Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 42 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x-\frac{1274}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-42& = & 91x-1274 \\\Leftrightarrow & 26x \color{red}{-42} \color{blue}{+42} \color{blue}{-91x} & = & \color{red}{91x} -1274 \color{blue}{-91x} \color{blue}{+42} \\\Leftrightarrow & -65x& = & -1232 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-1232}{-65} \\\Leftrightarrow & x = \frac{1232}{65} & & \\ & V = \left\{ \frac{1232}{65} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-18& = & 21x+84 \\\Leftrightarrow & 14x \color{red}{-18} \color{blue}{+18} \color{blue}{-21x} & = & \color{red}{21x} +84 \color{blue}{-21x} \color{blue}{+18} \\\Leftrightarrow & -7x& = & 102 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{102}{-7} \\\Leftrightarrow & x = \frac{-102}{7} & & \\ & V = \left\{ \frac{-102}{7} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{7}& = & \frac{3}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-30& = & 63x-252 \\\Leftrightarrow & 14x \color{red}{-30} \color{blue}{+30} \color{blue}{-63x} & = & \color{red}{63x} -252 \color{blue}{-63x} \color{blue}{+30} \\\Leftrightarrow & -49x& = & -222 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-222}{-49} \\\Leftrightarrow & x = \frac{222}{49} & & \\ & V = \left\{ \frac{222}{49} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{11}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x-\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-120& = & 396x-990 \\\Leftrightarrow & 55x \color{red}{-120} \color{blue}{+120} \color{blue}{-396x} & = & \color{red}{396x} -990 \color{blue}{-396x} \color{blue}{+120} \\\Leftrightarrow & -341x& = & -870 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{-870}{-341} \\\Leftrightarrow & x = \frac{870}{341} & & \\ & V = \left\{ \frac{870}{341} \right\} & \\\end{align}\)
5. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{8}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{24}{ \color{blue}{120} }x-\frac{360}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-75& = & 24x-360 \\\Leftrightarrow & 20x \color{red}{-75} \color{blue}{+75} \color{blue}{-24x} & = & \color{red}{24x} -360 \color{blue}{-24x} \color{blue}{+75} \\\Leftrightarrow & -4x& = & -285 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-285}{-4} \\\Leftrightarrow & x = \frac{285}{4} & & \\ & V = \left\{ \frac{285}{4} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -84x-1470 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+84x} & = & \color{red}{-84x} -1470 \color{blue}{+84x} \color{blue}{+60} \\\Leftrightarrow & 119x& = & -1410 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-1410}{119} \\\Leftrightarrow & x = \frac{-1410}{119} & & \\ & V = \left\{ \frac{-1410}{119} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{7}& = & \frac{1}{6}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 150 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-150& = & 35x+1260 \\\Leftrightarrow & 42x \color{red}{-150} \color{blue}{+150} \color{blue}{-35x} & = & \color{red}{35x} +1260 \color{blue}{-35x} \color{blue}{+150} \\\Leftrightarrow & 7x& = & 1410 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{1410}{7} \\\Leftrightarrow & x = \frac{1410}{7} & & \\ & V = \left\{ \frac{1410}{7} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x-\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-20& = & 45x-360 \\\Leftrightarrow & 18x \color{red}{-20} \color{blue}{+20} \color{blue}{-45x} & = & \color{red}{45x} -360 \color{blue}{-45x} \color{blue}{+20} \\\Leftrightarrow & -27x& = & -340 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-340}{-27} \\\Leftrightarrow & x = \frac{340}{27} & & \\ & V = \left\{ \frac{340}{27} \right\} & \\\end{align}\)
9. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{7}{3}x-6 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{84}{ \color{blue}{36} }x-\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+8& = & 84x-216 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{-84x} & = & \color{red}{84x} -216 \color{blue}{-84x} \color{blue}{-8} \\\Leftrightarrow & -75x& = & -224 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{-224}{-75} \\\Leftrightarrow & x = \frac{224}{75} & & \\ & V = \left\{ \frac{224}{75} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{4}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{24}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & 24x+210 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-24x} & = & \color{red}{24x} +210 \color{blue}{-24x} \color{blue}{+4} \\\Leftrightarrow & -19x& = & 214 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{214}{-19} \\\Leftrightarrow & x = \frac{-214}{19} & & \\ & V = \left\{ \frac{-214}{19} \right\} & \\\end{align}\)
11. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 45 }{ \color{blue}{195} })& = & (\frac{-156}{ \color{blue}{195} }x-\frac{585}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+45& = & -156x-585 \\\Leftrightarrow & 65x \color{red}{+45} \color{blue}{-45} \color{blue}{+156x} & = & \color{red}{-156x} -585 \color{blue}{+156x} \color{blue}{-45} \\\Leftrightarrow & 221x& = & -630 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{-630}{221} \\\Leftrightarrow & x = \frac{-630}{221} & & \\ & V = \left\{ \frac{-630}{221} \right\} & \\\end{align}\)
12. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{3}{2}x-8 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 56 }{ \color{blue}{126} })& = & (\frac{189}{ \color{blue}{126} }x-\frac{1008}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+56& = & 189x-1008 \\\Leftrightarrow & 18x \color{red}{+56} \color{blue}{-56} \color{blue}{-189x} & = & \color{red}{189x} -1008 \color{blue}{-189x} \color{blue}{-56} \\\Leftrightarrow & -171x& = & -1064 \\\Leftrightarrow & \frac{-171x}{ \color{red}{-171} }& = & \frac{-1064}{-171} \\\Leftrightarrow & x = \frac{56}{9} & & \\ & V = \left\{ \frac{56}{9} \right\} & \\\end{align}\)