IB Mathematics AI – Topic 1

General Form (2 equations, 2 unknowns):

\[ \begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases} \]

Types of Solutions:

• Unique solution: Lines intersect at one point (different slopes)
• No solution: Lines are parallel (same slope, different intercepts)
• Infinite solutions: Lines coincide (same line)

Solution Methods:

1. Substitution Method:

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

2. Elimination Method:

1. Multiply equations to make coefficients equal
2. Add or subtract equations to eliminate one variable
3. Solve for remaining variable
4. Substitute back to find other variable

3. Graphical Method:

Graph both lines and find intersection point using GDC

4. Using GDC (Technology):

Most efficient for IB exams - enter equations directly into calculator

⚠️ Common Pitfalls & Tips:

• Use GDC for IB exams - it's faster and reduces errors
• Always verify solution by substituting back into original equations
• Write equations in standard form before solving
• Check for parallel lines (no solution) or identical lines (infinite solutions)

Solution:

Method 1: Using Elimination

Multiply equation 2 by 2:

\[ 10x - 2y = 26 \]

Add to equation 1:

\[ (3x + 2y) + (10x - 2y) = 16 + 26 \]

\[ 13x = 42 \]

\[ x = \frac{42}{13} \]

Substitute back into \(y = 5x - 13\):

\[ y = 5\left(\frac{42}{13}\right) - 13 = \frac{210}{13} - \frac{169}{13} = \frac{41}{13} \]

Method 2: Using GDC

Enter into calculator's simultaneous equation solver:

Equation 1: 3x + 2y = 16

Equation 2: 5x - y = 13

Answer: x = 42/13 (≈ 3.23), y = 41/13 (≈ 3.15)

Verification:

Check equation 1: \(3(\frac{42}{13}) + 2(\frac{41}{13}) = \frac{126}{13} + \frac{82}{13} = \frac{208}{13} = 16\) ✓

General Form:

\[ \begin{cases} a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3 \end{cases} \]

Solution Methods:

• GDC/Technology: Most practical method for IB
• Matrix method: Using inverse matrices (covered in HL)
• Elimination: Eliminate one variable at a time (tedious)

Using GDC:

1. Enter all three equations into calculator
2. Use simultaneous equation solver
3. Record solutions for x, y, and z

⚠️ Common Pitfalls & Tips:

• Always use technology for 3-variable systems in exams
• Write "Using GDC" in your solution
• Double-check you've entered equations correctly
• Verify at least one equation with your solutions

Standard Form:

\[ ax^2 + bx + c = 0, \quad a \neq 0 \]

Quadratic Formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Discriminant:

\[ \Delta = b^2 - 4ac \]

• If \(\Delta > 0\): Two distinct real roots
• If \(\Delta = 0\): One repeated real root
• If \(\Delta < 0\): No real roots (two complex roots)

Other Methods:

• Factoring: \(ax^2 + bx + c = (px + q)(rx + s)\)
• Completing the square: \((x + p)^2 = q\)
• Graphing: Find x-intercepts using GDC
• GDC solver: Most efficient for IB exams

Linear-Quadratic Systems:

When solving a linear and quadratic equation simultaneously:

• Can have 0, 1, or 2 solutions
• Substitute linear equation into quadratic
• Or use GDC to find intersection points

⚠️ Common Pitfalls & Tips:

• Always write in standard form first: \(ax^2 + bx + c = 0\)
• Check discriminant to know how many solutions to expect
• Remember ± in quadratic formula gives TWO solutions
• Use GDC to verify solutions

Solution:

Method 1: Quadratic Formula

Identify: \(a = 2, b = -5, c = -3\)

Calculate discriminant:

\[ \Delta = (-5)^2 - 4(2)(-3) = 25 + 24 = 49 \]

Since \(\Delta > 0\), two distinct real roots exist

Apply formula:

\[ x = \frac{-(-5) \pm \sqrt{49}}{2(2)} = \frac{5 \pm 7}{4} \]

