Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{11}& = & \frac{2}{5}x+3 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 40 }{ \color{blue}{220} })& = & (\frac{88}{ \color{blue}{220} }x+\frac{660}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-40& = & 88x+660 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{-88x} & = & \color{red}{88x} +660 \color{blue}{-88x} \color{blue}{+40} \\\Leftrightarrow & -33x& = & 700 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{700}{-33} \\\Leftrightarrow & x = \frac{-700}{33} & & \\ & V = \left\{ \frac{-700}{33} \right\} & \\\end{align}\)
2. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{7}{3}x-7 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{308}{ \color{blue}{132} }x-\frac{924}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+60& = & 308x-924 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{-308x} & = & \color{red}{308x} -924 \color{blue}{-308x} \color{blue}{-60} \\\Leftrightarrow & -275x& = & -984 \\\Leftrightarrow & \frac{-275x}{ \color{red}{-275} }& = & \frac{-984}{-275} \\\Leftrightarrow & x = \frac{984}{275} & & \\ & V = \left\{ \frac{984}{275} \right\} & \\\end{align}\)
3. \(\text{120 is het kleinste gemene veelvoud van 3, 8 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{-7}{5}x-7 \\\Leftrightarrow & \color{blue}{120.} (\frac{40x}{ \color{blue}{120} }- \frac{ 45 }{ \color{blue}{120} })& = & (\frac{-168}{ \color{blue}{120} }x-\frac{840}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 40x-45& = & -168x-840 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{+168x} & = & \color{red}{-168x} -840 \color{blue}{+168x} \color{blue}{+45} \\\Leftrightarrow & 208x& = & -795 \\\Leftrightarrow & \frac{208x}{ \color{red}{208} }& = & \frac{-795}{208} \\\Leftrightarrow & x = \frac{-795}{208} & & \\ & V = \left\{ \frac{-795}{208} \right\} & \\\end{align}\)
4. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x+\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+20& = & 108x+180 \\\Leftrightarrow & 45x \color{red}{+20} \color{blue}{-20} \color{blue}{-108x} & = & \color{red}{108x} +180 \color{blue}{-108x} \color{blue}{-20} \\\Leftrightarrow & -63x& = & 160 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{160}{-63} \\\Leftrightarrow & x = \frac{-160}{63} & & \\ & V = \left\{ \frac{-160}{63} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{13}& = & \frac{6}{5}x-8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 150 }{ \color{blue}{390} })& = & (\frac{468}{ \color{blue}{390} }x-\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+150& = & 468x-3120 \\\Leftrightarrow & 65x \color{red}{+150} \color{blue}{-150} \color{blue}{-468x} & = & \color{red}{468x} -3120 \color{blue}{-468x} \color{blue}{-150} \\\Leftrightarrow & -403x& = & -3270 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{-3270}{-403} \\\Leftrightarrow & x = \frac{3270}{403} & & \\ & V = \left\{ \frac{3270}{403} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & -140x-420 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{+140x} & = & \color{red}{-140x} -420 \color{blue}{+140x} \color{blue}{+48} \\\Leftrightarrow & 161x& = & -372 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-372}{161} \\\Leftrightarrow & x = \frac{-372}{161} & & \\ & V = \left\{ \frac{-372}{161} \right\} & \\\end{align}\)
7. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{16}{ \color{blue}{80} }x-\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-25& = & 16x-240 \\\Leftrightarrow & 40x \color{red}{-25} \color{blue}{+25} \color{blue}{-16x} & = & \color{red}{16x} -240 \color{blue}{-16x} \color{blue}{+25} \\\Leftrightarrow & 24x& = & -215 \\\Leftrightarrow & \frac{24x}{ \color{red}{24} }& = & \frac{-215}{24} \\\Leftrightarrow & x = \frac{-215}{24} & & \\ & V = \left\{ \frac{-215}{24} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x-\frac{660}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & 396x-660 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{-396x} & = & \color{red}{396x} -660 \color{blue}{-396x} \color{blue}{+150} \\\Leftrightarrow & -341x& = & -510 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{-510}{-341} \\\Leftrightarrow & x = \frac{510}{341} & & \\ & V = \left\{ \frac{510}{341} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -294x-1680 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+294x} & = & \color{red}{-294x} -1680 \color{blue}{+294x} \color{blue}{-120} \\\Leftrightarrow & 329x& = & -1800 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-1800}{329} \\\Leftrightarrow & x = \frac{-1800}{329} & & \\ & V = \left\{ \frac{-1800}{329} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{-5}{6}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-90& = & -175x-840 \\\Leftrightarrow & 42x \color{red}{-90} \color{blue}{+90} \color{blue}{+175x} & = & \color{red}{-175x} -840 \color{blue}{+175x} \color{blue}{+90} \\\Leftrightarrow & 217x& = & -750 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{-750}{217} \\\Leftrightarrow & x = \frac{-750}{217} & & \\ & V = \left\{ \frac{-750}{217} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{7}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & 294x+630 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{-294x} & = & \color{red}{294x} +630 \color{blue}{-294x} \color{blue}{-150} \\\Leftrightarrow & -259x& = & 480 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{480}{-259} \\\Leftrightarrow & x = \frac{-480}{259} & & \\ & V = \left\{ \frac{-480}{259} \right\} & \\\end{align}\)
12. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{-2}{5}x+1 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{-52}{ \color{blue}{130} }x+\frac{130}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+20& = & -52x+130 \\\Leftrightarrow & 65x \color{red}{+20} \color{blue}{-20} \color{blue}{+52x} & = & \color{red}{-52x} +130 \color{blue}{+52x} \color{blue}{-20} \\\Leftrightarrow & 117x& = & 110 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{110}{117} \\\Leftrightarrow & x = \frac{110}{117} & & \\ & V = \left\{ \frac{110}{117} \right\} & \\\end{align}\)