Zadanie 1. a) [latex]-4(2x-5)=2(3x+7)[/latex] [latex]-8x+20=6x+14[/latex] [latex]-14x=-6[/latex] [latex]x=frac{-6}{-14}=frac{3}{7}[/latex] b) [latex]frac{2}{3}x-13=frac{1}{3}(x+9)[/latex] [latex]frac{2}{3}x-13=frac{1}{3}x+3[/latex] [latex]frac{2}{3}x-frac{1}{3}x=3+13[/latex] [latex]frac{1}{3}x=16[/latex] [latex]x=48[/latex] c) [latex]frac{3}{5}(2x-7)=frac{7}{15}(3+x)[/latex] Mnożymy obie strony razy 15 [latex]9(2x-7)=7(3+x)[/latex] [latex]18x-63=21+x[/latex] [latex]17x=84[/latex] [latex]x=frac{84}{17}[/latex] d) [latex]frac{5x-4}{6}-frac{7-2x}{2}=0[/latex] mnożymy wszystko strony razy 6 [latex]5x-4-3(7-2x)=0[/latex] [latex]5x-4-21+6x=0[/latex] [latex]11x-25=0[/latex] [latex]11x=25[/latex] [latex]x=frac{25}{11}[/latex] e) [latex]x(x-3)=(x+2)^2[/latex] [latex]x^2-3x=x^2+4x+4[/latex] [latex]-7x=4[/latex] [latex]x=-frac{4}{7}[/latex] f) [latex]frac{x(3-x)}{3}-frac{x-2x^2}{6}=2x[/latex] mnożymy wszystko razy 6 [latex]2x(3-x)-(3-2x^2)=12x[/latex] [latex]6x-2x^2-3+2x^2=12x[/latex] [latex]6x-3=12x[/latex] [latex]-6x=3[/latex] [latex]x=-frac{1}{2}[/latex] Zadanie 2. a) [latex]egin{cases} 6x-2y=9 \ 3x+y-4,5=0 end{cases}[/latex] [latex]egin{cases} 6x-2y=9 \ 3x+y=4,5end{cases}[/latex] mnożymy drugie równanie razy 2 [latex]egin{cases} 6x-2y=9 \ underline{6x+2y=9}end{cases}[/latex] [latex]12x=18[/latex] [latex]x=frac{18}{12}frac{3}{2}[/latex] [latex]6 cdot frac{3}{2} -2y = 9[/latex] [latex]9-2y=9[/latex] [latex]-2y=0[/latex] [latex]y=0[/latex] [latex]egin{cases} x=frac{3}{2} \ y=0 end{cases}[/latex] b) [latex]egin{cases} 0,1x-0,2y=0,7 \ -2x+4y=-14 end{cases}[/latex] mnożymy pierwsze równanie razy 20 [latex]egin{cases} 2x-4y=14 \ underline{-2x+4y=-14} end{cases}[/latex] [latex]0=0[/latex] Czyli układ równań tożsamościowy. c) [latex]egin{cases} 4a-7b=-19 \ 3a+2b=22end{cases}[/latex] mnożymy pierwsze równanie razy -3, a drugie razy 4 [latex]egin{cases} -12a+21b=57 \ underline{12a+8b=88}end{cases}[/latex] [latex]29b=145[/latex] [latex]b=frac{145}{29}=5[/latex] [latex]3a+2 cdot 5 = 22[/latex] [latex]3a+10=22[/latex] [latex]3a=12[/latex] [latex]a=4[/latex] [latex]egin{cases} a=4 \ b=5 end{cases}[/latex] d) [latex]egin{cases}frac{2}{3}x+y-7=0 \ 1frac{1}{3}x+4y=frac{7}{3}end{cases}[/latex] [latex]frac{2}{3}x+y=7 \ frac{4}{3}x+4y=frac{7}{3}end{cases}[/latex] mnożymy pierwsze równanie razy -2 [latex]egin{cases} -frac{4}{3}x-2y=-14 \ underline{frac{4}{3}+4y=frac{7}{3}}end{cases}[/latex] [latex]2y=-frac{35}{3}[/latex] [latex]y=-frac{35}{6}[/latex] [latex]frac{2}{3}x-frac{35}{6}=7[/latex] mnożymy wszystko razy 6 [latex]4x-35=42[/latex] [latex]4x=77[/latex] [latex]x=frac{77}{4}[/latex] [latex]egin{cases} x=frac{77}{4} \ y=-frac{35}{6}end{cases}[/latex] e) [latex]egin{cases}frac{m+n}{3}=5 \ frac{3m+n}{2}=20-frac{n}{3}end{cases}[/latex] Pierwsze równanie mnożymy razy 3, drugie razy 6 [latex]egin{cases} m+n=15 \ 9m+3n=120-2nend{cases}[/latex] [latex]egin{cases} n=15-m \ 9m+5n=120end{cases}[/latex] [latex]9m+5(15-m)=120[/latex] [latex]9m+75-5m=120[/latex] [latex]4m=45[/latex] [latex]m=frac{45}{4}[/latex] [latex]frac{45}{4}+n=15[/latex] [latex]n=15-frac{45}{4}[/latex] [latex]n=frac{60}{4}-frac{45}{4}[/latex] [latex]n=frac{15}{4}[/latex] [latex]egin{cases} m=frac{45}{4} \ n=frac{15}{4}end{cases}[/latex] f) [latex]egin{cases} 5(x-3y)=-7(3y-x) \ -3(x+4)+2(3-5y)=0end{cases}[/latex] [latex]egin{cases} 5x-15y=-21y+7x \ -3x-12+6-10y=0end{cases}[/latex] [latex]egin{cases} -2x+6y=0 \ -3x-10y=6end{cases}[/latex] z pierwszego równania wyznaczamy x [latex]-2x+6y=0[/latex] [latex]-2x=-6y[/latex] [latex]x=frac{-6}{-2}y[/latex] [latex]x=3y[/latex] wstawiamy do drugiego równania [latex]-3 cdot 3y-10y=6[/latex] [latex]-9y-10y=6[/latex] [latex]-19y=6[/latex] [latex]y=-frac{6}{19}[/latex] [latex]x=3*(-frac{6}{19})[/latex] [latex]x=-frac{18}{19}[/latex] [latex]egin{cases} x=-frac{18}{19} \ y=-frac{6}{19}[/latex] Zadanie 3. a) [latex]frac{1}{2}x^2=9[/latex] [latex]x^2=18[/latex] [latex]x=sqrt{18}[/latex] [latex]x=sqrt{2 cdot 9}[/latex] [latex]x=sqrt{2} cdot sqrt{9}[/latex] [latex]x=3sqrt{2}[/latex] lub [latex]x=-3sqrt{2}[/latex] b) [latex]2x^2-1=17[/latex] [latex]2x^2=18[/latex] [latex]x^2=9[/latex] [latex]x=3[/latex] lub [latex]x=-3[/latex] c) [latex]6x^2-1frac{3}{5}=0[/latex] [latex]6x^2-frac{8}{5}[/latex] [latex]6x^2=frac{8}{5}[/latex] [latex]x^2=frac{48}{5}[/latex] [latex]x=sqrt{frac{48}{5}}[/latex] lub [latex]x=-sqrt{frac{48}{5}}[/latex] d) [latex]1-2x^2=5+x^2[/latex] [latex]-3x^2=4[/latex] [latex]x^2=-frac{4}{3}[/latex] [latex]x=sqrt{-frac{4}{3}}[/latex] Fałsz - brak rozwiązań. e) [latex]12-3x^2=0[/latex] [latex]3x^2=12[/latex] [latex]x^2=4[/latex] [latex]x=2[/latex] lub [latex]x=-2[/latex] f) [latex]6x^2=8-x^2[/latex] [latex]7x^2=8[/latex] [latex]x^2=frac{8}{7}[/latex] [latex]x=sqrt{frac{8}{7}}[/latex] lub [latex]x=-sqrt{frac{8}{7}}[/latex] g) [latex]4(x^2+2)=8[/latex] [latex]x^2+2=2[/latex] [latex]x^2=0[/latex] [latex]x=0[/latex] h) [latex]frac{5+x^2}{2}-4=0[/latex] [latex]5+x^2-8=0[/latex] [latex]x^2-3=0[/latex] [latex]x^2=3[/latex] [latex]x=sqrt3[/latex] lub [latex]x=-sqrt3[/latex] i) [latex]3(1frac{1}{2}-frac{x^2}{6}=frac{1-2x^2}{2}[/latex] [latex]3(frac{3}{2}-frac{x^2}{6})=frac{1-2x^2}{2}[/latex] [latex]3(9-x^2)=3(1-2x^2)[/latex] [latex]27-3x^2=3-6x^2[/latex] [latex]3x^2=-24[/latex] [latex]x^2=-8[/latex] Fałsz - brak rozwiązania. Zadanie 4. a) [latex]x^2-2x=0[/latex] [latex]x(x-2)=0[/latex] [latex]x=0[/latex] lub [latex]x=2[/latex] b) [latex]3x^2+8x=0[/latex] [latex]x(3x+8)=0[/latex] [latex]x=0[/latex] lub [latex]3x+8=0[/latex] [latex]x=0[/latex] lub [latex]3x=-8[/latex] [latex]x=0[/latex] lub [latex]x=-frac{8}{3}[/latex] c) [latex]4x^2-frac{x}{2}=0[/latex] [latex]8x^2-x=0[/latex] [latex]x(8x-1)=0[/latex] [latex]x=0[/latex] lub [latex]8x-1=0[/latex] [latex]x=0[/latex] lub [latex]x=frac{1}{8}[/latex] d) [latex]x^2=-7x[/latex] [latex]x^2+7x=0[/latex] [latex]x(x+7)=0[/latex] [latex]x=0[/latex] lub [latex]x=-7[/latex] e) [latex]-5x-4x^2=0[/latex] [latex]-x(5+4x)=0[/latex] [latex]x=0[/latex] lub [latex]x=-frac{5}{4}[/latex] f) [latex]frac{1}{3}x^2=5x[/latex] [latex]x^2=15x[/latex] [latex]x^2-15x=0[/latex] [latex]x(x-15)=0[/latex] [latex]x=0[/latex] lub [latex]x=15[/latex] g) [latex]-frac{2}{7}x^2+2x=0[/latex] [latex]-2x^2+14x=0[/latex] [latex]-2x(x-7)=0[/latex] [latex]x=0[/latex] lub [latex]x=7[/latex] h) [latex]frac{x^2+2}{2}=4x+1[/latex] [latex]x^2+2=8x+2[/latex] [latex]x^2-8x=0[/latex] [latex]x(x-8)=0[/latex] [latex]x=0[/latex] lub [latex]x=8[/latex] Zadanie 5. a) [latex](5x+2)^2=20x[/latex] [latex]25x^2+20x+4=20x[/latex] [latex]25x^2+4=0[/latex] [latex]25x^2=-4[/latex] [latex]x^2=-frac{4}{25}[/latex] Fałsz - brak rozwiązania. b) [latex](2x-1)(x+2)=3x-2[/latex] [latex]2x^2+3x-2=3x-2[/latex] [latex]2x^2=0[/latex] [latex]x=0[/latex]
