 \} \), which translates to 'the set of all points (x, y) such that x equals 3 and y equals 2.' It's a succinct way to report the answer clearly indicating the values of x and y that solve both equations in the system.