Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{3}{5}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & 18x+30 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{-18x} & = & \color{red}{18x} +30 \color{blue}{-18x} \color{blue}{-4} \\\Leftrightarrow & -13x& = & 26 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{26}{-13} \\\Leftrightarrow & x = -2 & & \\ & V = \left\{ -2 \right\} & \\\end{align}\)
2. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{7}{4}x-4 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{84}{ \color{blue}{48} }x-\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 84x-192 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-84x} & = & \color{red}{84x} -192 \color{blue}{-84x} \color{blue}{+9} \\\Leftrightarrow & -68x& = & -183 \\\Leftrightarrow & \frac{-68x}{ \color{red}{-68} }& = & \frac{-183}{-68} \\\Leftrightarrow & x = \frac{183}{68} & & \\ & V = \left\{ \frac{183}{68} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 5, 14 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{14}& = & \frac{-3}{4}x+5 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 50 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x+\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-50& = & -105x+700 \\\Leftrightarrow & 28x \color{red}{-50} \color{blue}{+50} \color{blue}{+105x} & = & \color{red}{-105x} +700 \color{blue}{+105x} \color{blue}{+50} \\\Leftrightarrow & 133x& = & 750 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{750}{133} \\\Leftrightarrow & x = \frac{750}{133} & & \\ & V = \left\{ \frac{750}{133} \right\} & \\\end{align}\)
4. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{1}{4}x+3 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 16 }{ \color{blue}{28} })& = & (\frac{7}{ \color{blue}{28} }x+\frac{84}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+16& = & 7x+84 \\\Leftrightarrow & 4x \color{red}{+16} \color{blue}{-16} \color{blue}{-7x} & = & \color{red}{7x} +84 \color{blue}{-7x} \color{blue}{-16} \\\Leftrightarrow & -3x& = & 68 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{68}{-3} \\\Leftrightarrow & x = \frac{-68}{3} & & \\ & V = \left\{ \frac{-68}{3} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{5}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & 140x-672 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{-140x} & = & \color{red}{140x} -672 \color{blue}{-140x} \color{blue}{-48} \\\Leftrightarrow & -119x& = & -720 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{-720}{-119} \\\Leftrightarrow & x = \frac{720}{119} & & \\ & V = \left\{ \frac{720}{119} \right\} & \\\end{align}\)
6. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+9& = & 16x+336 \\\Leftrightarrow & 24x \color{red}{+9} \color{blue}{-9} \color{blue}{-16x} & = & \color{red}{16x} +336 \color{blue}{-16x} \color{blue}{-9} \\\Leftrightarrow & 8x& = & 327 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{327}{8} \\\Leftrightarrow & x = \frac{327}{8} & & \\ & V = \left\{ \frac{327}{8} \right\} & \\\end{align}\)
7. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{7}{6}x+2 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }- \frac{ 210 }{ \color{blue}{462} })& = & (\frac{539}{ \color{blue}{462} }x+\frac{924}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x-210& = & 539x+924 \\\Leftrightarrow & 66x \color{red}{-210} \color{blue}{+210} \color{blue}{-539x} & = & \color{red}{539x} +924 \color{blue}{-539x} \color{blue}{+210} \\\Leftrightarrow & -473x& = & 1134 \\\Leftrightarrow & \frac{-473x}{ \color{red}{-473} }& = & \frac{1134}{-473} \\\Leftrightarrow & x = \frac{-1134}{473} & & \\ & V = \left\{ \frac{-1134}{473} \right\} & \\\end{align}\)
8. \(\text{140 is het kleinste gemene veelvoud van 4, 14 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{14}& = & \frac{7}{5}x-6 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 30 }{ \color{blue}{140} })& = & (\frac{196}{ \color{blue}{140} }x-\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+30& = & 196x-840 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{-196x} & = & \color{red}{196x} -840 \color{blue}{-196x} \color{blue}{-30} \\\Leftrightarrow & -161x& = & -870 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{-870}{-161} \\\Leftrightarrow & x = \frac{870}{161} & & \\ & V = \left\{ \frac{870}{161} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & -56x+84 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+56x} & = & \color{red}{-56x} +84 \color{blue}{+56x} \color{blue}{+24} \\\Leftrightarrow & 77x& = & 108 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{108}{77} \\\Leftrightarrow & x = \frac{108}{77} & & \\ & V = \left\{ \frac{108}{77} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -168x-1470 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+168x} & = & \color{red}{-168x} -1470 \color{blue}{+168x} \color{blue}{-60} \\\Leftrightarrow & 203x& = & -1530 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1530}{203} \\\Leftrightarrow & x = \frac{-1530}{203} & & \\ & V = \left\{ \frac{-1530}{203} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{14}& = & \frac{7}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+45& = & 294x-630 \\\Leftrightarrow & 35x \color{red}{+45} \color{blue}{-45} \color{blue}{-294x} & = & \color{red}{294x} -630 \color{blue}{-294x} \color{blue}{-45} \\\Leftrightarrow & -259x& = & -675 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{-675}{-259} \\\Leftrightarrow & x = \frac{675}{259} & & \\ & V = \left\{ \frac{675}{259} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{13}& = & \frac{-5}{6}x-8 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }- \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-325}{ \color{blue}{390} }x-\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x-150& = & -325x-3120 \\\Leftrightarrow & 78x \color{red}{-150} \color{blue}{+150} \color{blue}{+325x} & = & \color{red}{-325x} -3120 \color{blue}{+325x} \color{blue}{+150} \\\Leftrightarrow & 403x& = & -2970 \\\Leftrightarrow & \frac{403x}{ \color{red}{403} }& = & \frac{-2970}{403} \\\Leftrightarrow & x = \frac{-2970}{403} & & \\ & V = \left\{ \frac{-2970}{403} \right\} & \\\end{align}\)