AB: Gemischte Rechenaufgaben II
II. Rechnen mit Termen / Rechnen mit Klammern
Fassen Sie zusammen und vereinfachen Sie die Terme.
Beispiel: \( 3 a+7 b-a+b=2 a+8 b=2(a+4 b) \)
\( 5 x+7 y-x+13 y \)
\( 5 x+7 y-x+13 y=4 x+20 y=4 \cdot(x+5 y) \)
\( 10 k+6 m-8 n+5 k-m-2 n \)
\( 10 k+6 m-8 n+5 k-m-2 n \\ =15 k+5 m-10 n=5 \cdot(3 k +m-2 n) \)
\( 1,8 x+2,3 y+3,2 z-0,9 x-1,1 y-1,4 z \)
\( \begin{aligned} 1,8 x+2,3 y+3,23-0, & 9 x-1,1 y-1,4 z \\ =0,9 x+1,2 y+1,8 z &=1,8 \cdot\left(\frac{1}{2} x+\frac{2}{3} y+z\right) \\ &=0,9 \cdot\left(x+\frac{4}{3} y+2 z\right) \end{aligned} \)
\( 5 \frac{1}{3} a+1 \frac{1}{6} b-\frac{1}{2} c+\frac{1}{3} b-\frac{5}{6} a+\frac{1}{3} c+9 \frac{1}{2} a-3 \frac{1}{2} b+1 \frac{1}{6} c \)
\( 5 \frac{1}{3} a+1 \frac{1}{6} b-\frac{1}{2} c+\frac{1}{3} b-\frac{5}{6} a+\frac{1}{3} c+9 \frac{1}{2} a-3 \frac{1}{2} b+1 \frac{1}{6}c \\ =\frac{16}{3} a+\frac{7}{6} b-\frac{1}{2} c+\frac{1}{3} b - \frac{5}{6} a+\frac{1}{3} c+\frac{19}{2} a - \frac{7}{2} b+\frac{7}{6} c \\ =\frac{32}{6} a - \frac{5}{6} a+\frac{57}{6} a+\frac{7}{6} b+\frac{2}{6} b-\frac{21}{6} b-\frac{3}{6} c+\frac{2}{6} c+\frac{7}{6} c \\ =\frac{84}{6} a+\frac{6}{6} c - \frac{12}{6} b \\ = 14a + c - 2b \)
Oder: \( = c + 2·(7a - b) \)
\( 7 \frac{1}{4} a x-3 \frac{1}{2} b x+5 \frac{2}{3} c x-2 \frac{1}{8} a x+4 \frac{5}{6} b x-2 \frac{1}{9} c x \)
\( 7 \frac{1}{4} a x-3 \frac{1}{2} b x+5 \frac{2}{3} c x-2 \frac{1}{8} a x+4 \frac{5}{6} b x-2 \frac{1}{9} cx \\ = \frac{23}{4} a x-\frac{17}{8} a x-\frac{7}{2} b x+\frac{25}{6} b x+\frac{17}{3}cx-\frac{19}{9} cx \\ = \frac{58}{8} a x-\frac{17}{8} a x-\frac{21}{6} b x+\frac{29}{6} b x+\frac{51}{9} cx - \frac{19}{9} cx \\ = \frac{41}{8} a x+\frac{8}{6} b x+\frac{32}{9}cx \\ = x \cdot\left(\frac{41}{8} a+\frac{4}{3}b+\frac{32}{9} c\right) \)
Lösen Sie die Klammern auf und vereinfachen Sie die Terme. Beispiel:
\( 3 x+2 y+9 z-[2 z-(4 x-9 y)] = 3 x+2 y+9 z-[2 z-4 x+9 y] \\ = 3 x+2 y+9 z-2 z+4 x-9 y = 7 x-7 y+7 z=7(x-y+z) \)
\( 4 x-[2 x-(x+18)-3]+3 \)
\( 4 x-[2 x-(x+18)-3]+3 \\ =4 x-(2 x-x-18-3)+3 \\ =3 x+24=3 \cdot(x+8)\\ = 4 x-2 x+x+18+3+3 \\ = 3x + 24 = 3·(x + 8) \)
\( [ (2 a-5 b)-12 b]-(7 a-5 b)-[-(4 a+b) ] \)
\( [ (2 a-5 b)-12 b ]-(7 a-5 b)-[-(4 a+b)] \\ = 2 a-5 b-12 b - 7 a+5 b+4 a+b \\ = -a-11 b \)
\( 16 u v-(5 v w+21 u v)-(13 w z-18 u v)-(26 w z+21 v w) \)
\( 16 u v-(5 v w+21 u v)-(13 w z-18 u v)-(26 w z+21 v w) \\ = 16 u v-5 v w-21 u v-13 w z+18 u v-26 w z-21 v w \\ = 13 u v-26 v w-39 w z \\ = 13 \cdot(u v-2 v w-3 w z) \)
\( 8 \frac{1}{2} x-\left[\left(3 \frac{1}{3} y-2 z\right)-4 x\right]-[4 y-(3 x-z)] \)
\( 8 \frac{1}{2} x-\left[\left(3 \frac{1}{3} y-2 z\right)-4 x\right]-[4 y-(3 x-z)] \\ = \frac{17}{2} x-\left(\frac{10}{3} y-2 z-4 x\right)-(4 y - 3 x+2) \\ = \frac{17}{2} x-\frac{10}{3} y+2 z+4 x-4 y+3 x - z \\ = 15,5 x-\frac{22}{3} y+z \)
\( 3 \frac{4}{5} m-\left[8 \frac{1}{3} n-\left(2 m-4 \frac{1}{6} n\right)\right]-\left[3 \frac{5}{6} n+\left(3 \frac{1}{2} m-2 \frac{1}{3} n\right)\right] \)
\( 3 \frac{4}{5} m-\left[8 \frac{1}{3} n-\left(2 m-4 \frac{1}{6} n\right)\right]-\left[3 \frac{5}{6} n+\left(3 \frac{1}{2} m-2 \frac{1}{3} n\right)\right] \\ =\frac{19}{5} m-\left(\frac{25}{3} n-2 m+\frac{25}{6} n\right)-\left(\frac{23}{6}n + \frac{7}{2} m-\frac{7}{3} n\right) \\ =\frac{19}{5} m-\frac{25}{3} n+2 m-\frac{25}{6} n-\frac{23}{6} n-\frac{7}{2} m+\frac{7}{3} n \\ =\frac{38}{10} m+\frac{20}{10} m - \frac{35}{10} m-\frac{50}{6} n-\frac{25}{6} n+\frac{14}{6} n-\frac{23}{6} n \\ =\frac{23}{10} m-\frac{84}{6} n \\ = 2,3 m-14 n \)
Multiplizieren Sie die Summen aus.
