Problem 56 - Prealgebra

Question

Solve each equation. $$\frac{t}{3}+5=2$$

Step-by-Step Solution

Answer

The solution to the equation is \(t = -9\).

Step 1: Isolate the Term Involving the Variable

First, we want to isolate the term that contains the variable \(t\). To do this, subtract 5 from both sides of the equation: \[\frac{t}{3} + 5 - 5 = 2 - 5\]This simplifies to: \[\frac{t}{3} = -3\]

Step 2: Eliminate the Fraction by Multiplying

Now that we have the fraction \(\frac{t}{3} = -3\), we want to get rid of the fraction to solve for \(t\). Multiply both sides of the equation by 3:\[3 \times \frac{t}{3} = 3 \times (-3)\]This simplifies to:\[t = -9\]

Step 3: Verify the Solution

To ensure our solution is correct, substitute \(t = -9\) back into the original equation:\[\frac{-9}{3} + 5 = 2\]Calculating \(\frac{-9}{3}\) gives \(-3\). So, we check:\[-3 + 5 = 2\]This verifies that our solution is correct.

Key Concepts

Understanding Algebra in Linear Equations

Algebra is a powerful branch of mathematics that helps us solve equations by finding unknown variables. For linear equations, we aim to balance the equation by performing the same operation on both sides. This rule helps maintain the equation's equality. In a linear equation like \(\frac{t}{3} + 5 = 2\), we want to find out what value of \(t\) makes this equation true.
To solve an algebraic equation, we need to understand a few key operations:

• Adding or subtracting a number from both sides
• Multiplying or dividing both sides by the same number

These operations allow us to manipulate the equation without changing its balance. Solving algebraically often involves a series of steps where the main task is to simplify the equation until we isolate the unknown variable.

Isolating the Variable

Isolating the variable means getting the variable by itself on one side of the equation. In our example, the equation is \(\frac{t}{3} + 5 = 2\). Here, the variable \(t\) is part of a fraction and added to a constant (5).
To isolate \(\frac{t}{3}\), we first need to eliminate the constant on its side. We do this by subtracting 5 from both sides, leading to:\[\frac{t}{3} = -3\]
This operation is a crucial first step in solving linear equations because it paves the way for further simplification and eventually revealing the value of the solution.

Eliminating Fractions

Fractions can sometimes make equations look more complex than they actually are. The aim is to simplify the equation by getting rid of the fraction. In our step, we have \(\frac{t}{3} = -3\).
To eliminate the fraction, we multiply both sides by the denominator, which, in this case, is 3. This action effectively cancels out the division on the left side, simplifying to:\[t = -9\]
This step of eliminating fractions is crucial because it transitions the equation from a fractional form that is often harder to understand into a simple linear format that reveals the value of the unknown variable.

Verifying the Solution

After calculating the value of \(t\), it is important to ensure our solution is correct. We do this by substituting \(t = -9\) back into the original equation \(\frac{t}{3} + 5 = 2\).
Substitute and simplify:\[\frac{-9}{3} + 5 = 2\]
The calculation for \(\frac{-9}{3}\) results in \(-3\), leading to the equation:\[-3 + 5 = 2\]
Since both sides of the equation equal 2, we confirm that our solution \(t = -9\) is indeed correct. Verifying solutions ensures that our steps were accurate and that the variable holds the correct value within the original equation's context.