Equation Solver — Solve Linear Equations with Steps
Solve linear equations with step-by-step solutions. Enter any equation like 2x + 3 = 7 and see every step explained. Includes real-world examples from break-even analysis to budgeting.
What is an Equation?
An equation is a mathematical statement that two expressions are equal. Solving an equation means finding the value(s) of the variable that make the statement true.
Types of Equations
Linear Equations: ax + b = 0
Example: 3x + 7 = 22 → x = 5. Always has exactly one solution.
Quadratic Equations: ax² + bx + c = 0
Methods: factoring, quadratic formula, completing the square. Can have 0, 1, or 2 solutions.
Rational Equations
Contains fractions with variables in denominator — multiply by LCD to clear fractions.
Systems of Equations
Two or more equations with two or more variables. Methods: substitution, elimination, graphing.
The 4-Step Method for Solving Linear Equations
Step 1: Simplify both sides
Combine like terms on each side of the equation. Distribute any coefficients if needed.
Step 2: Move variable terms to one side
Add or subtract to move all terms with x to one side and constants to the other.
Step 3: Isolate the variable
Divide both sides by the coefficient of x to find the value of x.
Step 4: Verify your answer
Substitute your solution back into the original equation to confirm both sides are equal.
Real-World Applications
Budgeting Example
Tutoring charges $50 registration + $35/hour. Budget $225: 35h + 50 = 225 → 35h = 175 → h = 5 hours.
Break-Even Analysis
Fixed costs $2,400, variable $4/mug, sell at $16: 16m = 2400 + 4m → 12m = 2400 → m = 200 mugs/month.
Projectile Motion
Ball thrown at 20 m/s: h = −4.9t² + 20t. Hits ground at t = 0 (start) or t ≈ 4.08 seconds.
Real-World Case Studies
These detailed case studies show how equations model real business and physics problems:
Case Study: Break-Even Analysis
A coffee shop has fixed costs of $2,400/month, variable cost of $4/mug, and sells each mug for $16. How many mugs must they sell to break even?
Revenue = Cost
16m = 2400 + 4m
12m = 2400
m = 200 mugs/month
At 200 mugs, revenue is $3,200 and total costs are $3,200 — exactly breaking even. Above 200 mugs, the shop generates profit. Below 200, it operates at a loss.
Case Study: Projectile Motion
A ball is thrown upward at 20 m/s from ground level. The height equation is h = −4.9t² + 20t. When does it hit the ground?
Set h = 0:
0 = −4.9t² + 20t
0 = t(−4.9t + 20)
t = 0 (launch) or t = 20/4.9 ≈ 4.08 seconds
The ball returns to ground level after approximately 4.08 seconds. Maximum height occurs at the midpoint: t = 10/4.9 ≈ 2.04 seconds, h ≈ 20.4 meters.
Common Mistakes
Forgetting Negative Signs
−(2x − 5) = −2x + 5, not −2x − 5. Always distribute the negative sign to every term inside parentheses.
Losing Solutions When Dividing by Variable
Dividing both sides by x can eliminate valid solutions. Always add/subtract to collect variable terms instead.
Not Checking for Extraneous Solutions
Rational and radical equations can produce solutions that don't work in the original equation. Always verify by substituting back.
Misapplying PEMDAS
When isolating a variable, work in reverse PEMDAS order: addition/subtraction first, then multiplication/division, then exponents.
Equation Solving Benchmarks
Typical difficulty levels and solution counts by equation type.
| Equation Type | Possible Solutions | Difficulty |
|---|---|---|
| Linear (ax + b = 0) | Exactly 1 | Beginner |
| Quadratic (ax² + bx + c = 0) | 0, 1, or 2 | Intermediate |
| Rational (fractions with variables) | 1 or more + extraneous | Intermediate |
| System (2 equations, 2 unknowns) | Exactly 1, none, or infinite | Intermediate |
| Absolute value |ax + b| = c | 0, 1, or 2 | Intermediate |
| Radical (√ax + b = c) | 0 or 1 (+ extraneous check) | Advanced |
Frequently Asked Questions About Equation Solving
How do you solve a linear equation?
To solve a linear equation, isolate the variable on one side. Move all x terms to one side and constants to the other using addition, subtraction, multiplication, or division. For example, for 2x + 3 = 7: subtract 3 from both sides (2x = 4), then divide by 2 (x = 2).
What is a linear equation?
A linear equation is an equation where the variable (like x) has a power of 1. It can be written in the form ax + b = 0, where a and b are constants. Linear equations always have exactly one solution.
How do you verify a solution?
To verify, substitute the solution back into the original equation. If both sides are equal, the solution is correct. For example, if x = 2 is the solution to 2x + 3 = 7: 2(2) + 3 = 4 + 3 = 7.
Can a linear equation have no solution?
Yes. If you get a false statement like 5 = 3 after simplifying, there is no solution. If you get a true statement like 5 = 5, there are infinitely many solutions.
What's the difference between solving and simplifying?
Simplifying reduces an expression to its most compact form. Solving finds the value(s) of the variable that make an equation true. Simplifying doesn't involve an equals sign with a specific value.
What's the first step in solving any equation?
Isolate the variable. Work backward through the order of operations (PEMDAS in reverse) to get the unknown on one side. Start by eliminating addition/subtraction, then multiplication/division, then exponents.
Why do some equations have no solution?
An equation has no solution when the variable cancels out and leaves a false statement (e.g., 0 = 5). This means no value of the variable can make the equation true.
When would I use a system of equations in real life?
Systems of equations appear whenever you have multiple constraints simultaneously — finding the optimal product mix, calculating break-even points, or solving circuit problems. For example, a business might need to find the price and quantity that maximize profit while meeting demand and budget constraints.
Equation Solver Pros & Cons
Pros
- Instant step-by-step solutions
- Automatic answer verification
- Handles linear, quadratic, and rational equations
- Shows work for every step
- Free forever — no subscriptions
- No sign-up or email required
Cons
- No graphing visualization
- Limited to algebraic equations
- No calculus or differential equations
Reviewed by Shahid
Content Reviewer & Calculator SpecialistContent reviewer specializing in marketing, finance, health, and math calculators on GM Calculator.