91Ó°ÊÓ

Substituting values into an equation is like finding a missing piece of a puzzle. When we have an equation with a variable, such as the given equation \(y = 3x + 2\), we can find the value of \(y\) if we know the value of \(x\). This process involves replacing the variable with a specific number—just like how we plug in headphones to listen to music!

When the problem states that \(x = 1\), we replace every \(x\) in our equation with 1, leading us to \(y = 3(1) + 2\). Remember, the order of operations (PEMDAS—Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) still applies when substituting values. This makes our equation simpler and prepares us for the next step—solving it.

Simplifying expressions is akin to cleaning up a messy room; it makes everything more manageable and clearer. When the equation \(y = 3(1) + 2\) results from substituting \(x = 1\), we must simplify it to find \(y\).

Here's our simplification strategy in action: first, perform the multiplication \(3 \times 1\), obtaining the result of 3. Next, add the 2, yielding a neat and tidy \(y = 3 + 2\). Simplifying helps us avoid errors and makes the process of solving equations almost as smooth as sliding down a well-polished slide.

Solving equations is the grand finale of the algebraic show, where we discover the value of the unknown. After substituting values and simplifying, we are left with an expression that needs to be solved.

From \(y = 3 + 2\), we add the numbers to find that \(y\) equals 5. Solving the equation uncovers the mystery of what \(y\) is when \(x\) is 1. It’s like hitting the last note in a song perfectly—it feels satisfying and resolves the melody, or in our case, the math problem.

Algebraic operations are the tools in a mathematician's toolkit. They consist of addition, subtraction, multiplication, and division, and are used not just in algebra but all areas of math.

In our exercise, we used multiplication \(3 \times 1\), and addition \(3 + 2\) to simplify and solve the equation. Mastery of these operations, just like a craftsman perfecting their use of tools, allows us to manipulate equations and unlock all sorts of mathematical doors.