Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{11}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 30 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{110}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+30& = & 55x+110 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-55x} & = & \color{red}{55x} +110 \color{blue}{-55x} \color{blue}{-30} \\\Leftrightarrow & -33x& = & 80 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{80}{-33} \\\Leftrightarrow & x = \frac{-80}{33} & & \\ & V = \left\{ \frac{-80}{33} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & -24x+240 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{+24x} & = & \color{red}{-24x} +240 \color{blue}{+24x} \color{blue}{-25} \\\Leftrightarrow & 29x& = & 215 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{215}{29} \\\Leftrightarrow & x = \frac{215}{29} & & \\ & V = \left\{ \frac{215}{29} \right\} & \\\end{align}\)
3. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{3}{2}x+1 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 70 }{ \color{blue}{154} })& = & (\frac{231}{ \color{blue}{154} }x+\frac{154}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-70& = & 231x+154 \\\Leftrightarrow & 22x \color{red}{-70} \color{blue}{+70} \color{blue}{-231x} & = & \color{red}{231x} +154 \color{blue}{-231x} \color{blue}{+70} \\\Leftrightarrow & -209x& = & 224 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{224}{-209} \\\Leftrightarrow & x = \frac{-224}{209} & & \\ & V = \left\{ \frac{-224}{209} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{-5}{6}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-35}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-24& = & -35x+252 \\\Leftrightarrow & 6x \color{red}{-24} \color{blue}{+24} \color{blue}{+35x} & = & \color{red}{-35x} +252 \color{blue}{+35x} \color{blue}{+24} \\\Leftrightarrow & 41x& = & 276 \\\Leftrightarrow & \frac{41x}{ \color{red}{41} }& = & \frac{276}{41} \\\Leftrightarrow & x = \frac{276}{41} & & \\ & V = \left\{ \frac{276}{41} \right\} & \\\end{align}\)
5. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{13}& = & \frac{5}{4}x-8 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }+ \frac{ 84 }{ \color{blue}{364} })& = & (\frac{455}{ \color{blue}{364} }x-\frac{2912}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x+84& = & 455x-2912 \\\Leftrightarrow & 52x \color{red}{+84} \color{blue}{-84} \color{blue}{-455x} & = & \color{red}{455x} -2912 \color{blue}{-455x} \color{blue}{-84} \\\Leftrightarrow & -403x& = & -2996 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{-2996}{-403} \\\Leftrightarrow & x = \frac{2996}{403} & & \\ & V = \left\{ \frac{2996}{403} \right\} & \\\end{align}\)
6. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{-2}{3}x+2 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x+\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-24& = & -52x+156 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{+52x} & = & \color{red}{-52x} +156 \color{blue}{+52x} \color{blue}{+24} \\\Leftrightarrow & 91x& = & 180 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{180}{91} \\\Leftrightarrow & x = \frac{180}{91} & & \\ & V = \left\{ \frac{180}{91} \right\} & \\\end{align}\)
7. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{7}{3}x+1 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{112}{ \color{blue}{48} }x+\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & 112x+48 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{-112x} & = & \color{red}{112x} +48 \color{blue}{-112x} \color{blue}{+15} \\\Leftrightarrow & -88x& = & 63 \\\Leftrightarrow & \frac{-88x}{ \color{red}{-88} }& = & \frac{63}{-88} \\\Leftrightarrow & x = \frac{-63}{88} & & \\ & V = \left\{ \frac{-63}{88} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{-5}{6}x+3 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-275}{ \color{blue}{330} }x+\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+120& = & -275x+990 \\\Leftrightarrow & 66x \color{red}{+120} \color{blue}{-120} \color{blue}{+275x} & = & \color{red}{-275x} +990 \color{blue}{+275x} \color{blue}{-120} \\\Leftrightarrow & 341x& = & 870 \\\Leftrightarrow & \frac{341x}{ \color{red}{341} }& = & \frac{870}{341} \\\Leftrightarrow & x = \frac{870}{341} & & \\ & V = \left\{ \frac{870}{341} \right\} & \\\end{align}\)
9. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 21 }{ \color{blue}{112} })& = & (\frac{56}{ \color{blue}{112} }x+\frac{448}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-21& = & 56x+448 \\\Leftrightarrow & 16x \color{red}{-21} \color{blue}{+21} \color{blue}{-56x} & = & \color{red}{56x} +448 \color{blue}{-56x} \color{blue}{+21} \\\Leftrightarrow & -40x& = & 469 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{469}{-40} \\\Leftrightarrow & x = \frac{-469}{40} & & \\ & V = \left\{ \frac{-469}{40} \right\} & \\\end{align}\)
10. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{9}& = & \frac{7}{3}x+3 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 20 }{ \color{blue}{36} })& = & (\frac{84}{ \color{blue}{36} }x+\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-20& = & 84x+108 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{-84x} & = & \color{red}{84x} +108 \color{blue}{-84x} \color{blue}{+20} \\\Leftrightarrow & -75x& = & 128 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{128}{-75} \\\Leftrightarrow & x = \frac{-128}{75} & & \\ & V = \left\{ \frac{-128}{75} \right\} & \\\end{align}\)
11. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{8}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-100}{ \color{blue}{120} }x-\frac{840}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x-75& = & -100x-840 \\\Leftrightarrow & 24x \color{red}{-75} \color{blue}{+75} \color{blue}{+100x} & = & \color{red}{-100x} -840 \color{blue}{+100x} \color{blue}{+75} \\\Leftrightarrow & 124x& = & -765 \\\Leftrightarrow & \frac{124x}{ \color{red}{124} }& = & \frac{-765}{124} \\\Leftrightarrow & x = \frac{-765}{124} & & \\ & V = \left\{ \frac{-765}{124} \right\} & \\\end{align}\)
12. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{3}{4}x-8 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 12 }{ \color{blue}{28} })& = & (\frac{21}{ \color{blue}{28} }x-\frac{224}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+12& = & 21x-224 \\\Leftrightarrow & 4x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} -224 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -17x& = & -236 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{-236}{-17} \\\Leftrightarrow & x = \frac{236}{17} & & \\ & V = \left\{ \frac{236}{17} \right\} & \\\end{align}\)