Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{4}{5}x-3 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{64}{ \color{blue}{80} }x-\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+25& = & 64x-240 \\\Leftrightarrow & 20x \color{red}{+25} \color{blue}{-25} \color{blue}{-64x} & = & \color{red}{64x} -240 \color{blue}{-64x} \color{blue}{-25} \\\Leftrightarrow & -44x& = & -265 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{-265}{-44} \\\Leftrightarrow & x = \frac{265}{44} & & \\ & V = \left\{ \frac{265}{44} \right\} & \\\end{align}\)
2. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{3}{2}x-1 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 28 }{ \color{blue}{126} })& = & (\frac{189}{ \color{blue}{126} }x-\frac{126}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+28& = & 189x-126 \\\Leftrightarrow & 18x \color{red}{+28} \color{blue}{-28} \color{blue}{-189x} & = & \color{red}{189x} -126 \color{blue}{-189x} \color{blue}{-28} \\\Leftrightarrow & -171x& = & -154 \\\Leftrightarrow & \frac{-171x}{ \color{red}{-171} }& = & \frac{-154}{-171} \\\Leftrightarrow & x = \frac{154}{171} & & \\ & V = \left\{ \frac{154}{171} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{12}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }- \frac{ 35 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x-35& = & -63x-504 \\\Leftrightarrow & 12x \color{red}{-35} \color{blue}{+35} \color{blue}{+63x} & = & \color{red}{-63x} -504 \color{blue}{+63x} \color{blue}{+35} \\\Leftrightarrow & 75x& = & -469 \\\Leftrightarrow & \frac{75x}{ \color{red}{75} }& = & \frac{-469}{75} \\\Leftrightarrow & x = \frac{-469}{75} & & \\ & V = \left\{ \frac{-469}{75} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{5}{2}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{75}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & 75x+120 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{-75x} & = & \color{red}{75x} +120 \color{blue}{-75x} \color{blue}{+9} \\\Leftrightarrow & -65x& = & 129 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{129}{-65} \\\Leftrightarrow & x = \frac{-129}{65} & & \\ & V = \left\{ \frac{-129}{65} \right\} & \\\end{align}\)
5. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 35 }{ \color{blue}{112} })& = & (\frac{56}{ \color{blue}{112} }x+\frac{448}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-35& = & 56x+448 \\\Leftrightarrow & 16x \color{red}{-35} \color{blue}{+35} \color{blue}{-56x} & = & \color{red}{56x} +448 \color{blue}{-56x} \color{blue}{+35} \\\Leftrightarrow & -40x& = & 483 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{483}{-40} \\\Leftrightarrow & x = \frac{-483}{40} & & \\ & V = \left\{ \frac{-483}{40} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & 108x-90 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{-108x} & = & \color{red}{108x} -90 \color{blue}{-108x} \color{blue}{-20} \\\Leftrightarrow & -93x& = & -110 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{-110}{-93} \\\Leftrightarrow & x = \frac{110}{93} & & \\ & V = \left\{ \frac{110}{93} \right\} & \\\end{align}\)
7. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{1}{3}x-2 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x-\frac{96}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+9& = & 16x-96 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{-16x} & = & \color{red}{16x} -96 \color{blue}{-16x} \color{blue}{-9} \\\Leftrightarrow & -4x& = & -105 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-105}{-4} \\\Leftrightarrow & x = \frac{105}{4} & & \\ & V = \left\{ \frac{105}{4} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{14}& = & \frac{5}{3}x+6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x+\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-18& = & 140x+504 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-140x} & = & \color{red}{140x} +504 \color{blue}{-140x} \color{blue}{+18} \\\Leftrightarrow & -119x& = & 522 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{522}{-119} \\\Leftrightarrow & x = \frac{-522}{119} & & \\ & V = \left\{ \frac{-522}{119} \right\} & \\\end{align}\)
9. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{3}{5}x-6 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{78}{ \color{blue}{130} }x-\frac{780}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+20& = & 78x-780 \\\Leftrightarrow & 65x \color{red}{+20} \color{blue}{-20} \color{blue}{-78x} & = & \color{red}{78x} -780 \color{blue}{-78x} \color{blue}{-20} \\\Leftrightarrow & -13x& = & -800 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-800}{-13} \\\Leftrightarrow & x = \frac{800}{13} & & \\ & V = \left\{ \frac{800}{13} \right\} & \\\end{align}\)
10. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{-5}{6}x+4 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-200}{ \color{blue}{240} }x+\frac{960}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-75& = & -200x+960 \\\Leftrightarrow & 48x \color{red}{-75} \color{blue}{+75} \color{blue}{+200x} & = & \color{red}{-200x} +960 \color{blue}{+200x} \color{blue}{+75} \\\Leftrightarrow & 248x& = & 1035 \\\Leftrightarrow & \frac{248x}{ \color{red}{248} }& = & \frac{1035}{248} \\\Leftrightarrow & x = \frac{1035}{248} & & \\ & V = \left\{ \frac{1035}{248} \right\} & \\\end{align}\)
11. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+30& = & 35x+350 \\\Leftrightarrow & 14x \color{red}{+30} \color{blue}{-30} \color{blue}{-35x} & = & \color{red}{35x} +350 \color{blue}{-35x} \color{blue}{-30} \\\Leftrightarrow & -21x& = & 320 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{320}{-21} \\\Leftrightarrow & x = \frac{-320}{21} & & \\ & V = \left\{ \frac{-320}{21} \right\} & \\\end{align}\)
12. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{6}{5}x+3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x+\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & 168x+420 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-168x} & = & \color{red}{168x} +420 \color{blue}{-168x} \color{blue}{-40} \\\Leftrightarrow & -133x& = & 380 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{380}{-133} \\\Leftrightarrow & x = \frac{-20}{7} & & \\ & V = \left\{ \frac{-20}{7} \right\} & \\\end{align}\)