Problem 52 - Prealgebra

Question

Solve each equation. Check your solution. $$3+y=16$$

Step-by-Step Solution

Answer

The solution is \(y = 13\).

Step 1: Isolate the Variable

To solve the equation \(3 + y = 16\), we need to isolate the variable \(y\). Start by subtracting 3 from both sides of the equation:\[3 + y - 3 = 16 - 3\]This simplifies to:\[y = 13\]

Step 2: Verify the Solution

After finding \(y = 13\), substitute \(y = 13\) back into the original equation to check if it satisfies the equation:\[3 + 13 = 16\]This simplifies to:\[16 = 16\]Since both sides of the equation are equal, \(y = 13\) is indeed the correct solution.

Key Concepts

Isolating the Variable

Isolating the variable is a crucial step when solving linear equations. When we say 'isolate the variable,' we mean that we want the variable to be alone on one side of the equation, typically the left side. This helps determine its value.
For the equation \(3 + y = 16\), our goal is to find the value of \(y\). To do this, we need to remove the constant term that is added or subtracted to the variable. Here, we're dealing with the addition of 3 to \(y\).
To isolate \(y\), subtract 3 from both sides of the equation. This keeps the equation balanced. Just imagine a balanced scale: If you remove a weight from one side, you’ll need to remove the same weight from the other side to keep the scale even.

• Step for isolation: \(3 + y - 3 = 16 - 3\)

This subtraction eliminates the number 3 on the side with \(y\), simplifying the equation to \(y = 13\). Now, \(y\) is isolated, and we know its value.

Checking Solutions

Checking your solutions is an essential step to ensure that you didn't make any mistakes while solving the equation. Once you've found a value for the variable, you should substitute it back into the original equation to verify the correctness of your solution.
In this case, after isolating \(y\), we found \(y = 13\). To check if this is correct, substitute \(13\) back into the original equation \(3 + y = 16\):

• Substitute and calculate: \(3 + 13 = 16\)
• Verify: Simplifies to \(16 = 16\)

Since both sides of the equation result in the same number, \(16\), we've confirmed that our solution \(y = 13\) is correct. This verification process helps you feel confident in your solution and ensures accuracy.

Substitution Method

Substitution, especially in this context, refers to replacing the variable in the original equation with the value you've found. This method works hand in hand with checking solutions, acting as a practical step to ensure the variable's value satisfies the equation.
Once you've isolated \(y\) and found that \(y = 13\), the substitution will show that the value truly works. By substituting \(13\) for \(y\) in the original equation, we follow:

• Substitution step: Replace \(y\) with \(13\) in the equation \(3 + y = 16\)
• Calculates to \(3 + 13 = 16\)

This process of substitution confirms that both sides of the equation are equal, proving that \(y = 13\) is a valid solution. By practicing the substitution method, you gain confidence in both solving and verifying linear equations.