System of Equations Calculator

This article explains how a System of Equations Calculator works, including the mathematical definition, solution formulas, worked examples, common mistakes, and practice problems with solutions. It is designed as companion content for an online system of equations calculator.

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What Is a System of Equations?

A system of equations is a collection of two or more equations that share the same variables. The goal is to find values of the variables that satisfy all equations simultaneously.

A general linear system can be written as:

$$ \begin{cases} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = b_1 \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = b_2 \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n = b_m \end{cases} $$

In matrix form, this system is written compactly as:

$$ A\mathbf{x} = \mathbf{b} $$

where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the vector of variables, and \(\mathbf{b}\) is the constant vector.

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Worked Examples

Example 1: Unique Solution

Solve the system:

$$ \begin{cases} 5x + 2y = 10 \\ 7x + 2y = 14 \end{cases} $$

Show Solution

Subtract the first equation from the second: \[ (7x+2y) - (5x+2y) = 14-10 \Rightarrow 2x=4 \Rightarrow x=2 \] Substitute \(x=2\) into the first equation: \[ 5*2 + 2y = 10 \Rightarrow 10 + 2y = 10 \Rightarrow 2y = 0 \Rightarrow y=0 \] Solution: \[ (x,y)=(2,0) \]

Example 2: Unique Solution

Solve the system:

$$ \begin{cases} 3x + y = 7 \\ 5x + y = 11 \end{cases} $$

Show Solution

Subtract the first equation from the second: \[ (5x+y)-(3x+y)=11-7 \Rightarrow 2x=4 \Rightarrow x=2 \] Substitute \(x=2\) into the first equation: \[ 3*2 + y =7 \Rightarrow 6+y=7 \Rightarrow y=1 \] Solution: \[ (x,y)=(2,1) \]

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Common Mistakes When Solving Systems of Equations

• Arithmetic errors during elimination or substitution.
• Forgetting to check whether equations are dependent or inconsistent.
• Using the matrix inverse method when the determinant is zero.
• Stopping before reaching reduced row echelon form.

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Practice Problems

Try solving the following systems using your system of equations calculator. The answers are hidden below each problem.

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