Problem 41 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 4. $$ 60 t-50=15 t-5 $$

Step-by-Step Solution

Answer

The solution is \(t = 1\).

Step 1: Move Terms with 't' to One Side

To solve for 't', we first need to get all the terms involving 't' on one side of the equation. Start by subtracting \(15t\) from both sides:\[60t - 50 - 15t = 15t - 5 - 15t\]Simplifying both sides gives us:\[45t - 50 = -5\]

Step 2: Move Constant Terms to the Other Side

Next, move the constant term (-50) to the right side of the equation by adding 50 to both sides:\[45t - 50 + 50 = -5 + 50\]This simplifies to:\[45t = 45\]

Step 3: Solve for 't'

Now, solve for 't' by dividing both sides by the coefficient of 't', which is 45:\[\frac{45t}{45} = \frac{45}{45}\]Thus, we find:\[t = 1\]

Step 4: Check the Solution

To verify that \(t = 1\) is the correct solution, substitute it back into the original equation:\[60(1) - 50 = 15(1) - 5\]Calculate both sides:\[60 - 50 = 15 - 5\]\[10 = 10\]Since both sides equal, the solution \(t = 1\) is confirmed correct.

Key Concepts

Equation Simplification

Equation simplification is a fundamental step in solving linear equations. To simplify an equation, you need to reduce it to its simplest form, making it easier to identify the solution. Simplification often involves:

• Combining like terms
• Clearing fractions
• Performing basic arithmetic operations

In the given exercise, we simplified the original equation by combining like terms on both sides. Once we moved the terms with the variable to one side, we simplified the equation by subtracting similar terms:\[60t - 50 - 15t = -5\] This further simplified to \[45t - 50 = -5\], showing how simplification helps isolate the terms that include variables.

Checking Solutions

After finding a solution for a variable, it's crucial to check if that solution is correct. This step ensures that no arithmetic errors were made while solving the equation. To check solutions, substitute the found value back into the original equation and verify if both sides are equal.
In our exercise, after determining that \(t = 1\), you substitute it back:\[60(1) - 50 = 15(1) - 5\] Simplifying both sides gives:\[10 = 10\] Since both sides are equal, it confirms that our solution \(t = 1\) is correct. Checking solutions is a vital step in verifying your work and reinforcing your understanding.

Variable Isolation

Variable isolation involves manipulating an equation so that the variable we want to solve is alone on one side of the equation. This is a critical step in the process of solving linear equations. Here's how variable isolation happens:

• Move terms containing the variable to one side
• Move constant terms to the opposite side
• Simplify whenever possible

In our problem, we isolated \(t\) by first moving all terms containing \(t\) to the left side of the equation, yielding:\[45t = 45\] This form makes it easy to divide both sides by 45 to solve for \(t\). Successfully isolating the variable can significantly simplify solving the equation.

Basic Algebra Steps

Mastering basic algebra steps is essential for solving equations effectively. These steps build the foundation for working with more complex problems. The basic steps usually include:

• Identifying the equation type
• Simplifying expressions
• Transposing terms to isolate variables
• Performing operations on both sides to maintain equality

In the given example, we used these steps to solve for \(t\). We started with identifying that this is a linear equation. Then, we simplified by combining like terms, moved terms with \(t\) to one side, and finally confirmed our solution by substituting it back into the original equation. Understanding and practicing these basic steps will make tackling algebra problems more straightforward for students.