If 3t 7 5t Then 6t

If 3t + 7 = 5t, Then 6t = ? Solving Algebraic Equations and Their Applications

This article delves into the solution of the algebraic equation 3t + 7 = 5t, ultimately determining the value of 6t. We'll explore the step-by-step process of solving this type of equation, discuss the underlying mathematical principles, and even explore real-world applications where similar equations might arise. This comprehensive guide is designed to be accessible to a wide range of readers, from those just beginning their algebra journey to those looking for a refresher on fundamental algebraic concepts.

Understanding the Equation: 3t + 7 = 5t

The equation 3t + 7 = 5t is a linear equation in one variable, 't'. This means that the highest power of the variable 't' is 1. Solving this equation involves isolating the variable 't' on one side of the equation to find its value.

Key Concepts:

Variables: The letter 't' represents an unknown value, called a variable. Our goal is to determine the numerical value of this variable.
Constants: The numbers '7' are constants; they are fixed values that do not change.
Coefficients: The numbers '3' and '5' are coefficients; they are the numbers multiplying the variable 't'.
Equality: The equals sign (=) indicates that the expression on the left-hand side (LHS) is equal to the expression on the right-hand side (RHS).

Solving the Equation Step-by-Step

To solve for 't', we need to manipulate the equation using algebraic rules. The primary goal is to isolate 't' on one side of the equation. Here's the step-by-step process:

Subtract 3t from both sides: This eliminates '3t' from the LHS.

3t + 7 - 3t = 5t - 3t

This simplifies to:

7 = 2t
Divide both sides by 2: This isolates 't'.

7/2 = (2t)/2

This simplifies to:

t = 3.5 or t = 7/2

Therefore, the solution to the equation 3t + 7 = 5t is t = 3.5 or t = 7/2.

Finding the Value of 6t

Now that we know the value of 't', we can easily find the value of 6t by substituting the value of 't' into the expression:

6t = 6 * 3.5 = 21

Alternatively:

6t = 6 * (7/2) = 21

Therefore, 6t = 21.

Verification: Checking Our Solution

It's crucial to verify our solution. We can do this by substituting t = 3.5 back into the original equation:

3t + 7 = 5t

3(3.5) + 7 = 5(3.5)

10.5 + 7 = 17.5

17.5 = 17.5

Since the equation holds true, our solution, t = 3.5, is correct.

Real-World Applications of Linear Equations

Linear equations, like the one we've solved, are incredibly common in various real-world scenarios. Here are a few examples:

1. Calculating Costs and Profits

Imagine a small business that produces handmade crafts. Let's say the cost of producing each craft is $3 (represented by 3t), and there's a fixed overhead cost of $7 (represented by +7). If the selling price of each craft is $5 (represented by 5t), the equation 3t + 7 = 5t represents the point where the revenue equals the total cost. Solving for 't' would tell the business owner how many crafts need to be sold to break even.

2. Determining Speed and Distance

In physics, the relationship between distance (d), speed (s), and time (t) is often expressed as d = st. Suppose a car travels a certain distance at a speed of 3 units per time unit, and another car travels the same distance at 5 units per time unit. If the slower car starts 7 units ahead, you could express the situation with an equation similar to 3t + 7 = 5t to calculate the time it takes for the faster car to catch up.

3. Analyzing Financial Growth

Linear equations are used extensively in finance to model growth. For example, if an investment grows at a constant rate over time, the total value after a certain period can be modeled using a linear equation.

Advanced Concepts and Extensions

While the equation 3t + 7 = 5t is relatively straightforward, the principles involved extend to more complex algebraic scenarios. Here are some related concepts:

1. Solving Equations with Multiple Variables

More complex problems may involve multiple variables and require more advanced techniques like substitution or elimination to find solutions.

2. Inequalities

Instead of an equals sign, we might have an inequality (>, <, ≥, ≤). Solving inequalities involves similar steps to solving equations, but with some crucial differences in handling the inequality symbols.

3. Systems of Equations

Several linear equations can form a system of equations, requiring techniques such as graphing, substitution, or elimination to find the values of the variables that satisfy all equations simultaneously.

4. Quadratic Equations

Equations where the highest power of the variable is 2 are called quadratic equations. Solving these requires different methods, such as factoring, the quadratic formula, or completing the square.

Conclusion: Mastering the Fundamentals

The seemingly simple equation 3t + 7 = 5t provides a foundation for understanding and solving more complex algebraic problems. By mastering the fundamental principles of solving linear equations, you equip yourself with valuable skills applicable to various fields, from finance and physics to engineering and computer science. Remember to practice regularly and explore more complex examples to build your confidence and expertise in algebra. The ability to solve these types of equations is essential for success in numerous academic and professional pursuits. Understanding how these simple equations reflect real-world situations allows you to apply this knowledge effectively and solve practical problems efficiently.