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In algebra, substitution is a method used to replace a variable with another expression that represents the same value. This is especially useful in scenarios where you're given multiple equations and want to solve for one variable in terms of another. In our exercise, we start with the equation - \(y = 3m - 1\) and the expression \(m = \frac{x+1}{3}\). Our first task is to substitute the expression for \(m\) into the equation for \(y\). This is step one in our example.
By substituting \(m\) with \(\frac{x+1}{3}\), we now rewrite the original equation to: \(y = 3 \bigg (\frac{x+1}{3} \bigg) - 1\). Understanding this concept is crucial as substitution often serves as the bridge to simplify and solve complex algebraic expressions.
The key takeaway is: whenever you're asked to substitute, carefully replace the variable and rewrite the equation.

Solving equations is the process of finding the value(s) of the variable(s) that make the equation true. After substituting, we're left with the equation - \(y= 3 \bigg (\frac{x+1}{3} \bigg) - 1\). Our next step is to solve this equation to find 'y' as a function of 'x'.
We start by simplifying inside the parentheses. Multiply out the 3: \(y = x + 1 - 1\). This operation is a simple multiplication and shows how different steps work together to simplify an equation. Our goal here is to isolate 'y' on one side of the equation, resulting in a simpler, more understandable format.
The important skill here is recognizing how to manipulate and reorganize equations to isolate and solve for a variable.

Simplifying expressions involves combining like terms and reducing the equation to its simplest form. From our equation \(y = x + 1 - 1\), we perform our last step: combining like terms. Here, \(+1\) and \(-1\) cancel each other out, leaving us with \(y = x\). This tells us that 'y' is directly equal to 'x'.
Combining like terms is a fundamental part of algebra that helps in getting the most straightforward form of an equation or expression. Always look to combine terms that are like, which often means they have the same variable raised to the same power.
Ultimately, simplifying expressions makes it easier to understand relationships between variables and solve equations correctly. By understanding how to simplify, you can tackle more complex algebraic problems with confidence.