Problem 47 - Introductory Algebra for College Students

Question

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{array}{l} 3 x+2 y=-3 \\ 2 x-5 y=17 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \{(x, y) | x = -1 , y = -1\}.

Step 1: Decision of the Method

Considering the system of equations, the Substitution Method is chosen for this exercise. The Substitution Method allows us to express one variable in terms of the other from one equation and then substitute this into the second equation. This makes it easier to isolate each of the variables.

Step 2: Express one Variable in terms of the other

There is no preference, but for the sake of explanation, let's isolate \(x\) in the first equation. By rearranging the first equation, it would look like this: \(3x = -2y - 3\), then divide each term by 3 to get \(x\) alone: \(x = -\frac{2}{3}y - 1\). This is the value of \(x\) in terms of \(y\).

Step 3: Substitute into the second Equation

Substitute \(x = -\frac{2}{3}y - 1\) into the second equation: \(2(-\frac{2}{3}y - 1) - 5y = 17\). This will turn our system of equations into a single equation with only one variable.

Step 4: Solve for the unknown Variable

Solve the equation for \(y\). Simplify the equation to get \(y\), which will result in \(y = -1\).

Step 5: Substitute y back into first Equation

Substitute \(y = -1\) back into the first equation where we isolated \(x\): \(x = -\frac{2}{3}*(-1) - 1\). Solve this equation to get \(x = -1\).

Step 6: Express in Set Notation

In set notation, the solution set to the system of equations is \{(x, y) | x = -1 , y = -1\} - the set of \(x\) and \(y\) satisfying both equations.

Key Concepts

Substitution Method

The Substitution Method is a powerful algebraic tool for solving systems of equations. This technique involves rearranging one of the equations to isolate one of the variables, and then substituting this expression into the other equation. This allows us to turn a system of equations into a single equation with just one variable to solve for.

In our example, we isolated variable x in the first equation and then substituted it into the second. This approach is particularly useful when equations can be easily manipulated to isolate a variable. For instance, if a variable has a coefficient of 1 or -1, it can quickly be isolated, making the substitution into the other equation straightforward.

It's also helpful when one equation is already solved for a variable or when a quick rearrangement can eliminate fractions or complex terms. This strategic decision-making is crucial for efficiency and simplicity in solving algebraic problems.

Set Notation

Set notation is a standardized method of expressing collections of objects, which in algebra often refer to solutions of equations. Expressing our solutions in set notation allows us to be concise, clear, and clean when stating which values of the variables satisfy the system of equations.

When representing solutions to a system of equations, set notation looks like this: \( \{ (x, y) | \text{condition(s)} \} \), where \((x, y)\) are the variables of our system and the 'condition(s)' are the relationships they must satisfy. In the example provided, after the Substitution Method was used to solve for x and y, the solution was presented in set notation as \( \{ (x, y) | x = -1 , y = -1 \} \). This neatly states that the only solution to this system is the point where x equals -1 and y also equals -1.

Understanding and using set notation is essential for students as it is common in higher-level mathematics and it enhances their ability to communicate mathematical ideas effectively.

Algebraic Solution

An algebraic solution refers to finding the values of variables that satisfy a set of equations using algebraic manipulations. When we talk about an algebraic solution in the context of systems of equations, we imply a step-by-step approach to deduce the specific values that the variables can take.

In the given problem, by using the Substitution Method, we first expressed x in terms of y and substituted it in the other equation to find y before finally determining the value of x. The algebraic solution process can involve various operations such as addition, subtraction, multiplication, division, factorization, or expansion.

It's critical for students to learn this process to not only solve the equations but also to develop logical reasoning and problem-solving skills. It hones their ability to deal with abstract concepts and apply them in practical situations, a keystone of mathematical education.