Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & -312x+3120 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{+312x} & = & \color{red}{-312x} +3120 \color{blue}{+312x} \color{blue}{-120} \\\Leftrightarrow & 377x& = & 3000 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{3000}{377} \\\Leftrightarrow & x = \frac{3000}{377} & & \\ & V = \left\{ \frac{3000}{377} \right\} & \\\end{align}\)
2. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x-\frac{528}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & -88x-528 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{+88x} & = & \color{red}{-88x} -528 \color{blue}{+88x} \color{blue}{-24} \\\Leftrightarrow & 121x& = & -552 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{-552}{121} \\\Leftrightarrow & x = \frac{-552}{121} & & \\ & V = \left\{ \frac{-552}{121} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{-5}{6}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-90& = & -175x-1260 \\\Leftrightarrow & 42x \color{red}{-90} \color{blue}{+90} \color{blue}{+175x} & = & \color{red}{-175x} -1260 \color{blue}{+175x} \color{blue}{+90} \\\Leftrightarrow & 217x& = & -1170 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{-1170}{217} \\\Leftrightarrow & x = \frac{-1170}{217} & & \\ & V = \left\{ \frac{-1170}{217} \right\} & \\\end{align}\)
4. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{2}{5}x-5 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{52}{ \color{blue}{130} }x-\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+40& = & 52x-650 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{-52x} & = & \color{red}{52x} -650 \color{blue}{-52x} \color{blue}{-40} \\\Leftrightarrow & 13x& = & -690 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-690}{13} \\\Leftrightarrow & x = \frac{-690}{13} & & \\ & V = \left\{ \frac{-690}{13} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 5, 14 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{14}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 25 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-25& = & 35x+560 \\\Leftrightarrow & 14x \color{red}{-25} \color{blue}{+25} \color{blue}{-35x} & = & \color{red}{35x} +560 \color{blue}{-35x} \color{blue}{+25} \\\Leftrightarrow & -21x& = & 585 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{585}{-21} \\\Leftrightarrow & x = \frac{-195}{7} & & \\ & V = \left\{ \frac{-195}{7} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & -70x+84 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+70x} & = & \color{red}{-70x} +84 \color{blue}{+70x} \color{blue}{+24} \\\Leftrightarrow & 91x& = & 108 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{108}{91} \\\Leftrightarrow & x = \frac{108}{91} & & \\ & V = \left\{ \frac{108}{91} \right\} & \\\end{align}\)
7. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{9}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 20 }{ \color{blue}{36} })& = & (\frac{-24}{ \color{blue}{36} }x+\frac{36}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-20& = & -24x+36 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{+24x} & = & \color{red}{-24x} +36 \color{blue}{+24x} \color{blue}{+20} \\\Leftrightarrow & 33x& = & 56 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{56}{33} \\\Leftrightarrow & x = \frac{56}{33} & & \\ & V = \left\{ \frac{56}{33} \right\} & \\\end{align}\)
8. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{7}{3}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & 196x-588 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{-196x} & = & \color{red}{196x} -588 \color{blue}{-196x} \color{blue}{-48} \\\Leftrightarrow & -175x& = & -636 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-636}{-175} \\\Leftrightarrow & x = \frac{636}{175} & & \\ & V = \left\{ \frac{636}{175} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{7}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 50 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-50& = & -56x-350 \\\Leftrightarrow & 35x \color{red}{-50} \color{blue}{+50} \color{blue}{+56x} & = & \color{red}{-56x} -350 \color{blue}{+56x} \color{blue}{+50} \\\Leftrightarrow & 91x& = & -300 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-300}{91} \\\Leftrightarrow & x = \frac{-300}{91} & & \\ & V = \left\{ \frac{-300}{91} \right\} & \\\end{align}\)
10. \(\text{140 is het kleinste gemene veelvoud van 4, 14 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{14}& = & \frac{4}{5}x+4 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 50 }{ \color{blue}{140} })& = & (\frac{112}{ \color{blue}{140} }x+\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+50& = & 112x+560 \\\Leftrightarrow & 35x \color{red}{+50} \color{blue}{-50} \color{blue}{-112x} & = & \color{red}{112x} +560 \color{blue}{-112x} \color{blue}{-50} \\\Leftrightarrow & -77x& = & 510 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{510}{-77} \\\Leftrightarrow & x = \frac{-510}{77} & & \\ & V = \left\{ \frac{-510}{77} \right\} & \\\end{align}\)
11. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{1}{6}x+4 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 28 }{ \color{blue}{126} })& = & (\frac{21}{ \color{blue}{126} }x+\frac{504}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+28& = & 21x+504 \\\Leftrightarrow & 18x \color{red}{+28} \color{blue}{-28} \color{blue}{-21x} & = & \color{red}{21x} +504 \color{blue}{-21x} \color{blue}{-28} \\\Leftrightarrow & -3x& = & 476 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{476}{-3} \\\Leftrightarrow & x = \frac{-476}{3} & & \\ & V = \left\{ \frac{-476}{3} \right\} & \\\end{align}\)
12. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{5}{2}x-8 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{325}{ \color{blue}{130} }x-\frac{1040}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-20& = & 325x-1040 \\\Leftrightarrow & 26x \color{red}{-20} \color{blue}{+20} \color{blue}{-325x} & = & \color{red}{325x} -1040 \color{blue}{-325x} \color{blue}{+20} \\\Leftrightarrow & -299x& = & -1020 \\\Leftrightarrow & \frac{-299x}{ \color{red}{-299} }& = & \frac{-1020}{-299} \\\Leftrightarrow & x = \frac{1020}{299} & & \\ & V = \left\{ \frac{1020}{299} \right\} & \\\end{align}\)