Solve The Equation 3x 1 10x X 4

Solving the Equation: 3x + 1 = 10x - x + 4

This comprehensive guide delves into the process of solving the algebraic equation 3x + 1 = 10x - x + 4. We'll break down the steps involved, explain the underlying mathematical principles, and explore potential variations and applications of similar equations. This article is designed to be helpful for students learning algebra, as well as anyone looking to refresh their skills in solving linear equations.

Understanding the Equation

Before we begin solving, let's analyze the equation: 3x + 1 = 10x - x + 4. This is a linear equation, meaning the highest power of the variable 'x' is 1. Linear equations represent a straight line when graphed. Our goal is to find the value of 'x' that makes the equation true.

Key Concepts

• Variables: The 'x' in the equation represents an unknown value. Our task is to isolate 'x' to find its numerical value.
• Constants: The numbers 1 and 4 are constants. They are fixed values and don't change.
• Coefficients: The numbers 3, 10, and -1 are coefficients. They are the numerical factors multiplying the variable 'x'.
• Equation: An equation shows that two expressions are equal. The equals sign (=) separates the left-hand side (LHS) from the right-hand side (RHS).

Solving the Equation Step-by-Step

The process of solving this linear equation involves manipulating the equation to isolate the variable 'x'. We'll achieve this using basic algebraic operations. Here's a step-by-step approach:

Step 1: Simplify the Right-Hand Side (RHS)

The RHS of the equation, 10x - x + 4, can be simplified by combining like terms. Both 10x and -x are terms containing the variable 'x'.

10x - x = 9x

Therefore, the simplified RHS becomes: 9x + 4

The equation now looks like this: 3x + 1 = 9x + 4

Step 2: Isolate the Variable 'x'

Our aim is to have all terms with 'x' on one side of the equation and all constant terms on the other. To achieve this, we can subtract 3x from both sides:

3x + 1 - 3x = 9x + 4 - 3x

This simplifies to:

1 = 6x + 4

Step 3: Isolate the Constant Term

Next, we need to isolate the term with 'x'. We can do this by subtracting 4 from both sides:

1 - 4 = 6x + 4 - 4

This simplifies to:

-3 = 6x

Step 4: Solve for 'x'

Finally, we need to solve for 'x' by dividing both sides of the equation by the coefficient of 'x', which is 6:

-3 / 6 = 6x / 6

This simplifies to:

x = -1/2 or x = -0.5

Therefore, the solution to the equation 3x + 1 = 10x - x + 4 is x = -1/2 or x = -0.5.

Verification of the Solution

It's always a good practice to verify the solution by substituting the value of 'x' back into the original equation. Let's check if x = -1/2 satisfies the equation:

3(-1/2) + 1 = 10(-1/2) - (-1/2) + 4

-3/2 + 1 = -5 + 1/2 + 4

-1/2 = -1/2

Since the left-hand side (LHS) equals the right-hand side (RHS), our solution x = -1/2 is correct.

Understanding the Underlying Principles

Solving linear equations relies on the fundamental principles of algebra. These principles ensure that we manipulate the equation without altering its equality. The key principles used in this example are:

• Addition Property of Equality: Adding the same quantity to both sides of an equation maintains the equality.
• Subtraction Property of Equality: Subtracting the same quantity from both sides of an equation maintains the equality.
• Multiplication Property of Equality: Multiplying both sides of an equation by the same non-zero quantity maintains the equality.
• Division Property of Equality: Dividing both sides of an equation by the same non-zero quantity maintains the equality.

Applications of Linear Equations

Linear equations have widespread applications across various fields, including:

• Physics: Calculating velocity, acceleration, and distance.
• Engineering: Modeling relationships between variables in mechanical and electrical systems.
• Economics: Analyzing supply and demand, cost functions, and profit maximization.
• Computer Science: Developing algorithms and solving problems related to optimization and resource allocation.
• Finance: Calculating interest, loan repayments, and investment growth.

Understanding how to solve linear equations is a fundamental skill with broad applicability.

Variations and Extensions

The equation we solved is a basic example. More complex linear equations may involve:

• Fractions: Equations containing fractions require additional steps to clear the denominators.
• Decimals: Equations with decimals can be simplified by multiplying both sides by a power of 10.
• Parentheses: Equations with parentheses require applying the distributive property before solving.
• Multiple Variables: While this example involved only one variable ('x'), some equations involve multiple variables, requiring techniques like substitution or elimination.

Example with Fractions:

Let's consider a similar equation but with fractions:

(1/2)x + 3 = (2/3)x - 1

To solve this, we would first eliminate the fractions by multiplying both sides by the least common multiple (LCM) of the denominators (which is 6 in this case). This would lead to a simpler equation without fractions that can be solved using the steps outlined previously.

Example with Parentheses:

Consider an equation with parentheses:

2(x + 1) = 3x - 5

Here, the distributive property would be used to expand the parentheses before proceeding with the steps to isolate 'x'.

Conclusion

Solving the equation 3x + 1 = 10x - x + 4 involves a systematic approach using basic algebraic operations. By understanding the underlying principles of equality and applying these operations correctly, we can efficiently find the solution, which in this case is x = -1/2. This fundamental skill has far-reaching applications in various fields, making it an essential element of mathematical literacy. Mastering linear equations provides a solid foundation for tackling more complex algebraic problems and mathematical concepts. Remember to always verify your solution by substituting the value back into the original equation. Practice is key to building confidence and proficiency in solving these types of problems.