Solving 2t + 5 = 15 How To Find The Value Of T Step By Step

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This article aims to provide a comprehensive guide on how to solve the equation 2t + 5 = 15 for the variable t. We will break down the process step-by-step, ensuring a clear understanding of the underlying mathematical principles. This will not only help you solve this specific equation but also equip you with the skills to tackle similar algebraic problems. The correct answer among the options provided (A) 5, B) 10, C) 15, D) 20) will be identified and explained in detail.

Understanding the Basics of Algebraic Equations

Before we dive into the solution, it's crucial to grasp the fundamental concepts of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions can involve constants, variables, and mathematical operations. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. In our case, the equation 2t + 5 = 15 contains one variable, t, and our objective is to determine the value of t that satisfies the equation.

The core principle behind solving equations is maintaining the balance. Imagine an equation as a balanced scale. To keep the scale balanced, any operation performed on one side of the equation must also be performed on the other side. This principle ensures that the equality remains valid throughout the solution process. Common operations used in solving equations include addition, subtraction, multiplication, and division. We will apply these operations strategically to isolate the variable t on one side of the equation.

Another important concept is the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). While PEMDAS is crucial for simplifying expressions, we often apply it in reverse when solving equations. This means we typically address addition and subtraction before multiplication and division. This approach helps us systematically isolate the variable. In the equation 2t + 5 = 15, we will first address the addition of 5 before dealing with the multiplication of 2 by t.

Step-by-Step Solution of the Equation 2t + 5 = 15

To solve the equation 2t + 5 = 15, we will follow a systematic approach involving inverse operations to isolate the variable t. This process will reveal the value of t that makes the equation a true statement.

Step 1: Isolate the term with the variable

Our initial goal is to isolate the term containing the variable t, which in this case is 2t. To do this, we need to eliminate the constant term, +5, from the left side of the equation. We can achieve this by performing the inverse operation of addition, which is subtraction. Subtracting 5 from both sides of the equation maintains the balance and allows us to isolate 2t.

2t + 5 - 5 = 15 - 5

Simplifying both sides, we get:

2t = 10

Now, the equation is simplified to 2t = 10, where the term with the variable is isolated on the left side.

Step 2: Isolate the variable

The next step is to isolate the variable t itself. Currently, t is being multiplied by 2. To isolate t, we need to perform the inverse operation of multiplication, which is division. Dividing both sides of the equation by 2 will eliminate the coefficient 2 from the left side and give us the value of t.

(2t) / 2 = 10 / 2

Simplifying both sides, we get:

t = 5

Therefore, the solution to the equation 2t + 5 = 15 is t = 5.

Verification of the Solution

It's always a good practice to verify the solution to ensure its accuracy. To verify our solution, we substitute the value of t we found (t = 5) back into the original equation and check if it satisfies the equation.

Original equation: 2t + 5 = 15

Substitute t = 5:

2(5) + 5 = 15

Simplify:

10 + 5 = 15

15 = 15

The equation holds true when t = 5. This confirms that our solution is correct.

Conclusion: The Value of t

In conclusion, by following a step-by-step approach involving inverse operations, we have successfully solved the equation 2t + 5 = 15 and found the value of t. The correct answer is:

A) 5

This process demonstrates the fundamental principles of solving algebraic equations, emphasizing the importance of maintaining balance and applying inverse operations to isolate the variable. Understanding these concepts will empower you to solve a wide range of algebraic problems.

Solving the equation 2t + 5 = 15 is a prime example of solving a linear equation. Linear equations are algebraic equations where the highest power of the variable is 1. They are fundamental in mathematics and have numerous applications in various fields, including science, engineering, and economics. This section will delve deeper into the characteristics of linear equations and the general strategies for solving them.

Characteristics of Linear Equations

Linear equations, also known as first-degree equations, have a distinctive form. They can be generally represented as:

ax + b = c

Where:

  • x represents the variable.
  • a, b, and c are constants, with a not equal to 0.

The key characteristic of a linear equation is that the variable x is raised to the power of 1. There are no terms involving x squared (x^2), x cubed (x^3), or any other higher powers. This simplicity allows for straightforward solution methods.

The graph of a linear equation is always a straight line. This graphical representation provides a visual understanding of the equation's solutions. The solution to a linear equation corresponds to the point where the line intersects the x-axis (in the case of a single-variable equation). Understanding the graphical representation can be helpful in visualizing the solutions and interpreting the equation's behavior.

