# Algebraïese oplossing

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## Algebraïese oplossings

Die oplos van gelyktydige vergelykings in algebra is deur middel van substitusie

Byvoorbeeld die oplossing van

$\begin{array}{c}\hfill y-2x=-4\\ \hfill {x}^{2}+y=4\end{array}$

is:

$\begin{array}{ccc}\hfill y& =& 2x-4\phantom{\rule{1.em}{0ex}}\mathrm{in tweede vergelyking}\hfill \\ \hfill {x}^{2}+\left(2x-4\right)& =& 4\hfill \\ \hfill {x}^{2}+2x-8& =& 0\hfill \\ \hfill \mathrm{Faktoriseer}:\phantom{\rule{1.em}{0ex}}\left(\mathrm{x}+4\right)\left(\mathrm{x}-2\right)& =& 0\hfill \\ \hfill \therefore \mathrm{Die}2\mathrm{oplossings vir x is}:\mathrm{x}=-4\mathrm{en}\mathrm{x}=2\end{array}$

Die ooreenstemmende oplossings vir $y$ word verkry deur substitusie van die $x$ -waardes in die eerste vergelyking

$\begin{array}{ccc}\hfill y=2\left(-4\right)-4& =& -12\phantom{\rule{3pt}{0ex}}\mathrm{vir}\phantom{\rule{3pt}{0ex}}\mathrm{x}=-4\hfill \\ \hfill \mathrm{en}:\phantom{\rule{3pt}{0ex}}\mathrm{y}=2\left(2\right)-4& =& 0\phantom{\rule{3pt}{0ex}}\mathrm{vir}\phantom{\rule{3pt}{0ex}}\mathrm{x}=2\hfill \end{array}$

Soos verwag, is hierdie oplossings identies aan die waardes verkry deur die grafiese oplossing

Los op algebraïes:

$\begin{array}{ccc}\hfill y-{x}^{2}+9& =& 0\hfill \\ \hfill y+3x-9& =& 0\hfill \end{array}$
1. $\begin{array}{ccc}\hfill y+3x-9& =& 0\hfill \\ \hfill y& =& -3x+9\hfill \end{array}$
2. $\begin{array}{ccc}\hfill \left(-3x+9\right)-{x}^{2}+9& =& 0\hfill \\ \hfill {x}^{2}+3x-18& =& 0\hfill \\ \hfill \mathrm{Faktoriseer}:\phantom{\rule{3pt}{0ex}}\left(\mathrm{x}+6\right)\left(\mathrm{x}-3\right)& =& 0\hfill \\ \hfill \therefore \phantom{\rule{3pt}{0ex}}\mathrm{die}\phantom{\rule{3pt}{0ex}}2\phantom{\rule{3pt}{0ex}}\mathrm{oplossings vir x is}:\phantom{\rule{3pt}{0ex}}\mathrm{x}=-6\phantom{\rule{3pt}{0ex}}\mathrm{and}\phantom{\rule{3pt}{0ex}}\mathrm{x}=3\end{array}$
3. $\begin{array}{ccc}\hfill y=-3\left(-6\right)+9& =& 27\phantom{\rule{1.em}{0ex}}\mathrm{vir}\phantom{\rule{3pt}{0ex}}\mathrm{x}=-6\hfill \\ \hfill \mathrm{en}:\phantom{\rule{1.em}{0ex}}\mathrm{y}=-3\left(3\right)+9& =& 0\phantom{\rule{1.em}{0ex}}\mathrm{vir}\phantom{\rule{3pt}{0ex}}\mathrm{x}=3\hfill \end{array}$
4. Die eerste waarde is $x=-6$ en $y=27$ . Die tweede waarde is $x=3$ en $y=0$ .

## Algebraïese oplossing

Los op die volgende probleme van algebraïese vergelykings. Waar toepaslik, los jou antwoord in wortelvorm.

 1. $a+b=5$ $a-{b}^{2}+3b-5=0$ 2. $a-b+1=0$ $a-{b}^{2}+5b-6=0$ 3. $a-\frac{\left(2b+2\right)}{4}=0$ $a-2{b}^{2}+3b+5=0$ 4. $a+2b-4=0$ $a-2{b}^{2}-5b+3=0$ 5. $a-2+3b=0$ $a-9+{b}^{2}=0$ 6. $a-b-5=0$ $a-{b}^{2}=0$ 7. $a-b-4=0$ $a+2{b}^{2}-12=0$ 8. $a+b-9=0$ $a+{b}^{2}-18=0$ 9. $a-3b+5=0$ $a+{b}^{2}-4b=0$ 10. $a+b-5=0$ $a-{b}^{2}+1=0$ 11. $a-2b-3=0$ $a-3{b}^{2}+4=0$ 12. $a-2b=0$ $a-{b}^{2}-2b+3=0$ 13. $a-3b=0$ $a-{b}^{2}+4=0$ 14. $a-2b-10=0$ $a-{b}^{2}-5b=0$ 15. $a-3b-1=0$ $a-2{b}^{2}-b+3=0$ 16. $a-3b+1=0$ $a-{b}^{2}=0$ 17. $a+6b-5=0$ $a-{b}^{2}-8=0$ 18. $a-2b+1=0$ $a-2{b}^{2}-12b+4=0$ 19. $2a+b-2=0$ $8a+{b}^{2}-8=0$ 20. $a+4b-19=0$ $8a+5{b}^{2}-101=0$ 21. $a+4b-18=0$ $2a+5{b}^{2}-57=0$

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