System of Equations Solver

A system of equations (also called simultaneous equations) is a set of two or more equations with the same variables that must all be satisfied at the same time. The solution is the set of values that makes every equation true simultaneously.

A system of two linear equations in two unknowns has the form:

$\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$

Geometrically, each equation represents a line in the plane. The solution is the point where the lines intersect.

A system can have:

• One unique solution: The lines intersect at exactly one point (consistent and independent).
• No solution: The lines are parallel (inconsistent).
• Infinitely many solutions: The lines are identical (consistent and dependent).

Systems of equations appear in countless applications: mixing problems, circuit analysis, supply and demand equilibrium, traffic flow, and optimization. Larger systems with 3+ variables arise in engineering and data science.

1. Substitution Method

Solve one equation for one variable, then substitute into the other equation.

Example: Solve $\begin{cases} x - y = 1 \\ 2x + 3y = 7 \end{cases}$

1. From equation 1: $x = y + 1$
2. Substitute into equation 2: $2(y + 1) + 3y = 7$
3. $2y + 2 + 3y = 7$ → $5y = 5$ → $y = 1$
4. Back-substitute: $x = 1 + 1 = 2$

2. Elimination Method

Add or subtract equations to eliminate one variable.

Example: Solve $\begin{cases} 2x + 3y = 7 \\ x - y = 1 \end{cases}$

1. Multiply equation 2 by 3: $3x - 3y = 3$
2. Add to equation 1: $5x = 10$ → $x = 2$
3. Substitute back: $2 - y = 1$ → $y = 1$

3. Matrix Method (Gaussian Elimination)

Write the system as an augmented matrix and row-reduce:

$\begin{pmatrix} 2 & 3 & | & 7 \\ 1 & -1 & | & 1 \end{pmatrix} \rightarrow \begin{pmatrix} 1 & 0 & | & 2 \\ 0 & 1 & | & 1 \end{pmatrix}$

4. Cramer's Rule

For a $2 \times 2$ system, if $D = a_1 b_2 - a_2 b_1 \neq 0$:

$x = \frac{c_1 b_2 - c_2 b_1}{D}, \quad y = \frac{a_1 c_2 - a_2 c_1}{D}$

5. Graphing

Plot each equation and identify the intersection point.

• Incorrect substitution: When substituting an expression, replace the variable everywhere it appears and use parentheses.
• Multiplying only part of an equation: When multiplying to eliminate, every term (including the constant) must be multiplied.
• Losing track of signs: Be extra careful with negative coefficients during elimination.
• Declaring no solution prematurely: Getting $0 = 0$ means infinitely many solutions (dependent system), not no solution. Only $0 = c$ (where $c \neq 0$) means no solution.
• Forgetting to find all variables: After finding one variable, always substitute back to find the others.