Linear Equation Solver
Solve linear equations and systems of linear equations with step-by-step solutions
Enter a linear equation in the form ax + b = cx + d
Equation: \(0 = 0\)
What is a Linear Equation?
A linear equation is an equation where each term is either a constant or the product of a constant and a single variable raised to the power of 1.
The standard form of a linear equation in one variable is:
\(ax + b = 0\)
where \(a\) and \(b\) are constants, and \(a \neq 0\).
Systems of Linear Equations
A system of linear equations consists of two or more linear equations with the same variables. For example, a system of two equations with two variables:
\(a_1x + b_1y = c_1\)
\(a_2x + b_2y = c_2\)
The solution to a system of linear equations is the set of values for the variables that satisfies all equations in the system simultaneously.
Methods for Solving Systems
There are several methods to solve systems of linear equations:
- Substitution Method: Solve for one variable in terms of others, then substitute.
- Elimination Method: Add or subtract equations to eliminate a variable.
- Cramer's Rule: Use determinants to find the values of variables.
- Matrix Method: Use matrix operations to solve the system.
Examples
Single Equation Example
Solve: \(3x + 5 = 2x - 7\)
Step 1: Rearrange to standard form
\(3x + 5 = 2x - 7\)
\(3x - 2x = -7 - 5\)
\(x = -12\)
2×2 System Example
Solve the system:
\(2x + y = 5\)
\(3x - 2y = 4\)
Using elimination method:
Multiply the first equation by 2: \(4x + 2y = 10\)
Add to the second equation: \(7x = 14\)
Therefore, \(x = 2\)
Substitute back: \(2(2) + y = 5\)
Therefore, \(y = 1\)
Solution: \(x = 2, y = 1\)
3×3 System Example
Solve the system:
\(x + y + z = 6\)
\(2x - y + z = 3\)
\(x + 2y - z = 2\)
Using elimination and substitution:
From equations 1 and 3: \(2z = 4\), so \(z = 2\)
Substitute into equation 1: \(x + y = 4\)
Substitute into equation 2: \(2x - y = 1\)
Solving these two equations: \(x = \frac{5}{3}, y = \frac{7}{3}\)
Solution: \(x = \frac{5}{3}, y = \frac{7}{3}, z = 2\)
Frequently Asked Questions
What's the difference between a linear and a quadratic equation?
A linear equation contains variables raised only to the power of 1 (e.g., \(ax + b = 0\)), while a quadratic equation contains at least one variable raised to the power of 2 (e.g., \(ax^2 + bx + c = 0\)). Linear equations graph as straight lines, while quadratic equations graph as parabolas.
Can a system of linear equations have no solution?
Yes, a system of linear equations can have no solution if the equations are inconsistent. Graphically, this means the lines representing the equations are parallel and never intersect. Such a system is called an inconsistent system.
Can a system of linear equations have infinitely many solutions?
Yes, a system of linear equations can have infinitely many solutions if the equations are dependent. Graphically, this means the lines representing the equations are the same line. In this case, any point on the line is a solution to the system.
What is the best method to solve a system of linear equations?
The best method depends on the specific system:
- For simple 2×2 systems, substitution or elimination works well.
- For systems with coefficients that make elimination easy, the elimination method is often fastest.
- For larger systems (3×3 or more), Gaussian elimination or matrix methods are more systematic.
- For systems with special patterns, Cramer's rule can be elegant.
Our calculator uses the most efficient method based on the specific system you enter.
Real-World Applications of Linear Equations
Physics
Describing motion, forces, and relationships between physical quantities. Used in kinematics, thermodynamics, and electrical circuits.
Engineering
Modeling circuits, structures, and systems. Essential for structural analysis, control systems, and signal processing.
Economics
Analyzing supply and demand, cost functions, and economic models. Used in budgeting, forecasting, and market analysis.
Computer Graphics
Representing lines, planes, and transformations. Fundamental for 3D rendering and computer-aided design.
Finance
Calculating interest, investments, and loan repayments. Used in portfolio optimization and risk assessment.
Data Analysis
Performing linear regression and modeling relationships between variables. Key for predictive analytics and machine learning.