Problem 29 - Prealgebra

Question

Solve each equation. Check your solution. $$-56=\frac{t}{9}$$

Step-by-Step Solution

Answer

The solution is \( t = -504 \).

Step 1: Isolate the Variable

To solve for \( t \), multiply both sides of the equation by 9 to clear the fraction:\[-56 \times 9 = \frac{t}{9} \times 9\]This simplifies to:\[t = -504\]

Step 2: Check the Solution

Substitute \( t = -504 \) back into the original equation to verify:\[-56 = \frac{-504}{9}\]Simplifying the right side, we have:\[-504 \div 9 = -56\]Since the original equation holds true with \( t = -504 \), our solution is correct.

Key Concepts

Checking Solutions

After finding a solution to an equation, it's essential to check if it's correct. Think of it as confirming your work. You substitute the solution back into the original equation to see if it satisfies the equation.
In our exercise, we found that \( t = -504 \). To check if this is correct, we substitute \( t \) back into the equation:
\[-56 = \frac{-504}{9}\]
Simplifying the right side, we perform the division of \(-504 \div 9\), which gives us \(-56\).
Because both sides of the equation are equal, our solution \( t = -504 \) is verified as correct.

• Always start by substituting your solution back into the original equation.
• Ensure both sides of the equation are equivalent.
• If they are not, re-evaluate your steps to find where you might have gone wrong.

Checking solutions is a crucial step to ensure accuracy in solving equations.

Isolating Variables

To solve an equation, you often need to get the variable by itself on one side of the equation. This is known as isolating the variable.
By isolating the variable, you aim to undo all operations surrounding the variable. This involves using inverse operations, which are simply the opposite operations:
adding versus subtracting, or multiplying versus dividing.
In our example, the original equation is \(-56 = \frac{t}{9}\). We needed to isolate \(t\) by performing the inverse operation of dividing by 9, which is to multiply by 9:
\[-56 \times 9 = \frac{t}{9} \times 9\]
This step cancels out the division and leaves us with \(t = -504\).

• Identify the operation being used on the variable.
• Apply the inverse operation to both sides of the equation.
• Continue simplifying until the variable is isolated.

Isolating variables is the key step in solving equations efficiently.

Fraction Operations

Fractions often appear in equations, representing parts of a whole. Learning how to handle them methodically makes solving equations easier.
Let's break down how we worked with the fraction in our exercise:
In the equation \(-56 = \frac{t}{9}\), the fraction \(\frac{t}{9}\) indicates that \(t\) is divided by 9. To clear fractions, we multiply by the denominator.

By multiplying both sides by 9, we eliminate the fraction:

• \(-56 \times 9 = \frac{t}{9} \times 9\)

Now, the equation simplifies to \(t = -504\). Consider this whenever you encounter fractions:

• Identify the denominator of the fraction.
• Multiply the entire equation by the denominator.
• Simplify to remove the fraction and isolate the variable.

Mastering fraction operations helps in making algebraic equations more straightforward and less intimidating.