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01-06 - Systems of Linear Equations

Phase: 1 | Subject: 01-06 Prerequisites: 01-02-linear-equations.md, 01-05-linear-functions.md Next subject: 01-07-quadratic-expressions.md

Learning Objectives

By the end of this subject, you will be able to:

  1. Understand what a system of linear equations represents
  2. Solve systems using the graphical method
  3. Solve systems using the substitution method
  4. Solve systems using the elimination method
  5. Classify systems as consistent, inconsistent, or dependent

Core Content

What is a System?

A system of linear equations is two or more equations that must be true SIMULTANEOUSLY. The solution is the point(s) where the lines intersect.

Example:

$y = 2x + 1
y = -x + 4
$

Both equations must be satisfied by the same (x, y) pair.

Graphical Method

Graph both lines and find where they cross.

Example:

$y = 2x + 1
y = -x + 4
$

Line 1: m = 2, b = 1 — goes through (0, 1) and (1, 3) Line 2: m = -1, b = 4 — goes through (0, 4) and (1, 3)

They intersect at (1, 3). Check: Line 1: y = 2(1) + 1 = 3 ✓. Line 2: y = -1 + 4 = 3 ✓

Limitation: Graphs are approximate. We need algebraic methods for precision.

Substitution Method

Steps: 1. Rearrange one equation to make one variable the subject (usually y = ...) 2. Substitute that expression into the other equation 3. Solve the resulting single equation 4. Substitute back to find the other variable

Example:

$y = 2x + 1     ... (1)
3x + y = 8     ... (2)
$
  1. Equation (1) already has y = 2x + 1
  2. Substitute into (2): 3x + (2x + 1) = 8
  3. 5x + 1 = 8
  4. 5x = 7, so x = 7/5 = 1.4
  5. Substitute back: y = 2(1.4) + 1 = 2.8 + 1 = 3.8

Solution: (1.4, 3.8) or (7/5, 19/5)

Elimination Method

Steps: 1. Arrange both equations in the same form (usually ax + by = c) 2. Multiply one or both equations so that adding them eliminates one variable 3. Solve for the remaining variable 4. Substitute back

Example:

$2x + 3y = 12     ... (1)
x - y = 1        ... (2)
$
  1. Multiply (2) by 3: 3x - 3y = 3 ... (3)
  2. Add (1) and (3): (2x + 3x) + (3y - 3y) = 12 + 3
  3. 5x = 15, so x = 3
  4. Substitute into (2): 3 - y = 1, so y = 2

Solution: (3, 2)

Classifying Systems

Type Graphical Algebraic Example
Consistent Lines intersect at one point One unique solution y = x + 1, y = 2x - 1 → (2, 3)
Inconsistent Parallel lines (same gradient) No solution (false statement) y = 2x + 1, y = 2x - 3
Dependent Same line (infinitely many points) Always true statement y = 2x + 1, 2y = 4x + 2

Example of inconsistent:

$y = 3x + 2
y = 3x - 5
$

Subtract: 0 = 7, which is FALSE → no solution.

Example of dependent:

$2x + y = 5
4x + 2y = 10
$

Multiply first by 2: 4x + 2y = 10. Both equations are identical → infinitely many solutions.

Common pitfall — elimination scaling: When multiplying an equation to prepare for elimination, you must multiply EVERY term on BOTH sides. For example, to eliminate y from x - y = 1, multiply by 3 to get 3x - 3y = 3 (not 3x - y = 3). Missing a term is the most common source of wrong answers in elimination.

