Introduction

Systems of linear equations appear everywhere in mathematics and the real world.
At this difficulty level, we focus on the simplest geometric idea behind them:

• A linear equation in two variables represents a line.
• A linear equation in three variables represents a plane.
• A system of such equations asks: Where do these geometric objects meet?

This article builds intuition using simple algebra and clear step‑by‑step reasoning.

What Is a Linear Equation?

A linear equation is one where:

• Variables appear only to the first power.
• Variables are not multiplied together.
• The graph is a straight line (in 2D) or a plane (in 3D).

Examples:

• In two variables: $2x + 3y = 6$
• In three variables: $x - y + 2z = 4$

Key features:

• The coefficients (the numbers in front of $x$, $y$, $z$) determine the slope or tilt.
• The constant term determines where the line or plane sits in space.

Systems of Linear Equations

A system is simply a collection of linear equations considered together.

Example (two equations, two variables):

• $x + y = 5$
• $x - y = 1$

A solution is a pair $(x,y)$ that satisfies both equations at the same time.

Systems can have:

• One solution (lines intersect at a point)
• No solution (lines are parallel)
• Infinitely many solutions (lines lie on top of each other)

Geometric Interpretation in Two Dimensions

Each equation in two variables corresponds to a line.

When solving a system:

• If the lines cross, the crossing point is the solution.
• If the lines are parallel, they never meet → no solution.
• If the lines are the same line, every point on the line is a solution.

Visual intuition:

• Changing the constant term shifts the line up or down.
• Changing the coefficients changes the slope.

Example:

• $x + y = 4$
• $x + y = 6$

These lines have the same slope but different intercepts → parallel → no solution.

Geometric Interpretation in Three Dimensions

In three variables, each equation represents a plane.

Two planes can:

• Intersect in a line
• Be parallel
• Be the same plane

Three planes can:

• Meet at a single point (one solution)
• Meet in a line (infinitely many solutions)
• Have no common intersection (no solution)
• Coincide in more complex ways (e.g., two planes intersect, the third is parallel)

Example:

Consider:

• $x + y + z = 6$
• $x - y + z = 2$
• $2x + z = 4$

These three planes intersect at exactly one point.

Solving Systems Algebraically

Even though geometry gives intuition, we still solve systems using algebra.

Common methods:

• Substitution
  Solve one equation for a variable and plug it into the other.
• Elimination
  Add or subtract equations to eliminate a variable.
• Graphing
  Useful for simple systems or visual intuition.

Example using elimination:

Solve:

• $x + y = 5$
• $x - y = 1$

Add the equations:

• $(x + y) + (x - y) = 5 + 1$
• $2x = 6$
• $x = 3$

Substitute back:

• $3 + y = 5$
• $y = 2$

Solution: $(3,2)$

Calculator

Exercises

1. Solve the system:
   $x + y = 7$
   $x - y = 1$

   Solution

   Add the equations:
   $(x + y) + (x - y) = 7 + 1$
   $2x = 8$ → $x = 4$
   Then $4 + y = 7$ → $y = 3$
   Solution: $(4,3)$

2. Determine whether the lines $2x + y = 4$ and $4x + 2y = 8$ intersect, are parallel, or coincide.

   Solution

   The second equation is exactly twice the first.
   This means the lines lie on top of each other.
   They coincide (infinitely many solutions).

3. Solve the system in three variables:
   $x + y + z = 6$
   $x - y + z = 2$
   $2x + z = 4$

   Solution

   Subtract the second equation from the first: $$(x + y + z) - (x - y + z) = 6 - 2$$ $$2y = 4$ → $y = 2$$ Use $2x + z = 4$ and $x - y + z = 2$:
   Substitute $y = 2$: $$x - 2 + z = 2$ → $x + z = 4$$ Now solve the system: $$x + z = 4$$ $$2x + z = 4$$ Subtract: $$(2x + z) - (x + z) = 4 - 4$$ $$x = 0$$ Then $$z = 4 - x = 4$$.

   Solution: $(0,2,4)$

4. Describe in words what it means for two lines to have no solution.

   Solution

   Two lines have no solution when they are parallel and never meet.
   They have the same slope but different intercepts.

5. Solve the system:
   $3x - y = 5$
   $6x - 2y = 10$

   Solution

   The second equation is just twice the first.
   This means the system has infinitely many solutions (same line).

6. Find the intersection point of the lines:
   $y = 2x + 1$
   $y = -x + 4$

   Solution

   Set the right‑hand sides equal: $$2x + 1 = -x + 4$$ $$3x = 3$ → $x = 1$$ Then $y = 2(1) + 1 = 3$
   Solution: $(1,3)$

7. Determine whether the planes
   $x + z = 3$
   $2x + 2z = 6$
   represent the same plane or different parallel planes.

   Solution

   The second plane is exactly twice the first.
   They represent the same plane, not different ones.