Problem 12 - Introductory Algebra for College Students

Question

Solve each system by the substitution method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}-4 x+y=-11 \\\2 x-3 y=5\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is the set \{\(x, y\)| x=2.8, y=0.2\}

Step 1: Isolate variable y in the first equation

Rewriting the first equation -4x + y = -11 in terms of y, we get: \(y = 4x - 11\).

Step 2: Substitute y into the second equation

By putting the term we found for y in place of y in the second equation, we obtain: \(2x - 3(4x - 11) = 5\). Distributing and simplifying, we get \(2x - 12x +33 = 5\), or \(-10x = -28\), leading to \(x = 2.8\).

Step 3: Find the value of y

Substituting \(x = 2.8\) into the first equation to solve for y. This gives \(-4*2.8 + y = -11\), which simplifies to \(y = -11 + 11.2 = 0.2\).

Step 4: Express solution in set notation

The solution set is {(2.8, 0.2)}, and it is denoted as \{\(x, y\)| x=2.8 and y=0.2\}.

Key Concepts

Solving Systems of Equations

When dealing with algebra, you'll often encounter systems of equations, which are just sets of two or more equations that you solve at the same time. The goal is to find the values of the variables that satisfy all the equations simultaneously. One of the methods to solve these systems is called the substitution method. This involves rearranging one of the equations to express one variable in terms of the others and then substituting this expression into the remaining equations. This process can eliminate variables and make it easier to find the solution.

As with most algebraic methods, the key to success with the substitution method is careful and precise manipulation of the equations. Once you substitute the expression from one equation into another, you need to simplify and solve for the remaining variable. Understanding how to effectively combine these steps is essential in solving systems of equations efficiently.

Isolation of Variables

The isolation of variables is a foundational technique used in algebra to solve equations for a specific variable. When you isolate a variable, you're essentially rearranging the equation so that the variable you're solving for is by itself on one side of the equals sign, with all other terms on the opposite side. This step is often necessary in solving systems of equations, especially when using the substitution method.

To isolate a variable, you might have to perform a series of algebraic operations such as adding, subtracting, multiplying, dividing, or distributing. For example, to isolate y in the equation \( -4x + y = -11 \), you would add 4x to both sides, resulting in \( y = 4x - 11 \). This new expression for y can then be used in other equations involved in the system, allowing you to solve for the other variables.

Set Notation in Algebra

Set notation in algebra is a way to concisely express a set of numbers that possess a certain property or condition. In the context of systems of equations, set notation is a formal way of presenting the solution set. For instance, after using the substitution method to get the solutions for x and y, we can express the solutions in set notation.

To write the solution \( x = 2.8 \) and \( y = 0.2 \) in set notation, you would enclose the ordered pair in curly brackets like this: \( \{ (2.8, 0.2) \} \) or use a vertical bar to describe the variables and their respective solutions: \( \{ (x, y) | x=2.8 \text{ and } y=0.2 \} \). Set notation is a standardized language in mathematics that allows for the clear and concise communication of sets.

Linear Equations Solution

The solution of linear equations, or finding the value of the variables that make the equation true, is a fundamental concept in algebra. When you are dealing with a single linear equation, such as \( -4x + y = -11 \), there are infinitely many solutions as you can pick any value for x and find a corresponding y. However, when you have a system of linear equations—two or more equations—in the case of our example, the solution is the point where the graphs of the equations intersect.

The process of finding this point involves either graphing the equations and seeing where the lines cross, or using algebraic methods like substitution or elimination. The substitution method, illustrated by the steps in solving the original exercise, effectively reduces the system to a single variable equation, allowing for a precise and exact solution. The power in solving linear equations lies in understanding these methods and knowing when and how to apply them.