Key Terms

Worked Examples

Example 1: Solve by substitution

$x + 2y = 7
3x - y = 5
$
  1. From first: x = 7 - 2y
  2. Substitute: 3(7 - 2y) - y = 5
  3. 21 - 6y - y = 5
  4. 21 - 7y = 5
  5. 7y = 16
  6. y = 16/7
  7. x = 7 - 2(16/7) = 7 - 32/7 = 49/7 - 32/7 = 17/7

Solution: (17/7, 16/7)

Example 2: Solve by elimination

$2x + 5y = 3
4x - y = 19
$
  1. Multiply second by 5: 20x - 5y = 95
  2. Add to first: (2x + 20x) + (5y - 5y) = 3 + 95
  3. 22x = 98
  4. x = 98/22 = 49/11
  5. Substitute: 4(49/11) - y = 19, 196/11 - y = 19
  6. y = 196/11 - 209/11 = -13/11

Solution: (49/11, -13/11)

Example 3: Classify and solve

$3x - 2y = 4
6x - 4y = 10
$
  1. Multiply first by 2: 6x - 4y = 8
  2. Compare: 6x - 4y = 8 vs 6x - 4y = 10
  3. 8 ≠ 10, so inconsistent system
  4. No solution — lines are parallel

Quiz

Q1: What does the concept of Consistent primarily refer to in this subject?

A) A historical anecdote about Consistent B) A computational error related to Consistent C) A visual representation of Consistent D) The definition and application of Consistent

Correct: D)

Q2: What is the primary purpose of Inconsistent?

A) It is used only in advanced research contexts B) It is used to inconsistent in mathematical analysis C) It replaces all other methods in this domain D) It is primarily a historical notation system

Correct: B)

Q3: Which statement about Dependent is TRUE?

A) Dependent is an advanced topic beyond this subject's scope B) Dependent is not related to this subject C) Dependent is mentioned only as a historical footnote D) Dependent is a fundamental concept covered in this subject

Correct: D)

Q4: Based on the worked examples in this subject, what is the correct result?

A) The inverse of the correct answer B) An unrelated numerical value C) A different result from a common mistake D) Solve by substitution

Correct: D)

Q5: How are Dependent and No solution related?

A) Dependent is the inverse of No solution B) Dependent and No solution are closely related concepts C) Dependent and No solution are completely unrelated topics D) Dependent is a special case of No solution

Correct: B)

Q6: What is a common pitfall when working with What Is A System??

A) A common mistake is confusing What Is A System? with a similar concept B) What Is A System? is always computed the same way in all contexts C) The main error with What Is A System? is using it when it is not needed D) What Is A System? has no common misconceptions

Correct: A)

Q7: When should you apply Graphical Method?

A) Avoid Graphical Method unless explicitly instructed B) Graphical Method is not practically useful C) Apply Graphical Method to solve problems in this subject's domain D) Use Graphical Method only in pure mathematics contexts

Correct: C)

Practice Problems

  1. Solve by substitution: y = 3x - 2 and x + y = 10
    Click for answer

x + 3x - 2 = 10, 4x = 12, x = 3. y = 3(3) - 2 = 7. Solution: (3, 7)

  1. Solve by elimination: 2x + y = 8 and x - y = 2
    Click for answer

Add: 3x = 10, x = 10/3. Substitute: 10/3 - y = 2, y = 4/3. Solution: (10/3, 4/3)

  1. Solve: 3x + 2y = 1 and 2x - y = 3
    Click for answer

Multiply second by 2: 4x - 2y = 6. Add: 7x = 7, x = 1. Then 2(1) - y = 3, y = -1. Solution: (1, -1)

  1. Classify: y = 2x + 3 and 4x - 2y = -6
    Click for answer

Second: -2y = -4x - 6, y = 2x + 3. Same line — dependent system, infinitely many solutions.

  1. Classify: x + y = 5 and x + y = 8
    Click for answer

Same left side, different right side. Inconsistent — no solution.

  1. Solve: x + 2y = 8 and 3x - y = 7
    Click for answer

From first: x = 8 - 2y. Substitute: 3(8 - 2y) - y = 7, 24 - 6y - y = 7, 17 = 7y, y = 17/7. x = 8 - 34/7 = 22/7. Solution: (22/7, 17/7)

  1. Solve: 2x + 3y = 8 and 4x - y = 7
    Click for answer

Multiply second by 3: 12x - 3y = 21. Add: 14x = 29, x = 29/14. y = 4(29/14) - 7 = 116/14 - 98/14 = 18/14 = 9/7. Solution: (29/14, 9/7)

Summary

Key takeaways:

Pitfalls

Next Steps

Next up: 01-07-quadratic-expressions.md