Systems of Equations Formula

The Formula

For \begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}: x = \frac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1}

When to use: Where two lines cross—the point that satisfies both equations.

Quick Example

x + y = 5 \quad \text{and} \quad x - y = 1 \to x = 3, \; y = 2

Notation

Systems are written with a brace: \begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}

What This Formula Means

Two or more equations sharing the same variables, where the solution must satisfy all equations simultaneously.

Where two lines cross—the point that satisfies both equations.

Formal View

A linear system A\mathbf{x} = \mathbf{b} with A \in \mathbb{R}^{m \times n} has solution set S = \{\mathbf{x} \in \mathbb{R}^n \mid A\mathbf{x} = \mathbf{b}\}. S is nonempty iff \mathrm{rank}(A) = \mathrm{rank}([A \mid \mathbf{b}]); |S| = 1 iff additionally \mathrm{rank}(A) = n.

Worked Examples

Example 1

easy
Solve the system: x + y = 10 and x - y = 4.

Solution

  1. 1
    Add the two equations to eliminate y: (x+y)+(x-y) = 10+4, giving 2x = 14.
  2. 2
    Solve for x: x = 7.
  3. 3
    Substitute back into x + y = 10: 7 + y = 10, so y = 3.
  4. 4
    Check in second equation: 7 - 3 = 4 ✓

Answer

x = 7, \quad y = 3
The elimination method adds or subtracts equations to remove one variable. This works well when coefficients of one variable are equal (or opposites).

Example 2

medium
Solve the system: y = 2x + 1 and 3x + y = 11.

Example 3

hard
Solve the system: 2x + 3y = 12 and x - y = 1.

Common Mistakes

  • Forgetting that a system can have no solution (parallel lines) or infinitely many (same line)
  • Making arithmetic errors during elimination — especially with negative coefficients
  • Solving for one variable but forgetting to substitute back to find the other

Why This Formula Matters

Systems of equations model situations with multiple constraints — budgeting with multiple expenses, mixing solutions in chemistry, or finding where supply meets demand in economics. They are fundamental to engineering, physics, and data science.

Frequently Asked Questions

What is the Systems of Equations formula?

Two or more equations sharing the same variables, where the solution must satisfy all equations simultaneously.

How do you use the Systems of Equations formula?

Where two lines cross—the point that satisfies both equations.

What do the symbols mean in the Systems of Equations formula?

Systems are written with a brace: \begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}

Why is the Systems of Equations formula important in Math?

Systems of equations model situations with multiple constraints — budgeting with multiple expenses, mixing solutions in chemistry, or finding where supply meets demand in economics. They are fundamental to engineering, physics, and data science.

What do students get wrong about Systems of Equations?

Choose the right method: graphing, substitution, or elimination.

What should I learn before the Systems of Equations formula?

Before studying the Systems of Equations formula, you should understand: linear functions, solving linear equations.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Solving Systems of Equations: Substitution, Elimination, and Matrices →
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