Solve the System- Methods for Solving Equations

Why You Need to Know Multiple Equation-Solving Methods

Here's the reality: no single method works for every equation you'll encounter. Some equations solve themselves with basic algebra. Others require factoring tricks. Some need substitution. A few demand the quadratic formula.

Knowing which method to use saves you from wasting 20 minutes on a problem that could've taken 2 minutes with the right approach.

This guide covers the main methods, when they work, and why some are faster than others.

Types of Equations You'll Actually Solve

Equations fall into categories. Each category has preferred solving methods.

• Linear equations — one variable, no exponents (ax + b = c)
• Quadratic equations — one variable, x² term (ax² + bx + c = 0)
• Polynomial equations — higher powers, like x³ or x⁴
• Systems of equations — multiple equations with multiple unknowns
• Non-linear equations — anything involving exponentials, logs, or trig

Solving Linear Equations: The Basics

Linear equations are the starting point. Most people learned this in middle school and forgot it by graduation.

The Balancing Method

Whatever you do to one side, do to the other. That's it.

Example: 3x + 7 = 22

Subtract 7 from both sides: 3x = 15

Divide both sides by 3: x = 5

You can check this by plugging 5 back in: 3(5) + 7 = 15 + 7 = 22 ✓

When This Works

This handles any single-variable linear equation. If your equation has only x to the first power and no x², you're working with a linear equation.

Solving Quadratic Equations: Four Methods

Quadratics trip up more students than any other equation type. Here are your options, ranked by speed when applicable.

1. Factoring

Fastest when it works. You're looking for two numbers that multiply to give the constant term and add to give the coefficient of x.

Example: x² + 5x + 6 = 0

Find two numbers that multiply to 6 and add to 5: 2 and 3

Factor: (x + 2)(x + 3) = 0

Set each factor to zero: x = -2 or x = -3

The catch: many quadratics don't factor nicely into integers.

2. Completing the Square

Works every time. Takes longer but guarantees an answer.

Example: x² + 6x + 5 = 0

Move the constant: x² + 6x = -5

Take half of 6 (which is 3), square it (9), add to both sides: x² + 6x + 9 = 4

Rewrite the left side: (x + 3)² = 4

Take square roots: x + 3 = ±2

Solve: x = -1 or x = -5

3. The Quadratic Formula

The universal backup. Works for every quadratic equation.

For ax² + bx + c = 0:

x = (-b ± √(b² - 4ac)) / 2a

Example: 2x² + 7x - 15 = 0

a = 2, b = 7, c = -15

x = (-7 ± √(49 - 4(2)(-15))) / 2(2)

x = (-7 ± √(49 + 120)) / 4

x = (-7 ± √169) / 4

x = (-7 ± 13) / 4

x = 6/4 = 1.5 or x = -20/4 = -5

Memorize this formula. You'll use it constantly.

4. Graphing

Useful for visual learners or when you need approximate answers. Find where the parabola crosses the x-axis.

Not practical for exact answers unless you have graphing software.

Comparing Quadratic Solving Methods

Solving Systems of Equations

Two equations, two unknowns. Three main approaches.

Substitution

Solve one equation for one variable, plug it into the other.

System:

2x + y = 10

x - y = 2

Solve the second equation for x: x = y + 2

Substitute into the first: 2(y + 2) + y = 10

2y + 4 + y = 10

3y = 6

y = 2

Plug back: x = 2 + 2 = 4

Solution: x = 4, y = 2

Elimination

Add or subtract equations to cancel one variable.

Same system:

2x + y = 10

x - y = 2

Add them: 3x = 12

x = 4

Plug into x - y = 2: 4 - y = 2, so y = 2

Elimination is faster when coefficients line up or can be multiplied to match.

Graphing

Plot both lines, find the intersection point. Works for visual confirmation but isn't precise without a graphing calculator.

Which System Method Should You Use?

• Use substitution when one equation already has a variable isolated, or when coefficients make elimination messy
• Use elimination when variables cancel easily or when you can multiply one equation to create opposite coefficients
• Use graphing for quick estimates or when working with word problems you want to visualize

Getting Started: A Practical How-To

When you face an equation you don't immediately recognize:

1. Identify the type. How many variables? What's the highest power of x?
2. Check for easy paths. Can you factor? Are coefficients set up for elimination?
3. Apply the standard method. Linear = balancing. Quadratic = try factoring first, fall back to quadratic formula. Systems = elimination usually fastest.
4. Check your answer. Plug it back into the original equation. If it doesn't work, you made an arithmetic error.

Common Mistakes That Waste Time

• Spending 10 minutes trying to factor something that doesn't factor cleanly — just use the quadratic formula
• Forgetting to apply operations to both sides of the equation
• Dropping negative signs when distributing
• Not checking solutions in the original equation (extraneous roots exist, especially with fractions or radicals)

When to Move On to Harder Methods

Some equations don't have closed-form solutions. Cubic and quartic equations have formulas, but they're brutal. For polynomials of degree 5 or higher, numerical methods or computer algebra systems are your only practical options.

For trig equations, exponentials, and logarithms, you typically isolate the function first, then apply inverse operations. These often require numerical approximation for real-world values.