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Chapter 3: Problem 22

Solve the equation. $$-16=\frac{t}{4}-14$$

Short Answer

Expert verified
t = -8

Step by step solution

01

Isolate the variable term

To isolate the term involving the variable, add 14 to both sides of the equation. This eliminates the constant term on the right side. \[-16 + 14 = \frac{t}{4} - 14 + 14\] Simplifying, we get: \[-2 = \frac{t}{4}\]
02

Solve for the variable

Multiply both sides of the equation by 4 to solve for \(t\). This step will help us isolate \(t\). \[-2 \times 4 = \frac{t}{4} \times 4\] Simplifying, we get: \[t = -8\]

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Key Concepts

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

Isolating the Variable
When solving a linear equation, the first goal is often to isolate the variable term. This means we need to get the variable (like \(t\) or \(x\)) by itself on one side of the equation.

Let's look at our example: $$-16 = \frac{t}{4} - 14$$ Here, the variable term is \frac{t}{4}, and we need to move the constant term -14 to the other side of the equation.

To do this, we perform the opposite operation. Since -14 is being subtracted, we add 14 to both sides: $$-16 + 14 = \frac{t}{4} - 14 + 14$$

This simplifies to: $$-2 = \frac{t}{4}$$. Now, we have isolated the variable term!
Adding Constants
Adding constants is a common step to help isolate the variable when solving equations.

In our example, the constant term -14 was on the same side as the variable term. To move it, we added its opposite (which is +14) to both sides of the equation. When you add the same number to both sides, you maintain the balance of the equation.

Here is the operation again for clarity: $$-16 + 14 = \frac{t}{4} - 14 + 14$$ This simplifies to: $$-2 = \frac{t}{4}$$. By adding constants, we made it easier to isolate and solve for the variable!
Multiplying to Solve for the Variable
After isolating the variable term, we often need to perform an additional operation to solve for the variable itself.

In our example, we have: $$-2 = \frac{t}{4}$$. Here, the variable t is divided by 4.
To isolate t completely, multiply both sides by 4 (the opposite of division): $$-2 \times 4 = \frac{t}{4} \times 4$$. <\br>This simplifies to: $$t = -8$$.

Multiplying both sides by 4 allowed us to eliminate the fraction, leaving t by itself. This simple operation gave us the solution to the equation!

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