3.2 Solving Systems of Linear Equations in Two Variables
Key Takeaways
A system of two linear equations in two variables represents two lines whose geometric relationship determines whether the system has one unique solution, no solution, or infinitely many solutions.
The substitution method is algebraically most direct when one variable already has a coefficient of 1 or -1; the elimination method is preferred when equations are arranged in standard form (Ax + By = C).
Inconsistent systems represent parallel lines with identical slopes and different y-intercepts, resulting in a false algebraic contradiction (such as 0 = 9) with no solution.
Dependent systems represent coincident lines with identical slopes and identical y-intercepts, resulting in a true algebraic identity (such as 0 = 0) with infinitely many solutions.
Word problems involving mixture concentrations, dual-price ticket sales, and wind or water currents are modeled by establishing one equation for quantity and a second equation for total value or rate.
3.2 Solving Systems of Linear Equations in Two Variables
Quick Answer: A system of two linear equations in two variables consists of two equations sharing common variables x and y. The solution is the ordered pair (x, y) that satisfies both equations simultaneously—geometrically representing the point where the two lines intersect. Systems are solved algebraically through substitution (isolating a variable and plugging it into the other equation) or elimination (multiplying equations to cancel out one variable when added). Systems are classified as independent (one solution), inconsistent (parallel lines, no solution), or dependent (coincident lines, infinitely many solutions).
The Geometry and Classification of 2 × 2 Linear Systems
Every linear equation in two variables, Ax + By = C, graphs as a straight line in the Cartesian coordinate plane. When two such lines are graphed simultaneously, exactly one of three geometric configurations must occur:
| Classification | Consistency & Dependency | Geometric Behavior | Slopes & Intercepts | Algebraic Result | Solution Set |
|---|---|---|---|---|---|
| Independent | Consistent & Independent | Lines intersect at a single point | Different slopes: m₁ ≠ m₂ | Yields unique values: x = p, y = q | One unique solution: {(p, q)} |
| Inconsistent | Inconsistent | Lines are parallel and never meet | Equal slopes, different y-intercepts: m₁ = m₂, b₁ ≠ b₂ | Yields a false contradiction: e.g., 0 = 8 | No solution: ∅ |
| Dependent | Consistent & Dependent | Lines are coincident (identical line) | Equal slopes, identical y-intercepts: m₁ = m₂, b₁ = b₂ | Yields a true identity: e.g., 0 = 0 | Infinitely many solutions: {(x, y) | Ax + By = C} |
Analytical Method 1: The Substitution Method
The substitution method is most efficient when at least one variable in either equation has a coefficient of 1 or -1, allowing isolation without generating fractional terms.
Step-by-Step Procedure for Substitution:
- Isolate: Select one equation and isolate one variable (e.g., solve for y in terms of x).
- Substitute: Substitute the resulting algebraic expression into the other equation in place of that variable.
- Solve: Solve the resulting single-variable linear equation.
- Back-Substitute: Substitute the obtained numerical value into the isolated expression from Step 1 to solve for the second variable.
- Verify: Check the ordered pair (x, y) in both original equations.
Worked Example: Solving via Substitution
Solve the system: Equation 1: 2x - y = 7 Equation 2: 3x + 4y = 5
Step 1: Isolate y in Equation 1. 2x - y = 7 → -y = -2x + 7 → y = 2x - 7
Step 2: Substitute (2x - 7) into Equation 2. 3x + 4(2x - 7) = 5
Step 3: Solve for x. 3x + 8x - 28 = 5 11x - 28 = 5 11x = 33 x = 3
Step 4: Back-substitute x = 3 into the isolated equation. y = 2(3) - 7 = 6 - 7 = -1
Step 5: Verify the ordered pair (3, -1). In Equation 1: 2(3) - (-1) = 6 + 1 = 7 (True) In Equation 2: 3(3) + 4(-1) = 9 - 4 = 5 (True) The unique solution is (3, -1).
Analytical Method 2: The Elimination (Linear Combination) Method
The elimination method is most advantageous when both equations are written in standard form Ax + By = C, especially when coefficients are integers greater than 1.
Step-by-Step Procedure for Elimination:
- Standard Form: Write both equations in Ax + By = C form, placing like terms in vertical columns.
- Determine Target Variable: Choose a variable to eliminate. Find the least common multiple (LCM) of its coefficients in both equations.
- Multiply Equations: Multiply one or both equations by non-zero constants so that the coefficients of the chosen variable become exact opposites (e.g., +12 and -12).
- Add Equations: Add the two equations vertically. The target variable is eliminated, leaving a single-variable equation.
