Linear Equation With One Variable Definition

Linear Equations with One Variable: A Comprehensive Guide

Linear equations are fundamental building blocks in algebra and have far-reaching applications across numerous fields. Understanding them thoroughly is crucial for success in mathematics and related disciplines. This comprehensive guide delves into the definition of linear equations with one variable, explores various methods for solving them, and illustrates their practical applications with real-world examples.

What is a Linear Equation with One Variable?

A linear equation with one variable is an algebraic equation that can be written in the standard form:

ax + b = 0

Where:

• a and b are constants (numbers), and
• x is the variable.

The key characteristic of a linear equation is that the highest power of the variable is 1. This means there are no squared terms (x²), cubed terms (x³), or any other higher powers of the variable. The presence of only the first power of the variable ensures the equation represents a straight line when graphed on a coordinate plane (hence the term "linear"). The equation is called a "one-variable" equation because it contains only one unknown quantity, represented by 'x' (it could also be represented by other letters such as y, z, etc.).

Examples of Linear Equations with One Variable:

• 3x + 5 = 8
• -2x = 10
• x/4 - 7 = 2
• 0.5x + 1 = 3.5

Examples that are NOT Linear Equations with One Variable:

• x² + 2x - 3 = 0 (Contains x²)
• √x + 1 = 4 (Contains a square root of x)
• 2x + 3y = 6 (Contains two variables, x and y)
• 1/x + 5 = 7 (x is in the denominator)

Solving Linear Equations: A Step-by-Step Approach

The primary goal when working with a linear equation is to isolate the variable, meaning to get the variable (x in our standard form) by itself on one side of the equation. This is achieved by applying algebraic manipulations, adhering to the fundamental principle that any operation performed on one side of the equation must also be performed on the other to maintain the equality.

Here's a systematic approach to solving linear equations:

1. Simplify Both Sides: Combine like terms on each side of the equation. This involves adding or subtracting similar terms. For example, in the equation 2x + 5 - x = 8, you would simplify the left side to x + 5 = 8.

2. Isolate the Variable Term: Use addition or subtraction to move the constant terms to one side of the equation and the variable term to the other side. In x + 5 = 8, subtract 5 from both sides to obtain x = 3.

3. Solve for the Variable: If the variable is multiplied by a coefficient (a number in front of the variable), divide both sides of the equation by that coefficient. For instance, if 2x = 6, divide both sides by 2 to get x = 3. If the variable is divided by a coefficient, multiply both sides by that coefficient. For example, if x/3 = 4, multiply both sides by 3 to get x = 12.

4. Check Your Solution: Substitute the solution back into the original equation to verify that it satisfies the equation. If the left side equals the right side after substitution, your solution is correct.

Worked Examples:

Example 1: Solve 3x + 5 = 14

1. Subtract 5 from both sides: 3x = 9
2. Divide both sides by 3: x = 3
3. Check: 3(3) + 5 = 14 (9 + 5 = 14) This is true, so x = 3 is the correct solution.

Example 2: Solve -2x + 7 = 1

1. Subtract 7 from both sides: -2x = -6
2. Divide both sides by -2: x = 3
3. Check: -2(3) + 7 = 1 (-6 + 7 = 1) This is true, so x = 3 is the correct solution.

Example 3: Solve (x/2) - 3 = 5

1. Add 3 to both sides: x/2 = 8
2. Multiply both sides by 2: x = 16
3. Check: (16/2) - 3 = 5 (8 - 3 = 5) This is true, so x = 16 is the correct solution.

Example 4: Solve 5x - 2 = 3x + 8

1. Subtract 3x from both sides: 2x - 2 = 8
2. Add 2 to both sides: 2x = 10
3. Divide both sides by 2: x = 5
4. Check: 5(5) - 2 = 3(5) + 8 (25 - 2 = 15 + 8) (23 = 23). This is true, so x = 5 is the correct solution.

Dealing with Fractions and Decimals

Solving linear equations involving fractions or decimals follows the same principles but may require extra steps to simplify the equation first.

Fractions: Find the least common denominator (LCD) of all the fractions in the equation and multiply both sides of the equation by the LCD. This will eliminate the fractions.

Decimals: You can either work with decimals directly or, if you prefer, multiply both sides of the equation by a power of 10 to eliminate the decimals and work with whole numbers.

Example with Fractions: Solve (x/3) + (x/2) = 5

1. Find the LCD (6) and multiply both sides by 6: 6((x/3) + (x/2)) = 6(5) which simplifies to 2x + 3x = 30
2. Combine like terms: 5x = 30
3. Divide both sides by 5: x = 6
4. Check: (6/3) + (6/2) = 5 (2 + 3 = 5) This is true, so x = 6 is the correct solution.

Example with Decimals: Solve 0.2x + 1.5 = 3.1

1. Multiply both sides by 10 to eliminate decimals: 10(0.2x + 1.5) = 10(3.1) which simplifies to 2x + 15 = 31
2. Subtract 15 from both sides: 2x = 16
3. Divide both sides by 2: x = 8
4. Check: 0.2(8) + 1.5 = 3.1 (1.6 + 1.5 = 3.1) This is true, so x = 8 is the correct solution.

Applications of Linear Equations

Linear equations with one variable have a vast array of applications across diverse fields:

• Physics: Calculating speed, distance, and time using the equation distance = speed × time. For example, if you travel at a constant speed of 60 km/h for 2 hours, you can use the linear equation to calculate the distance you've covered.

• Finance: Determining simple interest earned or owed with the equation I = PRT (Interest = Principal × Rate × Time).

• Chemistry: In stoichiometry, calculating the amounts of reactants and products in chemical reactions often involves linear equations.

• Engineering: Modeling various relationships such as stress-strain relationships in materials science.

• Economics: Analyzing supply and demand curves (though these often involve two variables, simplified scenarios might use one).

• Computer Science: In algorithms and data structures, linear equations might be used for estimations or calculations in various applications.

Real-world Example:

Imagine you're saving money to buy a new bicycle that costs $250. You already have $50 saved and plan to save $20 per week. You can use a linear equation to determine how many weeks (x) it will take to save enough money:

50 + 20x = 250

Solving this equation:

1. Subtract 50 from both sides: 20x = 200
2. Divide both sides by 20: x = 10

Therefore, it will take you 10 weeks to save enough money to buy the bicycle.

Conclusion

Linear equations with one variable are a fundamental concept in algebra, providing a powerful tool for solving a wide range of problems. Mastering the techniques for solving these equations is crucial for further progress in mathematics and its applications. Remember to always check your solutions by substituting them back into the original equation to ensure accuracy. The systematic approach outlined in this guide, along with practice, will build your confidence and skills in tackling linear equations and their diverse real-world applications. The more you practice, the more intuitive the process becomes, allowing you to quickly and accurately solve even complex-looking equations.