Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}+\frac{4}{7}=\frac{-5}{3}x-5\)
2. \(\frac{x}{2}+\frac{4}{9}=\frac{6}{5}x+8\)
3. \(\frac{x}{3}-\frac{3}{10}=\frac{7}{5}x-2\)
4. \(\frac{x}{2}-\frac{3}{16}=\frac{-2}{3}x-1\)
5. \(\frac{x}{5}-\frac{5}{14}=\frac{-5}{3}x+3\)
6. \(\frac{x}{7}-\frac{2}{7}=\frac{-3}{4}x+2\)
7. \(\frac{x}{2}-\frac{4}{11}=\frac{1}{3}x+2\)
8. \(\frac{x}{7}+\frac{5}{8}=\frac{3}{4}x-1\)
9. \(\frac{x}{5}+\frac{2}{15}=\frac{5}{6}x+4\)
10. \(\frac{x}{7}+\frac{5}{11}=\frac{7}{2}x-7\)
11. \(\frac{x}{6}+\frac{2}{15}=\frac{-2}{5}x-7\)
12. \(\frac{x}{3}-\frac{4}{13}=\frac{1}{2}x+1\)

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Reeks met breuken waarbij we ervoor kiezen om de breuken weg te werken. Neem je ZRM!

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -70x-210 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+70x} & = & \color{red}{-70x} -210 \color{blue}{+70x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & -234 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-234}{91} \\\Leftrightarrow & x = \frac{-18}{7} & & \\ & V = \left\{ \frac{-18}{7} \right\} & \\\end{align}\)
2. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x+\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+40& = & 108x+720 \\\Leftrightarrow & 45x \color{red}{+40} \color{blue}{-40} \color{blue}{-108x} & = & \color{red}{108x} +720 \color{blue}{-108x} \color{blue}{-40} \\\Leftrightarrow & -63x& = & 680 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{680}{-63} \\\Leftrightarrow & x = \frac{-680}{63} & & \\ & V = \left\{ \frac{-680}{63} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{7}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{42}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & 42x-60 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{-42x} & = & \color{red}{42x} -60 \color{blue}{-42x} \color{blue}{+9} \\\Leftrightarrow & -32x& = & -51 \\\Leftrightarrow & \frac{-32x}{ \color{red}{-32} }& = & \frac{-51}{-32} \\\Leftrightarrow & x = \frac{51}{32} & & \\ & V = \left\{ \frac{51}{32} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-32}{ \color{blue}{48} }x-\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-9& = & -32x-48 \\\Leftrightarrow & 24x \color{red}{-9} \color{blue}{+9} \color{blue}{+32x} & = & \color{red}{-32x} -48 \color{blue}{+32x} \color{blue}{+9} \\\Leftrightarrow & 56x& = & -39 \\\Leftrightarrow & \frac{56x}{ \color{red}{56} }& = & \frac{-39}{56} \\\Leftrightarrow & x = \frac{-39}{56} & & \\ & V = \left\{ \frac{-39}{56} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{14}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{-350}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-75& = & -350x+630 \\\Leftrightarrow & 42x \color{red}{-75} \color{blue}{+75} \color{blue}{+350x} & = & \color{red}{-350x} +630 \color{blue}{+350x} \color{blue}{+75} \\\Leftrightarrow & 392x& = & 705 \\\Leftrightarrow & \frac{392x}{ \color{red}{392} }& = & \frac{705}{392} \\\Leftrightarrow & x = \frac{705}{392} & & \\ & V = \left\{ \frac{705}{392} \right\} & \\\end{align}\)
6. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{-3}{4}x+2 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 8 }{ \color{blue}{28} })& = & (\frac{-21}{ \color{blue}{28} }x+\frac{56}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-8& = & -21x+56 \\\Leftrightarrow & 4x \color{red}{-8} \color{blue}{+8} \color{blue}{+21x} & = & \color{red}{-21x} +56 \color{blue}{+21x} \color{blue}{+8} \\\Leftrightarrow & 25x& = & 64 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{64}{25} \\\Leftrightarrow & x = \frac{64}{25} & & \\ & V = \left\{ \frac{64}{25} \right\} & \\\end{align}\)
7. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x+\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-24& = & 22x+132 \\\Leftrightarrow & 33x \color{red}{-24} \color{blue}{+24} \color{blue}{-22x} & = & \color{red}{22x} +132 \color{blue}{-22x} \color{blue}{+24} \\\Leftrightarrow & 11x& = & 156 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{156}{11} \\\Leftrightarrow & x = \frac{156}{11} & & \\ & V = \left\{ \frac{156}{11} \right\} & \\\end{align}\)
8. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{8}& = & \frac{3}{4}x-1 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }+ \frac{ 35 }{ \color{blue}{56} })& = & (\frac{42}{ \color{blue}{56} }x-\frac{56}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x+35& = & 42x-56 \\\Leftrightarrow & 8x \color{red}{+35} \color{blue}{-35} \color{blue}{-42x} & = & \color{red}{42x} -56 \color{blue}{-42x} \color{blue}{-35} \\\Leftrightarrow & -34x& = & -91 \\\Leftrightarrow & \frac{-34x}{ \color{red}{-34} }& = & \frac{-91}{-34} \\\Leftrightarrow & x = \frac{91}{34} & & \\ & V = \left\{ \frac{91}{34} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{5}{6}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{25}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+4& = & 25x+120 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-25x} & = & \color{red}{25x} +120 \color{blue}{-25x} \color{blue}{-4} \\\Leftrightarrow & -19x& = & 116 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{116}{-19} \\\Leftrightarrow & x = \frac{-116}{19} & & \\ & V = \left\{ \frac{-116}{19} \right\} & \\\end{align}\)
10. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{7}{2}x-7 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 70 }{ \color{blue}{154} })& = & (\frac{539}{ \color{blue}{154} }x-\frac{1078}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+70& = & 539x-1078 \\\Leftrightarrow & 22x \color{red}{+70} \color{blue}{-70} \color{blue}{-539x} & = & \color{red}{539x} -1078 \color{blue}{-539x} \color{blue}{-70} \\\Leftrightarrow & -517x& = & -1148 \\\Leftrightarrow & \frac{-517x}{ \color{red}{-517} }& = & \frac{-1148}{-517} \\\Leftrightarrow & x = \frac{1148}{517} & & \\ & V = \left\{ \frac{1148}{517} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & -12x-210 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{+12x} & = & \color{red}{-12x} -210 \color{blue}{+12x} \color{blue}{-4} \\\Leftrightarrow & 17x& = & -214 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-214}{17} \\\Leftrightarrow & x = \frac{-214}{17} & & \\ & V = \left\{ \frac{-214}{17} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-24& = & 39x+78 \\\Leftrightarrow & 26x \color{red}{-24} \color{blue}{+24} \color{blue}{-39x} & = & \color{red}{39x} +78 \color{blue}{-39x} \color{blue}{+24} \\\Leftrightarrow & -13x& = & 102 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{102}{-13} \\\Leftrightarrow & x = \frac{-102}{13} & & \\ & V = \left\{ \frac{-102}{13} \right\} & \\\end{align}\)