Kaflapróf

$\begin{array}{l}\\ \begin{array}{l}-5x-y=12\hfill \\ x+4y=9\hfill \end{array}\end{array}$ með $(-3,3)$

12⁢𝑥−13⁢𝑦=432⁢𝑥−𝑦=0$\begin{array}{r}\frac{1}{2}x-\frac{1}{3}y=4\\ \frac{3}{2}x-y=0\end{array}$

−12⁢𝑥−4⁢𝑦=42⁢𝑥+16⁢𝑦=2$\begin{array}{l}\\ -\frac{1}{2}x-4y=4\\ \hfill 2x+16y=2\end{array}$

5⁢𝑥−𝑦=1−10⁢𝑥+2⁢𝑦=−2$\begin{array}{r}\hfill 5x-y=1\\ \hfill -10x+2y=-2\end{array}$

4⁢𝑥−6⁢𝑦−2⁢𝑧=110𝑥−7⁢𝑦+5⁢𝑧=−143⁢𝑥+6⁢𝑦−9⁢𝑧=65$\begin{array}{l}4x-6y-2z=\frac{1}{10}\hfill \\ x-7y+5z=-\frac{1}{4}\hfill \\ 3x+6y-9z=\frac{6}{5}\hfill \end{array}$

𝑥+𝑧=20𝑥+𝑦+𝑧=20𝑥+2⁢𝑦+𝑧=10$\begin{array}{r}x+z=20\\ x+y+z=20\\ x+2y+z=10\end{array}$

5⁢𝑥−4⁢𝑦−3⁢𝑧=02⁢𝑥+𝑦+2⁢𝑧=0𝑥−6⁢𝑦−7⁢𝑧=0$\begin{array}{r}5x-4y-3z=0\\ 2x+y+2z=0\\ x-6y-7z=0\end{array}$

𝑦=𝑥2+2⁢𝑥−3𝑦=𝑥−1$\begin{array}{l}y=x^2+2x-3\\ y=x-1\end{array}$

𝑦2+𝑥2=25𝑦2−2⁢𝑥2=1$\begin{array}{l}{y}^2+x^2=25\\ y^2-2x^2=1\end{array}$

𝑦<𝑥2+9$y<x^2+9$

𝑥2+𝑦2>4𝑦<𝑥2+1$\begin{array}{l}{x}^2+y^2>4\\ y<x^2+1\end{array}$

−8⁢𝑥−30𝑥2+10⁢𝑥+25$\frac{-8x-30}{x^2+10x+25}$

13⁢𝑥+2(3⁢𝑥+1)₂$\frac{13x+2}{{(3x+1)}^2}$

𝑥4−𝑥3+2⁢𝑥−1𝑥⁢(𝑥2+1)₂$\frac{x^4-x^3+2x-1}{x{(x^2+1)}^2}$

5⁢[49−23]+12⁢[−6124−8]$5\left[\begin{array}{cc}4& 9\\ -2& 3\end{array}\right]+\frac{1}{2}\left[\begin{array}{cc}-6& 12\\ 4& -8\end{array}\right]$

⎡⎢ ⎢⎣14−7−295120−4⎤⎥ ⎥⎦⎡⎢ ⎢⎣3−413510⎤⎥ ⎥⎦$\left[\begin{array}{rrr}\hfill 1& \hfill 4& \hfill -7\\ \hfill -2& \hfill 9& \hfill 5\\ \hfill 12& \hfill 0& \hfill -4\end{array}\right]\left[\begin{array}{cc}3& -4\\ 1& 3\\ 5& 10\end{array}\right]$

[12131415]−1${\left[\begin{array}{rr}\hfill \frac{1}{2}& \hfill \frac{1}{3}\\ \hfill \frac{1}{4}& \hfill \frac{1}{5}\end{array}\right]}^{-1}$

det⁡∣004004,000∣$\det \left|\begin{array}{cc}0& 0\\ 400& 4\text{,}000\end{array}\right|$

det⁡∣ ∣ ∣ ∣12−120−120120120∣ ∣ ∣ ∣$\det \left|\begin{array}{rrr}\hfill \frac{1}{2}& \hfill -\frac{1}{2}& \hfill 0\\ \hfill -\frac{1}{2}& \hfill 0& \hfill \frac{1}{2}\\ \hfill 0& \hfill \frac{1}{2}& \hfill 0\end{array}\right|$

Ef $\det (A)=\mathrm{-6},$ hver yrði ákveðan ef línum 1 og 3 væri víxlað, önnur línan margfölduð með 12 og andhverfan síðan tekin?

Ritið línulega jöfnuhneppið sem aukið fylki.

Ritið aukna fylkið sem línulegt jöfnuhneppi.

𝑥−6⁢𝑦=42⁢𝑥−12⁢𝑦=0$\begin{array}{r}x-6y=4\\ 2x-12y=0\end{array}$

2⁢𝑥+𝑦+𝑧=−3𝑥−2⁢𝑦+3⁢𝑧=6𝑥−𝑦−𝑧=6$\begin{array}{r}\hfill 2x+y+z=-3\\ \hfill x-2y+3z=6\\ \hfill x-y-z=6\end{array}$

4⁢𝑥−5⁢𝑦=−50−𝑥+2⁢𝑦=80$\begin{array}{r}\hfill 4x-5y=-50\\ \hfill -x+2y=80\end{array}$

1100⁢𝑥−3100⁢𝑦+120⁢𝑧=−493100⁢𝑥−7100⁢𝑦−1100⁢𝑧=139100⁢𝑥−9100⁢𝑦−9100⁢𝑧=99$\begin{array}{r}\hfill \frac{1}{100}x-\frac{3}{100}y+\frac{1}{20}z=-49\\ \hfill \frac{3}{100}x-\frac{7}{100}y-\frac{1}{100}z=13\\ \hfill \frac{9}{100}x-\frac{9}{100}y-\frac{9}{100}z=99\end{array}$

200⁢𝑥−300⁢𝑦=2400⁢𝑥+715⁢𝑦=4$\begin{array}{l}200x-300y=2\\ 400x+715y=4\end{array}$

0.1⁢𝑥+0.1⁢𝑦−0.1⁢𝑧=−1.20.1⁢𝑥−0.2⁢𝑦+0.4⁢𝑧=−1.20.5⁢𝑥−0.3⁢𝑦+0.8⁢𝑧=−5.9$\begin{array}{l}0.1x+0.1y-0.1z=-1.2\\ 0.1x-0.2y+0.4z=-1.2\\ 0.5x-0.3y+0.8z=-5.9\end{array}$

Verksmiðja sem framleiðir farsíma hefur kostnaðar- og tekjuföllin $C(x)=x^2+75x+2\text{,}688$ og $R(x)=x^2+160x.$ Á hvaða bili á fjöldi framleiddra farsíma á dag að vera svo reksturinn skili hagnaði? Námundið að næsta fjölda sem gefur hagnað.

Lítil hátíð rukkar $1.50 fyrir nemendur, $1 fyrir börn og $2 fyrir fullorðna. Á einum degi komu þrefalt fleiri börn en fullorðnir. Alls seldust 800 miðar fyrir $1,050. Hve margir miðar af hverri gerð seldust?