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[h] IB Mathematics AI HL Flashcards- Systems of linear equations

[q] **Tech. to solve equations**

In this section, we will see how to solve certain tricky types of equations by GDC only. Systems w/ 3 variables can be solved manually (in A&A HL), and **polynomials** can be solved manually, especially quadratics, but we will now see how to solve cubics and above with our GDCs:

SYSTEMS — You will have seen systems of 2 variables with 2 equations. With these, you must find an ‘x’ & ‘y’ that fits both simultaneously. Now, you might have equations, all ‘x,’ ‘y’ & ‘z,’ that you need to solve:

[q]

MATRIX — There are multiple ways, but I will need you to be able to convert the eqs. to matrix form → with just the coefficients.

– E.G. 1: Solve:

\[

2x + y – z = 2 \\

3x + 3y + z = 1 \\

2x + 4y + 6z = 6

\]

[a]

Matrix:

\[

\begin{bmatrix}

2 & 1 & -1 \\

3 & 3 & 1 \\

2 & 4 & 6

\end{bmatrix}

\]

– TI-nspire → [1] [B] → [8]

Type (3×cols) → \(\begin{bmatrix} 2 & 1 & -1 \\ 3 & 3 & 1 \\ 2 & 4 & 6 \end{bmatrix}\) → ENTER → Type rref( )

[q]

→ Go up ▲ → Click the matrix above → ENTER → gives you

\[

\begin{bmatrix} 1 & 0 & 0 & | & 2 \\ 0 & 1 & 0 & | & -1 \\ 0 & 0 & 1 & | & 1 \end{bmatrix}

\]

Answer: x = 2, y = -1, z = 1

– TI-84 → (2ⁿᵈ) → [x⁻¹] → EDIT → ENTER on 1: [A] → Change to 3×4 → Type matrix

(2ⁿᵈ) → [x⁻¹] → MATH → (B: rref( ) → (2ⁿᵈ) → [x⁻¹] → (ENTER) on 1: [A] 3×4 → ENTER → Answer as above

[a]

POLYNOMIALS — Make sure, if you have expressions on both sides, change to the form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 = 0 \), such as \( 3x^3 – 8x + 11 = 0 \) or \( -2x^3 + 7x^2 + 3 = 0 \).

– E.G. 1: Solve \( 2x^3 + x^2 – 13x + 6 = 0 \):

– TI-nspire → [MENU] → (3: ALGEBRA) → (8: POLY.) (1: FIND ROOTS) → Degree: 3, because it is the highest power in the equation → OK

Type \( (2x^3 + x^2 – 13x + 6 = 0, x) \) → ENTER → Roots given

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– TI-84 → [APPS] → [ALPHA] 8 → (PlySmlt2) → POLY ROOT FINDER → Degree = 3 → (GRAPH) → (SOLVE)

\( a_3 = 2, a_2 = 1, a_1 = -13, a_0 = 6 \) → (GRAPH) / (SOLVE) → \( x_1 = 2, x_2 = 3, x_3 = \frac{1}{2} \)

If these capabilities are not there, or are off for an exam for whatever reason, you should also be comfortable graphing \( y = \text{polynomial} \) and asking for the roots / x-intercepts. This will be needed for tricky non-polynomials as well.

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