Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{8}& = & \frac{2}{3}x+2 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{16}{ \color{blue}{24} }x+\frac{48}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-15& = & 16x+48 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{-16x} & = & \color{red}{16x} +48 \color{blue}{-16x} \color{blue}{+15} \\\Leftrightarrow & -4x& = & 63 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{63}{-4} \\\Leftrightarrow & x = \frac{-63}{4} & & \\ & V = \left\{ \frac{-63}{4} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{1}{6}x-5 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x-\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-90& = & 65x-1950 \\\Leftrightarrow & 78x \color{red}{-90} \color{blue}{+90} \color{blue}{-65x} & = & \color{red}{65x} -1950 \color{blue}{-65x} \color{blue}{+90} \\\Leftrightarrow & 13x& = & -1860 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-1860}{13} \\\Leftrightarrow & x = \frac{-1860}{13} & & \\ & V = \left\{ \frac{-1860}{13} \right\} & \\\end{align}\)
3. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }+ \frac{ 140 }{ \color{blue}{455} })& = & (\frac{-637}{ \color{blue}{455} }x+\frac{1820}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x+140& = & -637x+1820 \\\Leftrightarrow & 65x \color{red}{+140} \color{blue}{-140} \color{blue}{+637x} & = & \color{red}{-637x} +1820 \color{blue}{+637x} \color{blue}{-140} \\\Leftrightarrow & 702x& = & 1680 \\\Leftrightarrow & \frac{702x}{ \color{red}{702} }& = & \frac{1680}{702} \\\Leftrightarrow & x = \frac{280}{117} & & \\ & V = \left\{ \frac{280}{117} \right\} & \\\end{align}\)
4. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{-8}{3}x+3 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{-48}{ \color{blue}{18} }x+\frac{54}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+8& = & -48x+54 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{+48x} & = & \color{red}{-48x} +54 \color{blue}{+48x} \color{blue}{-8} \\\Leftrightarrow & 57x& = & 46 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{46}{57} \\\Leftrightarrow & x = \frac{46}{57} & & \\ & V = \left\{ \frac{46}{57} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+4& = & 15x+150 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-15x} & = & \color{red}{15x} +150 \color{blue}{-15x} \color{blue}{-4} \\\Leftrightarrow & -9x& = & 146 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{146}{-9} \\\Leftrightarrow & x = \frac{-146}{9} & & \\ & V = \left\{ \frac{-146}{9} \right\} & \\\end{align}\)
6. \(\text{315 is het kleinste gemene veelvoud van 7, 9 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{315.} (\frac{45x}{ \color{blue}{315} }+ \frac{ 140 }{ \color{blue}{315} })& = & (\frac{-252}{ \color{blue}{315} }x+\frac{945}{ \color{blue}{315} }) \color{blue}{.315} \\\Leftrightarrow & 45x+140& = & -252x+945 \\\Leftrightarrow & 45x \color{red}{+140} \color{blue}{-140} \color{blue}{+252x} & = & \color{red}{-252x} +945 \color{blue}{+252x} \color{blue}{-140} \\\Leftrightarrow & 297x& = & 805 \\\Leftrightarrow & \frac{297x}{ \color{red}{297} }& = & \frac{805}{297} \\\Leftrightarrow & x = \frac{805}{297} & & \\ & V = \left\{ \frac{805}{297} \right\} & \\\end{align}\)
7. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }+ \frac{ 100 }{ \color{blue}{220} })& = & (\frac{-165}{ \color{blue}{220} }x+\frac{660}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x+100& = & -165x+660 \\\Leftrightarrow & 44x \color{red}{+100} \color{blue}{-100} \color{blue}{+165x} & = & \color{red}{-165x} +660 \color{blue}{+165x} \color{blue}{-100} \\\Leftrightarrow & 209x& = & 560 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{560}{209} \\\Leftrightarrow & x = \frac{560}{209} & & \\ & V = \left\{ \frac{560}{209} \right\} & \\\end{align}\)
8. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{7}{4}x+5 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{42}{ \color{blue}{24} }x+\frac{120}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-9& = & 42x+120 \\\Leftrightarrow & 8x \color{red}{-9} \color{blue}{+9} \color{blue}{-42x} & = & \color{red}{42x} +120 \color{blue}{-42x} \color{blue}{+9} \\\Leftrightarrow & -34x& = & 129 \\\Leftrightarrow & \frac{-34x}{ \color{red}{-34} }& = & \frac{129}{-34} \\\Leftrightarrow & x = \frac{-129}{34} & & \\ & V = \left\{ \frac{-129}{34} \right\} & \\\end{align}\)
9. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{8}& = & \frac{-2}{3}x-5 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{-16}{ \color{blue}{24} }x-\frac{120}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x+9& = & -16x-120 \\\Leftrightarrow & 6x \color{red}{+9} \color{blue}{-9} \color{blue}{+16x} & = & \color{red}{-16x} -120 \color{blue}{+16x} \color{blue}{-9} \\\Leftrightarrow & 22x& = & -129 \\\Leftrightarrow & \frac{22x}{ \color{red}{22} }& = & \frac{-129}{22} \\\Leftrightarrow & x = \frac{-129}{22} & & \\ & V = \left\{ \frac{-129}{22} \right\} & \\\end{align}\)
10. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{-3}{4}x-2 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 35 }{ \color{blue}{112} })& = & (\frac{-84}{ \color{blue}{112} }x-\frac{224}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-35& = & -84x-224 \\\Leftrightarrow & 16x \color{red}{-35} \color{blue}{+35} \color{blue}{+84x} & = & \color{red}{-84x} -224 \color{blue}{+84x} \color{blue}{+35} \\\Leftrightarrow & 100x& = & -189 \\\Leftrightarrow & \frac{100x}{ \color{red}{100} }& = & \frac{-189}{100} \\\Leftrightarrow & x = \frac{-189}{100} & & \\ & V = \left\{ \frac{-189}{100} \right\} & \\\end{align}\)
11. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 14 }{ \color{blue}{105} })& = & (\frac{-84}{ \color{blue}{105} }x+\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-14& = & -84x+210 \\\Leftrightarrow & 15x \color{red}{-14} \color{blue}{+14} \color{blue}{+84x} & = & \color{red}{-84x} +210 \color{blue}{+84x} \color{blue}{+14} \\\Leftrightarrow & 99x& = & 224 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{224}{99} \\\Leftrightarrow & x = \frac{224}{99} & & \\ & V = \left\{ \frac{224}{99} \right\} & \\\end{align}\)
12. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{-2}{3}x-3 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }- \frac{ 60 }{ \color{blue}{105} })& = & (\frac{-70}{ \color{blue}{105} }x-\frac{315}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x-60& = & -70x-315 \\\Leftrightarrow & 21x \color{red}{-60} \color{blue}{+60} \color{blue}{+70x} & = & \color{red}{-70x} -315 \color{blue}{+70x} \color{blue}{+60} \\\Leftrightarrow & 91x& = & -255 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-255}{91} \\\Leftrightarrow & x = \frac{-255}{91} & & \\ & V = \left\{ \frac{-255}{91} \right\} & \\\end{align}\)