Problem 5 - Elementary Algebra Within Reach

Question

In Exercises 5-14, solve the system by the method of substitution. $$ \left\\{\begin{array}{l} x=4 y-5 \\ x=3 y \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 15, y = 5\).

Step 1: Equate the expressions for \(x\)

Since both equations express \(x\), set them equal to each other.\nHence, \(4y - 5 = 3y\)

Step 2: Isolate \(y\)

Given the equation \(4y - 5 = 3y\), we need to manipulate this equation in a way that will allow us to solve for \(y\). To achieve this, we can subtract \(3y\) from both sides. This leaves us with \(y - 5 = 0\). Adding 5 to both sides, we then get \(y = 5\).

Step 3: Substitute \(y\) back to find \(x\)

Now that we have found \(y = 5\), we can substitute \(y\) back into one of the original equations to find \(x\). For example, using the second equation, where \(x = 3y\), if we substitute \(y = 5\), this gives us \(x = 3*5 = 15\). Thus, the solution to the system of equations is \(x = 15, y = 5\).

Key Concepts

Solving Systems of Equations

To solve systems of equations like the one provided in the exercise, we need to find values for the variables that satisfy all equations simultaneously. In this case, we have a system with two equations involving the variables \(x\) and \(y\). The goal is to determine specific values where both equations are true at the same time.
One reliable way to solve systems of equations is by the substitution method. This involves expressing one of the variables in terms of the other, which simplifies the process. With the system given, both equations already express \(x\) in terms of \(y\):

• \(x = 4y - 5\)
• \(x = 3y\)

Easy right? By recognizing these expressions, we equate them to simplify further. This step is crucial to transition into obtaining a single equation that will solve for one variable at a time. Let's remember, our purpose for solving systems is to find a common solution for \(x\) and \(y\) that works in both equations.

Isolation of Variables

Isolation of variables is a critical step in simplifying and solving equations. Once we equate the two expressions for \(x\), we get: \(4y - 5 = 3y\). Here, our task is to isolate the variable \(y\) to solve for its value. This process involves rearranging the equation to have \(y\) alone on one side.
The steps to isolate \(y\) are straightforward:

• Subtract \(3y\) from both sides to simplify the equation to \(y - 5 = 0\).
• Add 5 to both sides to get \(y = 5\).

By isolating \(y\), we've simplified the system and made it possible to find a concrete numerical solution for \(y\). Understanding the isolation process is hugely beneficial, as it is applied in myriad scenarios outside of just solving equations, such as simplifying complex expressions or inequalities.

Step-by-Step Solution

Following a step-by-step approach helps build a clear path to solving complex problems. After isolating \(y\) and finding \(y = 5\), our next journey is discovering the value of \(x\). By substituting the value of \(y\) into any one of the original equations, we find \(x\). Here’s how it plays out:

• Start with an equation that contains both \(x\) and \(y\), let's use \(x = 3y\).
• Substitute \(y = 5\) into this equation, giving \(x = 3 \times 5 = 15\).

With \(x = 15\) and \(y = 5\), we've arrived at the solution to the system of equations. This step-by-step method is advantageous because it simplifies the task into more manageable parts. It offers a scaffold that ensures each logical step is correctly followed, resulting in an accurate and reliable solution.