General Strategies for Solving Linear Equations

Solving linear equations involves isolating the variable on one side of the equation. This is achieved by applying inverse operations systematically. The general strategy can be summarized as follows:

  1. Simplify both sides of the equation: Combine like terms and eliminate any parentheses or brackets.
  2. Isolate the term with the variable: Use addition or subtraction to move constant terms to the side of the equation opposite the variable term.
  3. Isolate the variable: Use multiplication or division to eliminate the coefficient of the variable.
  4. Verify the solution: Substitute the obtained value of the variable back into the original equation to check if it satisfies the equation.

This general strategy can be applied to a wide range of linear equations, regardless of their complexity. The key is to maintain balance by performing the same operations on both sides of the equation and to apply inverse operations strategically.

Example of Solving Another Linear Equation

Let's consider another example to illustrate the general strategy for solving linear equations:

Solve the equation: 3y - 7 = 8

  1. Simplify both sides: Both sides are already simplified.

  2. Isolate the term with the variable: Add 7 to both sides:

    3y - 7 + 7 = 8 + 7

    3y = 15

  3. Isolate the variable: Divide both sides by 3:

    (3y) / 3 = 15 / 3

    y = 5

  4. Verify the solution: Substitute y = 5 back into the original equation:

    3(5) - 7 = 8

    15 - 7 = 8

    8 = 8

    The equation holds true, confirming that y = 5 is the correct solution.

Common Mistakes to Avoid

When solving linear equations, it's crucial to avoid common mistakes that can lead to incorrect solutions. Some common errors include:

  • Not maintaining balance: Failing to perform the same operation on both sides of the equation.
  • Incorrectly applying inverse operations: Using the wrong operation to isolate the variable.
  • Arithmetic errors: Making mistakes in calculations, especially with negative numbers.
  • Forgetting to distribute: Not properly distributing a number across parentheses.
  • Not verifying the solution: Skipping the verification step, which can catch errors.

By being mindful of these common mistakes and practicing regularly, you can improve your accuracy and confidence in solving linear equations.

Linear equations are not just abstract mathematical concepts; they have numerous real-world applications. They are used to model various situations and solve practical problems in diverse fields. Understanding these applications can enhance your appreciation for the usefulness of linear equations.

Applications in Science and Engineering

In science and engineering, linear equations are used to model physical phenomena, analyze data, and design systems. Some examples include:

  • Ohm's Law: This fundamental law in electronics relates voltage (V), current (I), and resistance (R) in a circuit using a linear equation: V = IR. This equation can be used to calculate any of the three variables if the other two are known.
  • Kinematics: Linear equations are used to describe the motion of objects in physics. For example, the equation d = vt relates distance (d), velocity (v), and time (t) for an object moving at a constant velocity.
  • Material Science: Linear equations can model the stress-strain relationship in materials, allowing engineers to predict how materials will behave under load.
  • Chemical Reactions: Linear equations can be used to balance chemical equations, ensuring that the number of atoms of each element is the same on both sides of the equation.

Applications in Economics and Finance

Linear equations are also widely used in economics and finance to model economic relationships and make financial decisions. Some examples include:

  • Supply and Demand: Linear equations can represent supply and demand curves, which show the relationship between the price of a product and the quantity supplied or demanded. The intersection of these curves determines the equilibrium price and quantity.
  • Cost-Benefit Analysis: Linear equations can be used to model costs and benefits in a business setting, helping decision-makers evaluate the profitability of different projects.
  • Simple Interest: The formula for simple interest, I = PRT, is a linear equation that relates interest (I), principal (P), interest rate (R), and time (T). This equation can be used to calculate the interest earned on an investment or the cost of a loan.
  • Budgeting: Linear equations can be used to create budgets and track expenses, helping individuals and organizations manage their finances effectively.

Applications in Everyday Life

Linear equations are also present in many everyday situations, often without us even realizing it. Some examples include:

  • Calculating the cost of a taxi ride: The fare often consists of a fixed initial charge plus a per-mile charge, which can be modeled using a linear equation.
  • Determining the total cost of a purchase: If an item has a fixed price and a sales tax rate, the total cost can be calculated using a linear equation.
  • Converting between temperature scales: The relationship between Celsius and Fahrenheit is linear and can be represented by the equation F = (9/5)C + 32.
  • Calculating the distance traveled: If you know your speed and travel time, you can use the linear equation d = vt to calculate the distance traveled.

Examples of Problem Solving Using Linear Equations

Let's consider a few examples of how linear equations can be used to solve real-world problems:

  1. Problem: A taxi charges a $3 initial fee plus $2 per mile. If a ride costs $15, how many miles was the ride?

    Solution: Let m be the number of miles. The equation is 3 + 2m = 15. Solving for m, we get m = 6 miles.

  2. Problem: A store is selling shirts for $15 each and has a sales tax of 8%. What is the total cost of buying 3 shirts?

    Solution: The cost of 3 shirts is 3 * $15 = $45. The sales tax is 0.08 * $45 = $3.60. The total cost is $45 + $3.60 = $48.60.

