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Chapter 2: Problem 32

Solve each equation, and check your solution. $$ 8 t+5=7 t $$

Short Answer

Expert verified
The solution is \(t = -5\).

Step by step solution

01

Isolate the Variable

First, subtract \(7t\) from both sides of the equation to isolate the variable term on one side.\[8t + 5 - 7t = 7t - 7t\]This simplifies to:\[t + 5 = 0\]
02

Solve for the Variable

Next, subtract 5 from both sides of the equation to solve for \(t\).\[t + 5 - 5 = 0 - 5\]This simplifies to:\[t = -5\]
03

Check the Solution

Substitute \(t = -5\) back into the original equation to verify the solution.\[8(-5) + 5 = 7(-5)\]This simplifies to:\[-40 + 5 = -35\]\[-35 = -35\]Since both sides of the equation are equal, the solution \(t = -5\) is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variable Isolation
Variable isolation is the process of getting the variable by itself on one side of the equation. This helps to simplify solving the equation. In the equation from the example, we start with:

8t + 5 = 7t

To isolate the variable t, we need to get rid of the 7t on the right side of the equation. We can do this by subtracting 7t from both sides. This uses the principle that what you do to one side of the equation, you must also do to the other:

t + 5 = 0

This step is crucial because it makes the next steps easier. Now that we have t all by itself on one side, we can proceed to solve the equation.
Equation Simplification
Equation simplification involves making the equation easier to work with by reducing it to its simplest form. In the example, once we have isolated the variable, we simplify the equation to solve for it. We have:

t + 5 = 0

Next, we need to solve for t. To do that, we subtract 5 from both sides of the equation:

t + 5 - 5 = 0 - 5

This simplifies to:

t = -5

By reducing the equation step-by-step, we make it easier to identify the value of the variable. Simplification is essential for clarity and precision in mathematics.
Solution Verification
Solution verification is the process of confirming that our solution is correct. This ensures accuracy and confidence in our answer. To verify our solution for t = -5, we substitute it back into the original equation:

8(-5) + 5 = 7(-5)

This simplifies to:

-40 + 5 = -35

-35 = -35

Since both sides of the equation are equal, we know that our solution t = -5 is correct. Verification helps prevent errors and confirms that our steps were executed correctly.

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