Beispiel: \( 3(2 a+b)=6 a+3 b \)
\( 2,5(4 x+2 y) \)
\( 2,5 \cdot(4 x+2 y)= 10 x+5 y \)
\( 2,4 a\left(a^{2}-5 a\right) \)
\( 2,4 a \cdot\left(a^{2}-5 a\right)=2,4 a^{3}-12 a^{2} \)
\( \frac{1}{2}(4 x-12 y) \)
\( \frac{1}{2} \cdot(4 x-12 y)=2 x-6 y \)
\( 3 m(2 a-2 b+4 c) \)
\( 3 m \cdot(2 a-2 b+4 c)=6 a m-6 b m+12 c m \)
\( 6 m(3 m-1,5 n-4 m n) \)
\( 6 m \cdot(3 m-1,5 n-4 m n)=18 m^{2}-9 m n-24 m^2 n \)
Multiplizieren Sie und fassen Sie zusammen.
Beispiel: \( 3(u+2 v)+2 u=3 u+6 v+2 u=5 u+6 v \)
\( 9 x-2(x-3 y)+4(y+4 x) \)
\( 9 x-2(x-3 y)+4(y+4 x) \\ = 9 x-2 x+6 y+4 y+16 x \\ = 23 x+10 y \)
\( u(3 u-2 v)-2 v(1,5 u-5,5 v)+2 u v \)
\( u \cdot (3 u-2 v)-2 v(1,5 u-5,5 v)+2 u v \\ =3 u^{2}-2 u v-3 u v + 11 v^{2}+2 u v \\ =3 u^{2} - 3 u v + 11 v^2 \)
\( 2 x^{2}-4 x(x+2 y)+3 y^{2}+2 y(2 x-y)+3 x y \)
\( 2 x^{2}-4 x \cdot(x+2 y)+3 y^{2}+2 y \cdot(2 x-y)+3 x y \\ = 2 x^{2}-4 x^{2}-8 x y+3 y^{2}+4 x y-2 y^{2}+3 x y \\ = -2 x^{2}-x y+y^{2} \)
Multiplizieren Sie und fassen Sie zusammen.
Beispiel: \( (2 a+3 b)(4 m+2 n)=8 a m+4 a n+12 b m+6 b n \)
\( (2,5 r-1,5 s)(-4 r-2 s) \)
\( (2,5 r-1,5) \cdot(-4 r-2 s) \\ =-10 r^{2}-5 r s+6 r s+3 s^{2} \\ =-10 r^{2} + rs+3 s^{2} \)
\( \left(\frac{5}{6} u+\frac{1}{3} v\right)\left(\frac{9}{10} u-\frac{3}{5} v\right) \)
\( \left(\frac{5}{6} u+\frac{1}{3} v\right)\left(\frac{9}{10} u-\frac{3}{5} v\right) \\ = \frac{3}{4} u^{2}-\frac{1}{2} u v+\frac{3}{10} u v-\frac{1}{5} v^{2} \\ = \frac{3}{4} u^{2}-\frac{2}{10} u v-\frac{1}{5} v^{2} \\ = \frac{3}{4} u^{2}-\frac{1}{5} u v-\frac{1}{3} v^{2} \)
\( (2 a+5 b-c)(3 a-b) \)
\( (2 a+5 b-c)(3 a-b) \\ = 6 a^{2}-2 a b+15 a b+5 b^{2}-3 a c+b c \\ = 6 a^{2}+13 a b-5 b^{2}-3 a c+b c \)
\( (1,5 x-3,5 y+6 z)(6 x+10 y-2 z) \)
\( (1,5 x-3,5 y+6 z)(6 x+10 y-2 z) \\ = 9 x^{2}+15 x y-3 x z-35 y^{2}-21 x y+7 y z+36 x z+60 y z-12 z^{2} \\ = 9 x^{2}-6 x y+33 x z-35 y^{2}+67 y z-12 z^{2} \)
\( (-2)(5 a-7 b)(2 a+3 b) + 3\left( \frac{2}{3} a+\frac{1}{3} b\right) \cdot 6\left(\frac{5}{6} a-\frac{2}{3} b \right) \)
\( (-2) \cdot(5 a-7 b) \cdot(2 a+3 b)+3 \cdot\left(\frac{2}{3} a+\frac{1}{3} b\right) \cdot 6\left(\frac{5}{6} a-\frac{2}{3} b\right) \\ = (-10 a+14 b) \cdot(2 a+3 b)+(2 a+b) \cdot(5 a-4 b) \\ = -20 a^{2}-30 a b+28 a b+42 b^{2}+10 a^{2}-8 a b+5 ab -4 b^{2} \\ = -10 a^{2}-5 a b+38 b^{2} \)