MaxWong

\(f(x) = 2x^2 +3x\\ g(x) = 5x^2 - 2x - 6\\ (f + g)(x) = 7x^2 + x - 6\)

The function you typed with your question is \((g - f)(x)\).

3. \(\begin{cases} 2x + y= 1\\2x+y = 5\end{cases}\)

Substituting equation 1 into equation 2, we get 5 = 1, which means no solution.

4. \(\begin{cases}2x + y = 1\\4x+2y = 2\end{cases}\)

Substituting equation 1 into equation 2, we get 2 = 2 which is always true. Therefore, any pair of (x, y) satisfies this equation and thus it has infinite solutions.

\(\text{Explicit formula: } a_n = 3n - 7\).

If we plot the graph of \(a_n\) against \(n\) of this explicit formula, we get a straight line with slope = 3 and y-intercept = -7. If we plug in 1,2,3,..., we get the corresponding term of the sequence.

3b.

\(a_{n+1} = a_n + 3\)

Because the next term is always greater than the last term by 3.

3a.

a_n = a₁ + (n - 1) d.

a₈ = -4 + 7(3) = 17

Therefore the 8^th term of the sequence is 17.

(a) I would choose to eliminate x from the equation. (Actually it is your choice to eliminate which variable)

(b) I chose equation 2 and equation 3 because this makes life easier :P.

\(x + 3y + 2z = 5 -----(2)\\ 4x + 4y + 3z = 13----\;(3)\\ \text{From equation 2: } 4x + 12y + 8z = 20-----(4)\\ \text{Subtract equation 3 from equation 4: } 8y + 5z = 7\)

(c) This time I would choose equation 1 and equation 2.

\(-3x+2y+z=-6 -----(1)\\ x+3y+2z =5 \quad\;\;-----(2)\\ \text{From equation 2: }3x + 9y + 6z = 15 -----(5)\\ \text{Add equation 1 and equation 5: }11y + 7z = 9\)

(d)

 \(\begin{cases}8y+5z = 7\\11y+7z=9\end{cases}\\ 88y + 55z = 77\\ 88y + 56z = 72\\ z = 72 - 77 = -5\\ 88y + 55(-5) = 77\\ y =4\)

(e) Because (y, z) = (4, -5),

\(x + 3y + 2z = 5\\ x + 12 - 10 = 5\\ x = 3\)

Therefore (x, y, z) = (3, 4, -5). 

:D

\(\left(\dfrac{3}{6}\right)^2 + 7\times 4 - 5\\ = \left(\dfrac{1}{2}\right)^2 + 28 - 5\\ = \dfrac{1}{4} + 23\\ = 23.25 \quad \text{ OR } \quad \dfrac{93}{4}\)

Actually, it is better to attach an image so that everyone can help even while you are offline.

Anyways, I want the image.

\(108^{\circ} + z^{\circ} = 180^{\circ}\\ z^{\circ} = 72^{\circ}\\ z = 72\) (angles on a straight line adds up to 180 degrees)

\(z^{\circ} + y^{\circ} + 102^{\circ}+84^{\circ} = 360^{\circ}\\ 72^{\circ} + 102^{\circ} + 84^{\circ} + y^{\circ} = 360^{\circ}\\ y^{\circ} = 102^{\circ}\\ y = 102\)  (interior angle of a quadrilateral adds up to 360 degrees)

\(s = 90^{\circ} - u\\ u = 90^{\circ} - s\\ t = 90^{\circ} - q\\ q = 90^{\circ} - t\\ p = 90^{\circ} - r\\ r = 90^{\circ} - p\\\)

Actually, by the properties of rectangle:

\(q = r = u = 90^{\circ} - t = 90^{\circ} - p = 90^{\circ} - s\).

\(\text{Therefore the following are true:}\\ \cos p = \sin u\\ \cos t = \sin u\\ \cos p = \sin q\\ \cos q = \sin t\\ \)