Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{9}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 70 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x+\frac{1008}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-70& = & 63x+1008 \\\Leftrightarrow & 18x \color{red}{-70} \color{blue}{+70} \color{blue}{-63x} & = & \color{red}{63x} +1008 \color{blue}{-63x} \color{blue}{+70} \\\Leftrightarrow & -45x& = & 1078 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{1078}{-45} \\\Leftrightarrow & x = \frac{-1078}{45} & & \\ & V = \left\{ \frac{-1078}{45} \right\} & \\\end{align}\)
2. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{7}{2}x-5 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{392}{ \color{blue}{112} }x-\frac{560}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & 392x-560 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{-392x} & = & \color{red}{392x} -560 \color{blue}{-392x} \color{blue}{-35} \\\Leftrightarrow & -376x& = & -595 \\\Leftrightarrow & \frac{-376x}{ \color{red}{-376} }& = & \frac{-595}{-376} \\\Leftrightarrow & x = \frac{595}{376} & & \\ & V = \left\{ \frac{595}{376} \right\} & \\\end{align}\)
3. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{-5}{6}x-6 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-200}{ \color{blue}{240} }x-\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+75& = & -200x-1440 \\\Leftrightarrow & 48x \color{red}{+75} \color{blue}{-75} \color{blue}{+200x} & = & \color{red}{-200x} -1440 \color{blue}{+200x} \color{blue}{-75} \\\Leftrightarrow & 248x& = & -1515 \\\Leftrightarrow & \frac{248x}{ \color{red}{248} }& = & \frac{-1515}{248} \\\Leftrightarrow & x = \frac{-1515}{248} & & \\ & V = \left\{ \frac{-1515}{248} \right\} & \\\end{align}\)
4. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{-3}{4}x+4 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }- \frac{ 84 }{ \color{blue}{364} })& = & (\frac{-273}{ \color{blue}{364} }x+\frac{1456}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x-84& = & -273x+1456 \\\Leftrightarrow & 52x \color{red}{-84} \color{blue}{+84} \color{blue}{+273x} & = & \color{red}{-273x} +1456 \color{blue}{+273x} \color{blue}{+84} \\\Leftrightarrow & 325x& = & 1540 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{1540}{325} \\\Leftrightarrow & x = \frac{308}{65} & & \\ & V = \left\{ \frac{308}{65} \right\} & \\\end{align}\)
5. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x+\frac{144}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-9& = & 12x+144 \\\Leftrightarrow & 8x \color{red}{-9} \color{blue}{+9} \color{blue}{-12x} & = & \color{red}{12x} +144 \color{blue}{-12x} \color{blue}{+9} \\\Leftrightarrow & -4x& = & 153 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{153}{-4} \\\Leftrightarrow & x = \frac{-153}{4} & & \\ & V = \left\{ \frac{-153}{4} \right\} & \\\end{align}\)
6. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{-7}{4}x-1 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }- \frac{ 56 }{ \color{blue}{420} })& = & (\frac{-735}{ \color{blue}{420} }x-\frac{420}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x-56& = & -735x-420 \\\Leftrightarrow & 60x \color{red}{-56} \color{blue}{+56} \color{blue}{+735x} & = & \color{red}{-735x} -420 \color{blue}{+735x} \color{blue}{+56} \\\Leftrightarrow & 795x& = & -364 \\\Leftrightarrow & \frac{795x}{ \color{red}{795} }& = & \frac{-364}{795} \\\Leftrightarrow & x = \frac{-364}{795} & & \\ & V = \left\{ \frac{-364}{795} \right\} & \\\end{align}\)
7. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{8}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-168}{ \color{blue}{120} }x-\frac{600}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+75& = & -168x-600 \\\Leftrightarrow & 20x \color{red}{+75} \color{blue}{-75} \color{blue}{+168x} & = & \color{red}{-168x} -600 \color{blue}{+168x} \color{blue}{-75} \\\Leftrightarrow & 188x& = & -675 \\\Leftrightarrow & \frac{188x}{ \color{red}{188} }& = & \frac{-675}{188} \\\Leftrightarrow & x = \frac{-675}{188} & & \\ & V = \left\{ \frac{-675}{188} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{1}{6}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-4& = & 5x-90 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-5x} & = & \color{red}{5x} -90 \color{blue}{-5x} \color{blue}{+4} \\\Leftrightarrow & x& = & -86 \\ & V = \left\{ -86 \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{-8}{3}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-160}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & -160x+360 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{+160x} & = & \color{red}{-160x} +360 \color{blue}{+160x} \color{blue}{-16} \\\Leftrightarrow & 175x& = & 344 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{344}{175} \\\Leftrightarrow & x = \frac{344}{175} & & \\ & V = \left\{ \frac{344}{175} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -168x-1680 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+168x} & = & \color{red}{-168x} -1680 \color{blue}{+168x} \color{blue}{-120} \\\Leftrightarrow & 203x& = & -1800 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1800}{203} \\\Leftrightarrow & x = \frac{-1800}{203} & & \\ & V = \left\{ \frac{-1800}{203} \right\} & \\\end{align}\)
11. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x+\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+4& = & 9x+108 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-9x} & = & \color{red}{9x} +108 \color{blue}{-9x} \color{blue}{-4} \\\Leftrightarrow & -3x& = & 104 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{104}{-3} \\\Leftrightarrow & x = \frac{-104}{3} & & \\ & V = \left\{ \frac{-104}{3} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -70x-168 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+70x} & = & \color{red}{-70x} -168 \color{blue}{+70x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & -192 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-192}{91} \\\Leftrightarrow & x = \frac{-192}{91} & & \\ & V = \left\{ \frac{-192}{91} \right\} & \\\end{align}\)