  3. Problem: Convert 25 degrees Celsius to Fahrenheit.

    Solution: Using the equation F = (9/5)C + 32, we substitute C = 25: F = (9/5)(25) + 32 = 45 + 32 = 77 degrees Fahrenheit.

These examples demonstrate the versatility of linear equations in solving practical problems. By understanding the principles of linear equations and their applications, you can develop valuable problem-solving skills that are applicable in various contexts.

While the basic techniques for solving linear equations are fundamental, there are more advanced techniques that can be used to solve more complex equations or systems of equations. This section will explore some of these techniques, providing a glimpse into the broader world of equation solving.

Solving Systems of Linear Equations

A system of linear equations is a set of two or more linear equations that share the same variables. Solving a system of equations means finding the values of the variables that satisfy all equations simultaneously. There are several methods for solving systems of linear equations, including:

  • Substitution: This method involves solving one equation for one variable and then substituting that expression into the other equation(s). This reduces the system to a single equation in one variable, which can then be solved. The solution is then substituted back into the original equations to find the values of the other variables.
  • Elimination: This method involves adding or subtracting multiples of the equations to eliminate one of the variables. This also reduces the system to a single equation in one variable. The solution is then substituted back into the original equations to find the values of the other variables.
  • Graphing: This method involves graphing the equations on a coordinate plane. The solution to the system is the point(s) where the graphs intersect. This method is particularly useful for systems of two equations in two variables.
  • Matrix Methods: For larger systems of equations, matrix methods, such as Gaussian elimination and matrix inversion, can be used to solve the system efficiently. These methods are often implemented using computer software.

Solving Quadratic Equations

A quadratic equation is an equation of the form:

ax^2 + bx + c = 0

Where a, b, and c are constants and a is not equal to 0. Quadratic equations have a higher degree of complexity than linear equations and require different solution techniques. Some common methods for solving quadratic equations include:

  • Factoring: This method involves factoring the quadratic expression into two linear factors. The solutions to the equation are then the values of x that make each factor equal to zero.

  • Completing the Square: This method involves manipulating the equation to create a perfect square trinomial on one side. The solutions can then be found by taking the square root of both sides.

  • Quadratic Formula: This formula provides a general solution for any quadratic equation:

    x = (-b ± √(b^2 - 4ac)) / (2a)

The quadratic formula can be used to solve any quadratic equation, regardless of whether it can be factored easily.

Solving Other Types of Equations

Beyond linear and quadratic equations, there are many other types of equations that can be encountered in mathematics and its applications. These include:

  • Polynomial Equations: Equations involving polynomials of degree higher than 2. These equations can be solved using various techniques, including factoring, the rational root theorem, and numerical methods.
  • Rational Equations: Equations involving rational expressions (fractions with polynomials in the numerator and denominator). These equations can be solved by multiplying both sides by the least common denominator to eliminate the fractions.
  • Radical Equations: Equations involving radicals (square roots, cube roots, etc.). These equations can be solved by isolating the radical term and then raising both sides to the appropriate power to eliminate the radical.
  • Exponential and Logarithmic Equations: Equations involving exponential and logarithmic functions. These equations can be solved by using the properties of exponents and logarithms to isolate the variable.
  • Trigonometric Equations: Equations involving trigonometric functions (sine, cosine, tangent, etc.). These equations can be solved by using trigonometric identities and inverse trigonometric functions.

Numerical Methods for Solving Equations

For equations that cannot be solved algebraically, numerical methods can be used to approximate the solutions. These methods involve iterative algorithms that generate a sequence of approximations that converge to the solution. Some common numerical methods include:

  • Newton's Method: An iterative method that uses the derivative of a function to find its roots.
  • Bisection Method: A method that repeatedly divides an interval in half to find a root of a function.
  • Fixed-Point Iteration: A method that rewrites an equation in the form x = g(x) and then iterates the function g(x) to find a fixed point.

Numerical methods are often implemented using computer software and are essential for solving complex equations in science, engineering, and other fields.

Mastering the skill of solving equations is crucial for success in mathematics and its applications. This article has provided a comprehensive guide to solving the equation 2t + 5 = 15, as well as a broader overview of linear equations, their applications, and advanced techniques for solving other types of equations. By understanding the fundamental principles and practicing regularly, you can develop your equation-solving skills and confidently tackle a wide range of mathematical problems. The journey of learning mathematics is continuous, and the ability to solve equations is a powerful tool that will serve you well in your academic and professional pursuits. Remember to always verify your solutions and to seek out additional resources and practice problems to further enhance your understanding.