Problem 33 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$3 y-2=2 y$$

Step-by-Step Solution

Answer

The solution to the equation is \(y = 2\).

Step 1: Rearrange the equation

Subtract \(2 y\) from both sides of the equation to re-arrange it. This will produce the following equation: \(3 y - 2 - 2 y = 2 y - 2 y\), which simplifies to \(y - 2 = 0\).

Step 2: Solve for y

Add 2 to both sides of the equation \(y-2 = 0\), to isolate \(y\). After performing this operation, we get \(y = 2\).

Step 3: Check the solution

Replace \(y\) with 2 in the original equation \(3 y - 2 = 2 y\) to check the validity of the solution. This produces \(3 * 2 - 2 = 2 * 2\). After simplifying, both sides of the equation result in 4, confirming correct solution.

Key Concepts

Equation Rearrangement

Rearranging an equation is like preparing a puzzle before you solve it. The aim is to make the equation simpler, so it's easier to find the solution. Let's take a look at our exercise:

• We start with the equation: \(3y - 2 = 2y\).
• To make the equation simpler, we move all the terms with \(y\) from one side so that they are organized together.
• Subtract \(2y\) from both sides, resulting in \(3y - 2 - 2y = 2y - 2y\). This simplifies our equation to the much friendlier: \(y - 2 = 0\).

Adjusting the equation in this way is like clearing out the noise, so that our focus is only on finding the value of \(y\). Having \(y\) isolated to one side is key for the next steps.

Isolation of Variable

Isolating the variable is a significant step in solving any equation. Here, our goal is to have the variable \(y\) by itself on one side of the equation:

• We have already rearranged the equation to \(y - 2 = 0\).
• To isolate \(y\), we need to get rid of whatever is subtracted or added to it. Here, we add 2 to both sides to cancel out the \(-2\).
• Doing this gives us \(y = 2\).

When the variable is by itself on one side, like \(y = 2\), it means you have found the solution. This is a satisfying moment, as it confirms that \(y\) has a definite value!

Solution Verification

Always check your work! Verifying the solution ensures the result is correct and reliable. Let's check if \(y = 2\) truly satisfies the original equation:

• Substitute \(y = 2\) back into the original equation: \(3y - 2 = 2y\).
• The left side becomes \(3 \times 2 - 2\), simplifying to \(6 - 2 = 4\).
• The right side is \(2 \times 2\), which is also \(4\).
• Both sides equal 4, confirming that \(y = 2\) is indeed the correct solution.

Verification not only confirms accuracy, but it also builds confidence that you’re solving problems correctly. It's a habit worth forming whenever you solve equations!