Solving Systems of Equations: Substitution, Elimination, and Matrices

What Systems of Equations Represent

A system of equations is two or more equations whose variables must be satisfied simultaneously. The solution is a set of values that makes every equation true at once.

Example:

Solving this system means finding the values of x and y that satisfy both equations. Here (3, 1) works: 2(3) + 1 = 7 ✓ and 3 - 1 = 2 ✓.

Systems arise whenever multiple functions must be matched — from finding intersections of lines, to mixing solutions, to solving application problems that involve multiple constraints. Systems involving quadratic equations also appear in calculus and physics.

Graphical Interpretation

Each equation in a 2-variable system graphs as a curve (a line for linear equations). The solution is the point of intersection — where all curves meet.

Three possibilities for linear systems:

• Exactly one solution: the lines cross at one point (different slopes).
• No solution: the lines are parallel (same slope, different intercepts).
• Infinitely many solutions: the lines are identical (same slope AND intercept).

Systems with more variables or nonlinear equations can have more intersection points (e.g., a line and a parabola can meet at 0, 1, or 2 points).

Substitution Method

Substitution works best when one equation is already solved for a variable (or can be easily rearranged). Steps:

1. Solve one equation for one variable.
2. Substitute that expression into the other equation.
3. Solve for the remaining variable.
4. Back-substitute to find the first variable.

Example: Solve

The first equation gives y explicitly. Substitute into the second:

Now back-substitute:

Elimination Method

Elimination adds or subtracts the equations to cancel one variable. It works best when neither equation is easily solvable for a variable.

Example: Solve

The y-coefficients are already opposites. Add the equations:

Back-substitute to find y:

If no coefficient pair is ready to cancel, multiply one (or both) equations through by a constant to create matching coefficients first.

Systems with Three Variables

Systems with three equations in three unknowns are common in physics, economics, and engineering. The strategy: reduce to a 2-variable system by eliminating one variable using two pairs of equations, solve the resulting 2-variable system, then back-substitute.

Example:

Eliminate z from equations 1 and 2 (subtract), then from equations 1 and 3 (add). That gives a 2-variable system in x and y, solve that, then back-substitute for z. (Solution: x = 1, y = 2, z = 3.)

Matrix Method Preview

Any linear system can be written in matrix form Ax = b:

For 2×2 systems, the solution is x = A⁻¹b, where A⁻¹ is the inverse matrix. For larger systems, Gaussian elimination (row reduction) is the standard algorithm.

Matrices become essential once you have 3+ unknowns or need to solve many systems with the same coefficient structure — they're the foundation of linear algebra and the basis for computational solvers used in engineering, graphics, and machine learning.

Applications and Word Problems

Most application problems give two constraints involving two unknowns — a classic setup for a system of equations.

Example (mixing): A theater sells adult tickets for $12 and child tickets for $8. A showing sold 200 tickets for total revenue of $1960. How many of each?

Let a = adult tickets, c = child tickets:

Solve by elimination or substitution: a = 90, c = 110. Always interpret the answer in context — negative or non-integer solutions should be checked against the problem's constraints.

Common Mistakes

Practice Problems

Solve each system using any method. Check your answer by substituting into both equations.

1. \begin{cases} x + y = 10 \\ x - y = 4 \end{cases}
2. \begin{cases} 2x + 3y = 12 \\ 4x - y = 10 \end{cases}
3. \begin{cases} y = 2x + 1 \\ y = -x + 7 \end{cases}
4. \begin{cases} 3x - 2y = 6 \\ 6x - 4y = 12 \end{cases}
5. \begin{cases} x + y = 5 \\ x + y = 8 \end{cases}

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