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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

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Key Concepts: 

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MAI HL and SL Notes – All topics

 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

$ 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

Find \( X, Y, Z \) for \( A(t) = Xt^2 + Yt + Z \) given:
\( A(1) = 9 \)
\( A(2) = 18 \)
\( A(-1) = 3 \)

▶️Answer/Explanation

Solution:

$ X + Y + Z= 9 $
$ 4X + 2Y + Z = 18 $
$ X – Y + Z = 3 $
\( X = 2 \), \(Y = 3 \), \( Z = 4 \).

♦

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

$h(x) = x^3 – kx^2 + 2x – 1$

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

$h(x) = x^3 + 3x^2 + 6x + 8$

▶️Answer/Explanation

Solution:

Try possible factors from:
Factors of 1 (leading coefficient): ±1
Factors of 8 (constant): ±1, ±2, ±4, ±8

\( x = -2 \Rightarrow h(-2) = (-2)^3 + 3(-2)^2 + 6(-2) + 8 = -8 + 12 – 12 + 8 = 0 \)
So \( x + 2 \) is a factor.
Then divide \( h(x) \) by \( x + 2 \) using synthetic or polynomial division to find other factors.

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