Problem 67 - Algebra 1: Concepts and Skills

Question

Use mental math to solve the equation. \(t+6=10\)

Step-by-Step Solution

Answer

The solution to the equation \(t + 6 = 10\) is \(t = 4\).

Step 1: Understand the equation

The equation provided is \(t + 6 = 10\). This means that some number \(t\), when added to 6, gives the result 10.

Step 2: Isolate the variable

To find the value of \(t\), we must isolate \(t\). We do this by 'undoing' the addition in the initial equation. The inverse operation of addition is subtraction. So, to cancel out the '+6' on the left side, we do the same thing on both sides of the equation: we subtract 6. This gives us \(t + 6 - 6 = 10 - 6\). Simplifying that gives us \(t = 4\).

Step 3: Check the solution

It is a good practice to verify our solution. We replace \(t\) with 4 in the original equation: \(4 + 6\). This simplifies to 10, which is the right hand side of the original equation, hence our solution is correct.

Key Concepts

Solving Equations

Solving equations is all about finding the unknown value. In this case, we're looking for the value of \( t \) that will make both sides of the equation equal. The equation given is \( t + 6 = 10 \). This means we have a starting number \( t \), and when we add 6 to it, the result should be 10.
To solve the equation, we need to figure out what \( t \) is. This process involves isolating the variable, which means getting \( t \) by itself on one side of the equation. It's important to understand what the equation is telling us: a number, when increased by 6, gives us 10.
The key is to "undo" what's being done to \( t \) using mathematical operations, which in this case leads us to the next step of inverse operations.

Inverse Operations

Inverse operations are mathematical functions that can "undo" each other. They are vital when solving equations, as they help us isolate the variable. For the given equation \( t + 6 = 10 \), the operation on \( t \) is addition.

To solve for \( t \), we need to do the opposite of adding 6. The inverse of addition is subtraction. So, we subtract 6 from both sides of the equation. Here's how it works:

• Subtract 6 from the left side: \( t + 6 - 6 \)
• Subtract 6 from the right side: \( 10 - 6 \)

This simplifies to \( t = 4 \). Subtraction effectively cancels out the addition, leaving \( t \) alone on one side, which helps us find the solution easily. Understanding inverse operations allows us to tackle various equations with confidence.

Checking Solutions

Checking solutions is an essential step to ensure that the answer we found is correct. It works like quality control for math problems. After solving \( t + 6 = 10 \) and finding \( t = 4 \), it's time to verify this result.

To check, we substitute \( t = 4 \) back into the original equation:

• Original equation: \( t + 6 = 10 \)
• Substitute \( t = 4 \): \( 4 + 6 \)

This simplifies to 10, which matches the right side of the equation: \( 4 + 6 = 10 \). This confirms that our solution is correct.
This verification step helps guarantee accuracy and builds confidence in solving more complex problems in the future. Always take a moment to check your work.