Problem 22 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 7 and 8 . \(8 t+5=5\)

Step-by-Step Solution

Answer

The solution is \(t = 0\). Substituting back verifies the solution is correct.

Step 1: Isolate the variable term

Start by isolating the variable term on one side of the equation. In the equation \(8t + 5 = 5\), we need to eliminate the \(+5\) on the left side. Subtract \(5\) from both sides to maintain the balance of the equation:\[8t + 5 - 5 = 5 - 5\]This simplifies to:\[8t = 0\]

Step 2: Solve for the variable

Now, solve for \(t\) by isolating it fully. Since \(t\) is being multiplied by \(8\), divide both sides by \(8\):\[\frac{8t}{8} = \frac{0}{8}\]This results in:\[t = 0\]

Step 3: Check the solution

Substitute \(t = 0\) back into the original equation to verify if it holds true:\[8(0) + 5 = 5\]Simplify the left side:\[0 + 5 = 5\]Since both sides are equal, \(t = 0\) is indeed the correct solution.

Key Concepts

Solving Equations

When tackling linear equations, it's important to understand what solving means. Solving an equation involves finding the value of the variable that makes the equation true.
To do this successfully, you need to perform operations that simplify the equation, such as addition, subtraction, multiplication, or division.
These operations must be applied to both sides of the equation to keep it balanced, much like a scale that needs even weight on both sides to stay level.In our example, the equation began as \(8t + 5 = 5\).
The first step was to simplify it by eliminating the constant on the same side as the variable.
This allowed us to clearly see the variable term \(8t\), which we could then work to isolate.

Isolating Variables

Isolating the variable is a crucial step in solving equations. It's like peeling away the layers to reveal the core unknown that you're trying to uncover.
In the equation \(8t + 5 = 5\), the term \(8t\) is our variable term, which is surrounded by other numbers and operations. To isolate \(t\), we first subtracted 5 from both sides, removing the constant attached to the variable term and simplifying the equation to \(8t = 0\).
With the variable term alone, it's easier to see how to make it equal to the answer.The next step was to isolate \(t\) completely by dividing both sides of the equation by 8.

• This action undid the multiplication of \(t\) by 8, leaving us the solution \(t = 0\).
• It's always important to remember that the same operation must be done to both sides of the equation to keep it balanced.

This principle of balancing maintains the truth of the original equation throughout the process of solving it.

Checking Solutions

After solving an equation, checking your solution ensures accuracy. This step is all about verification.
You substitute the found value back into the original equation to confirm it satisfies the equation.In our example, we found that \(t = 0\).
We then checked it by substituting \(0\) into the original equation \(8t + 5 = 5\), which transformed to \(8(0) + 5 = 5\).Simplifying gave us \(0 + 5 = 5\), and since both sides are indeed equal, the solution is verified as correct.

• This step reassures us that we've solved the equation correctly and that no mistakes were made in our arithmetic.
• Checking solutions, although it might seem tedious, is crucial for error-free math problem-solving.

It not only confirms the solution but also provides an opportunity to catch any potential errors early on.