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IB Mathematics AI SL Systems of linear equations Study Notes - New Syllabus

IB Mathematics AI SL Systems of linear equations Study Notes

LEARNING OBJECTIVE

  • Systems of linear equations

Key Concepts: 

  • Systems of linear equations in up to 3 variables
  •  Polynomial equations

MAI HL and SL Notes – All topics

A Pair of Linear Equations in Two Variables

♦General Form:

$
\begin{aligned}
a_1x + b_1y &= c_1 \\
a_2x + b_2y &= c_2
\end{aligned}
$

♦Classification Based on Solution Type

(i) Unique Solution

♦Condition:

$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$

The equations represent intersecting lines.  
The system has one unique solution.

(ii) Infinitely Many Solutions

♦Condition:

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

The equations represent coincident lines.
The system has infinitely many solutions.

(iii) Inconsistent System (Has no solution)

♦Condition:

$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

The equations represent parallel lines.
The system has no solution.

Example

Consider the following pairs of linear equations in two variables. For each pair:

  • Determine whether the system is consistent or inconsistent.
  • If consistent, state whether the system has a unique solution or infinitely many solutions.
  • Justify your answer by comparing the ratios \(\frac{a_1}{a_2}\), \(\frac{b_1}{b_2}\), and \(\frac{c_1}{c_2}\).
  1. $2x + 3y = 6; 4x + 5y = 8$
  2. $2x + 4y = 6; 4x + 8y = 12$
  3. $2x + 4y = 6; 4x + 8y = 10$
▶️Answer/Explanation

Solution:

1. Equations: $2x + 3y = 6$ and $4x + 5y = 8$
Ratios: $\frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}$, $\frac{b_1}{b_2} = \frac{3}{5}$
Since $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, the system is consistent and has a unique solution (intersecting lines).

2. Equations: $2x + 4y = 6$ and $4x + 8y = 12$
Ratios: $\frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}$, $\frac{b_1}{b_2} = \frac{4}{8} = \frac{1}{2}$, $\frac{c_1}{c_2} = \frac{6}{12} = \frac{1}{2}$
Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the system is consistent and has infinitely many solutions (coincident lines).

3. Equations: $2x + 4y = 6$ and $4x + 8y = 10$
Ratios: $\frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}$, $\frac{b_1}{b_2} = \frac{4}{8} = \frac{1}{2}$, $\frac{c_1}{c_2} = \frac{6}{10} = \frac{3}{5}$
Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the system is inconsistent and has no solution (parallel lines).

SYSTEMS OF LINEAR EQUATIONS

 ♦Systems of 2 Linear Equations:

A system of 2 linear equations in 2 unknowns:
$ a_1x + b_1y = c_1 $
$ a_2x + b_2y = c_2 $

Example

Given Systems of 2 Linear Equations.

Find x and y .

$ 5x + 13y = 23 $
$ 3x – y = 5 $

▶️Answer/Explanation

Solution:

\( x = 2 \), \( y = 1 \) (using GDC).

Example

Gia buys 3 burgers and 5 sandwiches for 21.4 euros. 

Cathy buys 2 burgers and 3 sandwiches for 13.6 euros.

Find the prices of a burger (\( B \)) and a sandwich (\( S \)).

▶️Answer/Explanation

Solution:

$ 3B + 5S = 21.4 $
$ 2B + 3S = 13.6 $

\( B = 3.8 \) euros, \( S = 2 \) euros.

♦Systems of 3 Linear Equations:

$ a_1x + b_1y + c_1z = d_1 $
$ a_2x + b_2y + c_2z = d_2 $
$ a_3x + b_3y + c_3z = d_3 $

Example(USING GDC)

Find 

\( X, Y, Z \)

for

$\begin{align*} 5x + y – 4z &= 5 \\ 2x – 3y – 5z &= 2 \\ 7x + 2y – 6z &= 5 \end{align*}$

▶️Answer/Explanation

Solution:

$\begin{align*} 5x + y – 4z &= 5 \\ 2x – 3y – 5z &= 2 \\ 7x + 2y – 6z &= 5 \end{align*}$

So, in the next screen, after pressing q, I choose w (3)

 

This screen shows the type of equation and the number of variables.
We edit the entries in the window to fit our system of equations. [If the gdc has been used
before, you can either overwrite the numbers there, or clear the matrix entries]

Then press $l ~($you can also press $q -\text{SOLVE})$

The solution is thus
$\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 3 \\ -2 \\ 2 \end{bmatrix}$

Polynomial Function

♦Definition

A polynomial of degree \( n \) is:

$p(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$

Where:
 \( a_n, a_{n-1}, \dots, a_0 \) are real numbers
\( a_n \ne 0 \)
Powers of \( x \) are non-negative integers

♦Degree of a Polynomial
The degree is the highest power of \( x \) with a non-zero coefficient.

♦Synthetic Division (Quick Recap)

Used for dividing a polynomial \( p(x) \) by a linear divisor of the form \( x – k \).

If dividing:

$p(x) = a_3x^3 + a_2x^2 + a_1x + a_0 \text{ by } (x – k)$

Then we get:

Quotient: \( a_3x^2 + b_1x + b_0 \)
Remainder: \( R \)

♦Remainder Theorem

For a polynomial \( p(x) \), the remainder when divided by \( x – \alpha \) is:
$\text{Remainder} = p(\alpha)$

Example

Find the remainder when:

$p(x) = 3x^4 + 4x^2 – 2x + 1$

is divided by \( x + 2 \)

▶️Answer/Explanation

Solution:

Since \( x + 2 = x – (-2) \), we evaluate:

$p(-2) = 3(-2)^4 + 4(-2)^2 – 2(-2) + 1 = 3(16) + 4(4) + 4 + 1 = 48 + 16 + 4 + 1 = \rm{69}$

♦Factor Theorem

\( x – \alpha \) is a factor of \( p(x) \) if and only if:
$p(\alpha) = 0$

Example

Find K 

$h(x) = x^3 – kx^2 + 2x – 1$

If \( x – 1 \) is a factor.

▶️Answer/Explanation

Solution:

$h(1) = 0$
$\Rightarrow 1^3 – k(1)^2 + 2(1) – 1 = 0$
$\Rightarrow 1 – k + 2 – 1 = 0$
$\Rightarrow k = \rm{2}$

♦Rational Root Theorem (Factor Guessing)

Given a polynomial:

$p(x) = a_nx^n + \dots + a_0$

Then any rational factor of the form \( (px – q) \) must satisfy:
 \( p \) divides \( a_n \)
\( q \) divides \( a_0 \)

Example (Using GDC)

Find all the Factors

$h(x) = 2x^3 -x^2 -7x + 6$

 $\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix}  \\  \\ \end{bmatrix}$

▶️Answer/Explanation

Solution:

Again, open the Equation application, press w for polynomial

 

Press w for degree 3, and type the coefficients in

Choose q for SOLVE, and here are All solutions.

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