Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{11}& = & \frac{1}{4}x-3 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }- \frac{ 140 }{ \color{blue}{308} })& = & (\frac{77}{ \color{blue}{308} }x-\frac{924}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x-140& = & 77x-924 \\\Leftrightarrow & 44x \color{red}{-140} \color{blue}{+140} \color{blue}{-77x} & = & \color{red}{77x} -924 \color{blue}{-77x} \color{blue}{+140} \\\Leftrightarrow & -33x& = & -784 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-784}{-33} \\\Leftrightarrow & x = \frac{784}{33} & & \\ & V = \left\{ \frac{784}{33} \right\} & \\\end{align}\)
2. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{13}& = & \frac{7}{4}x+6 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }+ \frac{ 140 }{ \color{blue}{364} })& = & (\frac{637}{ \color{blue}{364} }x+\frac{2184}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x+140& = & 637x+2184 \\\Leftrightarrow & 52x \color{red}{+140} \color{blue}{-140} \color{blue}{-637x} & = & \color{red}{637x} +2184 \color{blue}{-637x} \color{blue}{-140} \\\Leftrightarrow & -585x& = & 2044 \\\Leftrightarrow & \frac{-585x}{ \color{red}{-585} }& = & \frac{2044}{-585} \\\Leftrightarrow & x = \frac{-2044}{585} & & \\ & V = \left\{ \frac{-2044}{585} \right\} & \\\end{align}\)
3. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{28}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-4& = & 7x-28 \\\Leftrightarrow & 2x \color{red}{-4} \color{blue}{+4} \color{blue}{-7x} & = & \color{red}{7x} -28 \color{blue}{-7x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & -24 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-24}{-5} \\\Leftrightarrow & x = \frac{24}{5} & & \\ & V = \left\{ \frac{24}{5} \right\} & \\\end{align}\)
4. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{6}{5}x-7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{468}{ \color{blue}{390} }x-\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & 468x-2730 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{-468x} & = & \color{red}{468x} -2730 \color{blue}{-468x} \color{blue}{+120} \\\Leftrightarrow & -403x& = & -2610 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{-2610}{-403} \\\Leftrightarrow & x = \frac{2610}{403} & & \\ & V = \left\{ \frac{2610}{403} \right\} & \\\end{align}\)
5. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 40 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{770}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+40& = & 55x+770 \\\Leftrightarrow & 22x \color{red}{+40} \color{blue}{-40} \color{blue}{-55x} & = & \color{red}{55x} +770 \color{blue}{-55x} \color{blue}{-40} \\\Leftrightarrow & -33x& = & 730 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{730}{-33} \\\Leftrightarrow & x = \frac{-730}{33} & & \\ & V = \left\{ \frac{-730}{33} \right\} & \\\end{align}\)
6. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{-7}{4}x-3 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-273}{ \color{blue}{156} }x-\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-36& = & -273x-468 \\\Leftrightarrow & 52x \color{red}{-36} \color{blue}{+36} \color{blue}{+273x} & = & \color{red}{-273x} -468 \color{blue}{+273x} \color{blue}{+36} \\\Leftrightarrow & 325x& = & -432 \\\Leftrightarrow & \frac{325x}{ \color{red}{325} }& = & \frac{-432}{325} \\\Leftrightarrow & x = \frac{-432}{325} & & \\ & V = \left\{ \frac{-432}{325} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{3}{2}x-4 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{105}{ \color{blue}{70} }x-\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-20& = & 105x-280 \\\Leftrightarrow & 14x \color{red}{-20} \color{blue}{+20} \color{blue}{-105x} & = & \color{red}{105x} -280 \color{blue}{-105x} \color{blue}{+20} \\\Leftrightarrow & -91x& = & -260 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-260}{-91} \\\Leftrightarrow & x = \frac{20}{7} & & \\ & V = \left\{ \frac{20}{7} \right\} & \\\end{align}\)
8. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{8}& = & \frac{4}{3}x-5 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{32}{ \color{blue}{24} }x-\frac{120}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x+15& = & 32x-120 \\\Leftrightarrow & 6x \color{red}{+15} \color{blue}{-15} \color{blue}{-32x} & = & \color{red}{32x} -120 \color{blue}{-32x} \color{blue}{-15} \\\Leftrightarrow & -26x& = & -135 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{-135}{-26} \\\Leftrightarrow & x = \frac{135}{26} & & \\ & V = \left\{ \frac{135}{26} \right\} & \\\end{align}\)
9. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{13}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 50 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x+\frac{780}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-50& = & 65x+780 \\\Leftrightarrow & 26x \color{red}{-50} \color{blue}{+50} \color{blue}{-65x} & = & \color{red}{65x} +780 \color{blue}{-65x} \color{blue}{+50} \\\Leftrightarrow & -39x& = & 830 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{830}{-39} \\\Leftrightarrow & x = \frac{-830}{39} & & \\ & V = \left\{ \frac{-830}{39} \right\} & \\\end{align}\)
10. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{13}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 75 }{ \color{blue}{195} })& = & (\frac{234}{ \color{blue}{195} }x-\frac{975}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+75& = & 234x-975 \\\Leftrightarrow & 65x \color{red}{+75} \color{blue}{-75} \color{blue}{-234x} & = & \color{red}{234x} -975 \color{blue}{-234x} \color{blue}{-75} \\\Leftrightarrow & -169x& = & -1050 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-1050}{-169} \\\Leftrightarrow & x = \frac{1050}{169} & & \\ & V = \left\{ \frac{1050}{169} \right\} & \\\end{align}\)
11. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{11}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 30 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{880}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+30& = & 55x-880 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-55x} & = & \color{red}{55x} -880 \color{blue}{-55x} \color{blue}{-30} \\\Leftrightarrow & -33x& = & -910 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-910}{-33} \\\Leftrightarrow & x = \frac{910}{33} & & \\ & V = \left\{ \frac{910}{33} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x-\frac{400}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-25& = & 40x-400 \\\Leftrightarrow & 16x \color{red}{-25} \color{blue}{+25} \color{blue}{-40x} & = & \color{red}{40x} -400 \color{blue}{-40x} \color{blue}{+25} \\\Leftrightarrow & -24x& = & -375 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{-375}{-24} \\\Leftrightarrow & x = \frac{125}{8} & & \\ & V = \left\{ \frac{125}{8} \right\} & \\\end{align}\)