Systems of two linear equations

Systems of two linear equations consist of two equations with two variables (usually $x$ and $y$). Solving a system means finding the ordered pair $(x, y)$ that satisfies both equations simultaneously, representing the geometric intersection of their lines on a coordinate plane.

Algebraic Methods for Solving Systems

Systems can be solved algebraically using two main techniques depending on how the equations are structured.

• Substitution Method: Solves one equation for a single variable, then substitutes that expression into the second equation.

  Example: Given $y = 2x - 1$ and $3x + y = 9$, substitute $(2x - 1)$ into the second equation:

  $$3x + (2x - 1) = 9 \implies 5x - 1 = 9 \implies 5x = 10 \implies x = 2$$

  Substitute $x = 2$ back to find $y$: $y = 2(2) - 1 = 3$. Solution: $(2, 3)$.

• Elimination Method: Multiplies one or both equations by constants so that adding or subtracting them cancels out one variable.

  Example: Given $2x + 3y = 12$ and $4x - 3y = 6$, add the two equations to eliminate $y$:

  $$(2x + 3y) + (4x - 3y) = 12 + 6 \implies 6x = 18 \implies x = 3$$

  Substitute $x = 3$ back into $2(3) + 3y = 12 \implies 6 + 3y = 12 \implies y = 2$. Solution: $(3, 2)$.

Structural Analysis of Solutions (Intersections)

The number of solutions to a linear system in two variables depends on the graphical relationship between the two lines:

• One Solution (Independent/Consistent): Occurs when the lines have different slopes ($m_1 \neq m_2$). The lines intersect at a single point $(x, y)$.

  Example: $y = 2x + 1$ and $y = -x + 4$ intersect at $(1, 3)$.

• No Solution (Inconsistent): Occurs when the lines have equal slopes ($m_1 = m_2$) but different $y$-intercepts ($b_1 \neq b_2$). The lines are parallel and never intersect.

  Example: $y = 3x + 2$ and $y = 3x - 5$ yield the false statement $2 = -5$ when set equal.

• Infinitely Many Solutions (Dependent/Consistent): Occurs when the lines have equal slopes ($m_1 = m_2$) and equal $y$-intercepts ($b_1 = b_2$). The equations represent the exact same line, meaning every point on the line is a solution.

  Example: $2x + 4y = 8$ and $x + 2y = 4$ simplify to identical equations ($0 = 0$).

Modeling Systems in Context

Systems are frequently used to model scenarios with two unknown quantities governed by two distinct constraints (such as cost and quantity).

• Quantity Constraint: Sets up an equation based on the count or total number of items: $x + y = \text{total count}$.

  Example: A mix of $x$ adult tickets and $y$ child tickets totaling $150$ tickets $\implies x + y = 150$.

• Value/Cost Constraint: Sets up an equation based on individual values or rates: $Ax + By = \text{total value}$.

  Example: Adult tickets cost $\$10$ and child tickets cost $\$6$, generating $\$1,200$ in total sales $\implies 10x + 6y = 1200$.

1. Use elimination when both equations are in standard form ($Ax + By = C$).

   Example: For $3x + 2y = 11$ and $5x - 2y = 13$, add them directly to get $8x = 24 \implies x = 3$.

2. Use substitution when one equation already isolates a variable (e.g., $y = \dots$).

   Example: For $y = 4x - 2$ and $2x + 3y = 8$, plug $(4x - 2)$ directly into the second equation for $y$.

3. Compare slope ratios ($-\frac{A}{B}$) to quickly check for "No Solution" or "Infinitely Many.

   Example: For $2x + 3y = 5$ and $4x + 6y = k$, the left sides match ($2 \times 2 = 4$ and $3 \times 2 = 6$). If $k = 10$, there are infinitely many solutions. If $k \neq 10$, there is no solution.

4. Watch for questions asking for target expressions rather than $x$ or $y$ individually.

   Example: If $x + y = 7$ and $x - y = 3$, what is $x^2 - y^2$? Shortcut: Recognize $x^2 - y^2 = (x + y)(x - y) = 7 \times 3 = 21$ without solving for $x$ and $y$.

5. Check boundary points and intersections in system word problems to ensure realistic domain constraints.

   Example: In ticket pricing problems, both $x \ge 0$ and $y \ge 0$ must hold true; negative ticket counts are extraneous solutions.