Two solutions:

\[ x_1 = \frac{5 + 7}{4} = \frac{12}{4} = 3 \]

\[ x_2 = \frac{5 - 7}{4} = \frac{-2}{4} = -0.5 \]

Method 2: Factoring

\[ 2x^2 - 5x - 3 = (2x + 1)(x - 3) = 0 \]

So \(2x + 1 = 0\) or \(x - 3 = 0\)

Therefore \(x = -0.5\) or \(x = 3\)

Answer: x = 3 or x = -0.5

Polynomial Equation:

\[ a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0 \]

where n is the degree (highest power)

Cubic Equation (Degree 3):

\[ ax^3 + bx^2 + cx + d = 0 \]

Can have 1, 2, or 3 real roots

Key Properties:

• A polynomial of degree n has at most n real roots
• Odd-degree polynomials (cubic, quintic) have at least one real root
• Even-degree polynomials (quadratic, quartic) may have no real roots

Solution Methods:

• GDC (Recommended): Use equation solver or find zeros graphically
• Factor theorem: If \(f(a) = 0\), then \((x-a)\) is a factor
• Graphing: Find x-intercepts

Using GDC:

1. Enter polynomial equation
2. Use "Solve" or "Zeros" function
3. Or graph and find x-intercepts

⚠️ Common Pitfalls & Tips:

• Always use GDC for cubic and higher degree equations
• Check how many solutions to expect based on degree
• Some roots may be repeated (e.g., \((x-2)^2 = 0\) has root x = 2 twice)
• Verify solutions by substitution

Solution:

Using GDC:

Enter equation: \(x^3 - 6x^2 + 11x - 6 = 0\)

Use polynomial solver or graph to find zeros

Results from GDC:

\(x_1 = 1\)

\(x_2 = 2\)

\(x_3 = 3\)

Verification (optional):

Check \(x = 1\):

\[ (1)^3 - 6(1)^2 + 11(1) - 6 = 1 - 6 + 11 - 6 = 0 \] ✓

Factored form:

\[ x^3 - 6x^2 + 11x - 6 = (x-1)(x-2)(x-3) \]

Answer: x = 1, x = 2, or x = 3

Steps for Solving Word Problems:

1. Read carefully: Identify what you're asked to find
2. Define variables: Let x = ... , y = ... , etc.
3. Write equations: Translate words into mathematical equations
4. Solve the system: Use appropriate method (usually GDC)
5. Interpret results: Answer in context with units
6. Check reasonableness: Does answer make sense?

Common Application Types:

• Mixture problems: Combining solutions of different concentrations
• Rate/distance problems: Speed, time, distance relationships
• Cost/pricing problems: Finding prices or quantities
• Age problems: Relationships between ages
• Geometry: Area, perimeter, volume problems
• Projectile motion: Quadratic height functions

⚠️ Common Pitfalls & Tips:

• Always define variables clearly
• Include units in final answer
• Check if answer makes sense in context (e.g., can't have negative people)
• Show all steps: defining variables, equations, solution, interpretation

Solution:

Step 1: Define variables

Let x = price of adult ticket ($)

Let y = price of child ticket ($)

Step 2: Write equations

Family 1: 3 adults + 5 children = $165

\[ 3x + 5y = 165 \]

Family 2: 2 adults + 3 children = $105

\[ 2x + 3y = 105 \]

Step 3: Solve system

Using GDC:

Enter: 3x + 5y = 165

Enter: 2x + 3y = 105

Solve

GDC Result:

x = 30

y = 15

Step 4: Interpret

Answer: Adult ticket costs $30, child ticket costs $15

Step 5: Verify

Family 1: 3(30) + 5(15) = 90 + 75 = 165 ✓

Family 2: 2(30) + 3(15) = 60 + 45 = 105 ✓

Linear Systems

• 2 variables: substitution/elimination
• 3+ variables: use GDC
• Always verify solution

Quadratic

• \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)
• Check discriminant
• Factor when possible

Cubic/Polynomial

• Always use GDC
• Graph to find zeros
• Check all roots

Applications

• Define variables
• Write equations
• Interpret with units