- Solve & Back-Substitute: Solve for the remaining variable and substitute the result into either original equation to find the other variable.
Worked Example: Solving via Elimination
Solve the system: Equation 1: 3x + 4y = 10 Equation 2: 2x - 3y = 18
Step 1: Choose a variable to eliminate. Let us eliminate y. The coefficients of y are +4 and -3. The LCM of 4 and 3 is 12. We can make the coefficients +12 and -12.
Step 2: Multiply the equations. Multiply Equation 1 by 3: 3 · (3x + 4y) = 3 · (10) → 9x + 12y = 30
Multiply Equation 2 by 4: 4 · (2x - 3y) = 4 · (18) → 8x - 12y = 72
Step 3: Add the two equations vertically. (9x + 12y) = 30
- (8x - 12y) = 72
17x + 0y = 102 17x = 102
Step 4: Solve for x. x = 102 / 17 = 6
Step 5: Back-substitute x = 6 into Equation 1. 3(6) + 4y = 10 18 + 4y = 10 4y = -8 y = -2
Step 6: Verify in Equation 2. 2(6) - 3(-2) = 12 + 6 = 18 (True) The unique solution is (6, -2).
Recognizing Special Systems Algebraically
When performing substitution or elimination, variables can drop out entirely:
- Case 1: Contradiction → Inconsistent System. Consider 2x - 4y = 6 and x - 2y = 5. Multiply the second equation by -2: -2x + 4y = -10. Add to the first equation: (2x - 2x) + (-4y + 4y) = 6 - 10 → 0 = -4. Because 0 = -4 is false, the lines are parallel. There is no solution.
- Case 2: Identity → Dependent System. Consider 3x - 6y = 12 and x - 2y = 4. Multiply the second equation by -3: -3x + 6y = -12. Add to the first equation: (3x - 3x) + (-6y + 6y) = 12 - 12 → 0 = 0. Because 0 = 0 is unconditionally true, the equations represent the exact same line. There are infinitely many solutions.
Applied Systems: Contextual Word Problems
Contextual word problems are modeled through a standard two-equation architecture:
- Quantity Equation: Totals the physical items, volume, or hours: x + y = Total Quantity.
- Value/Rate Equation: Weights each quantity by its unit rate or cost: (Rate₁ · x) + (Rate₂ · y) = Total Value.
Category 1: Mixture and Concentration Problems
Mixture problems combine two substances with different concentrations to produce a target concentration.
- Total Volume Equation: x + y = V_total
- Pure Substance Equation: c₁ · x + c₂ · y = c_target · V_total
Category 2: Dual-Rate Revenue and Ticket Problems
Events selling two tiers of admission (e.g., student vs. general admission) require setting up:
- Headcount Equation: S + G = Total Tickets
- Revenue Equation: (Price_S · S) + (Price_G · G) = Total Revenue
Category 3: Motion with Wind or Water Currents
When an object travels with or against a uniform medium (current c, vehicle speed r in still conditions):
- Downstream / With the wind: Effective speed is (r + c). Distance = (r + c) · t_with.
- Upstream / Against the wind: Effective speed is (r - c). Distance = (r - c) · t_against.
TSIA2 Exam Traps & Strategic Checkpoints
- Trap 1: The Partial Multiplication Blunder. When multiplying an entire equation by a constant for elimination, candidates frequently multiply the variable terms on the left side but forget to multiply the constant term on the right side.
- Trap 2: Answering for the Wrong Variable. If a question asks 'What is the value of y?' or 'What is the cost of a student ticket?', solving for x and selecting x from the answer choices is one of the most common distractors. Always circle what the prompt specifically requests.
- Trap 3: Sign Errors During Subtraction of Equations. Rather than subtracting one equation from another (which leads to frequent negative sign dropped errors), always multiply by a negative number and add the equations.
- Trap 4: Incorrect Current Modeling. In uniform motion problems, always express relative speed as (r + c) and (r - c), where r is the vehicle speed in still conditions and c is the speed of the current. It is impossible for the current speed to exceed the vehicle speed in upstream travel.
Given the system of equations 3x - 2y = 14 and 5x + 4y = 16, what is the value of x - y?
3
-5
1
5
A laboratory technician needs to prepare 50 liters of a 24% saline solution by mixing a 15% saline solution with a 30% saline solution. How many liters of the 15% saline solution must be used?
20 liters
30 liters
25 liters
18 liters
Consider the linear system kx + 4y = 12 and 3x + 2y = 6. For what value of constant k will the system possess infinitely many solutions?
k = 3
k = 6
k = 2
k = -6
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