Solving Equations with Variables on Both Sides

Introduction

Key Concepts

Understanding Equations with Variables on Both Sides

An equation with variables on both sides is one where the unknown variable appears in more than one term on both sides of the equality sign. For example: $$3x + 5 = 2x + 12$$ Solving such equations involves isolating the variable on one side to determine its value.

Step-by-Step Method to Solve

1. Identify the Variable Terms: Locate all terms containing the variable on both sides of the equation.
2. Collect Like Terms: Use addition or subtraction to gather all variable terms on one side and constant terms on the opposite side.
3. Simplify the Equation: Combine like terms to simplify both sides of the equation.
4. Isolate the Variable: Divide or multiply to solve for the variable.
5. Verify the Solution: Substitute the found value back into the original equation to ensure its validity.

Example 1: Simple Linear Equation

Solve for \( x \): $$3x + 5 = 2x + 12$$

Solution:

• Subtract \( 2x \) from both sides: $$3x - 2x + 5 = 12$$ $$x + 5 = 12$$
• Subtract 5 from both sides: $$x = 7$$
• Verification: $$3(7) + 5 = 2(7) + 12$$ $$21 + 5 = 14 + 12$$ $$26 = 26 \quad \checkmark$$

Example 2: Equation Involving Fractions

Solve for \( y \): $$\frac{2y}{3} + 4 = \frac{y}{2} + 10$$

Solution:

• Find a common denominator to eliminate fractions. The least common multiple of 3 and 2 is 6. Multiply each term by 6: $$6 \times \frac{2y}{3} + 6 \times 4 = 6 \times \frac{y}{2} + 6 \times 10$$ $$4y + 24 = 3y + 60$$
• Subtract \( 3y \) from both sides: $$4y - 3y + 24 = 60$$ $$y + 24 = 60$$
• Subtract 24 from both sides: $$y = 36$$
• Verification: $$\frac{2(36)}{3} + 4 = \frac{36}{2} + 10$$ $$24 + 4 = 18 + 10$$ $$28 = 28 \quad \checkmark$$

Handling Brackets in Equations

When dealing with equations that contain brackets, apply the distributive property to eliminate them before isolating the variable.

Example:

• Solve for \( x \): $$2(x + 3) = 4x - 6$$
• Distribute the 2: $$2x + 6 = 4x - 6$$
• Subtract \( 2x \) from both sides: $$6 = 2x - 6$$
• Add 6 to both sides: $$12 = 2x$$
• Divide by 2: $$x = 6$$
• Verification: $$2(6 + 3) = 4(6) - 6$$ $$18 = 24 - 6$$ $$18 = 18 \quad \checkmark$$

Equations with Variables on Both Sides and Fractions

Combining variables on both sides with fractional coefficients requires careful manipulation to clear fractions before solving.

Example:

• Solve for \( z \): $$\frac{5z}{4} - 2 = \frac{3z}{2} + 6$$
• Multiply each term by 4 to eliminate denominators: $$5z - 8 = 6z + 24$$
• Subtract \( 5z \) from both sides: $$-8 = z + 24$$
• Subtract 24 from both sides: $$z = -32$$
• Verification: $$\frac{5(-32)}{4} - 2 = \frac{3(-32)}{2} + 6$$ $$-40 - 2 = -48 + 6$$ $$-42 = -42 \quad \checkmark$$

Special Cases and Considerations

• No Solution: If simplifying leads to a false statement, such as \( 0 = 5 \), the equation has no solution.
• Infinite Solutions: If simplifying results in a true statement for all values of the variable, such as \( 0 = 0 \), the equation has infinitely many solutions.
• Variables on Both Sides: Always aim to collect all variable terms on one side to simplify the solving process.

Equations with Multiple Variables

While this topic primarily focuses on single-variable equations, the principles can extend to systems of equations with multiple variables. Solving such systems often involves combining equations to eliminate one variable and solve for the others sequentially.

Applications in Real Life

Understanding how to solve equations with variables on both sides is essential in various real-life scenarios, such as:

• Financial Planning: Determining break-even points where costs equal revenues.
• Physics: Balancing forces in equilibrium problems.
• Engineering: Calculating dimensions and materials needed for construction projects.

Common Mistakes to Avoid

• Forgetting to distribute multiplication over addition or subtraction when brackets are present.
• Incorrectly combining like terms, leading to errors in isolating the variable.
• Neglecting to verify the solution by substituting it back into the original equation.
• Mismanaging negative signs during the simplification process.

Tips for Success

• Always perform the same operation on both sides of the equation to maintain equality.
• Carefully check each step to avoid calculation errors.
• Practice with a variety of problems to build confidence and proficiency.
• Use inverse operations systematically to isolate the variable.

Advanced Techniques

For more complex equations involving exponents or higher-degree polynomials, additional techniques such as factoring or using the quadratic formula may be necessary after isolating the variable terms.

Comparison Table

Summary and Key Takeaways