Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x+\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-18& = & -52x+468 \\\Leftrightarrow & 39x \color{red}{-18} \color{blue}{+18} \color{blue}{+52x} & = & \color{red}{-52x} +468 \color{blue}{+52x} \color{blue}{+18} \\\Leftrightarrow & 91x& = & 486 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{486}{91} \\\Leftrightarrow & x = \frac{486}{91} & & \\ & V = \left\{ \frac{486}{91} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{13}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x-\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-90& = & -156x-2340 \\\Leftrightarrow & 65x \color{red}{-90} \color{blue}{+90} \color{blue}{+156x} & = & \color{red}{-156x} -2340 \color{blue}{+156x} \color{blue}{+90} \\\Leftrightarrow & 221x& = & -2250 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{-2250}{221} \\\Leftrightarrow & x = \frac{-2250}{221} & & \\ & V = \left\{ \frac{-2250}{221} \right\} & \\\end{align}\)
3. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 28 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x+\frac{1456}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-28& = & 91x+1456 \\\Leftrightarrow & 26x \color{red}{-28} \color{blue}{+28} \color{blue}{-91x} & = & \color{red}{91x} +1456 \color{blue}{-91x} \color{blue}{+28} \\\Leftrightarrow & -65x& = & 1484 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{1484}{-65} \\\Leftrightarrow & x = \frac{-1484}{65} & & \\ & V = \left\{ \frac{-1484}{65} \right\} & \\\end{align}\)
4. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 45 }{ \color{blue}{195} })& = & (\frac{-78}{ \color{blue}{195} }x-\frac{1365}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+45& = & -78x-1365 \\\Leftrightarrow & 65x \color{red}{+45} \color{blue}{-45} \color{blue}{+78x} & = & \color{red}{-78x} -1365 \color{blue}{+78x} \color{blue}{-45} \\\Leftrightarrow & 143x& = & -1410 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-1410}{143} \\\Leftrightarrow & x = \frac{-1410}{143} & & \\ & V = \left\{ \frac{-1410}{143} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-8}{3}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & -112x+84 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{+112x} & = & \color{red}{-112x} +84 \color{blue}{+112x} \color{blue}{+12} \\\Leftrightarrow & 133x& = & 96 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{96}{133} \\\Leftrightarrow & x = \frac{96}{133} & & \\ & V = \left\{ \frac{96}{133} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 56 }{ \color{blue}{210} })& = & (\frac{105}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+56& = & 105x-210 \\\Leftrightarrow & 30x \color{red}{+56} \color{blue}{-56} \color{blue}{-105x} & = & \color{red}{105x} -210 \color{blue}{-105x} \color{blue}{-56} \\\Leftrightarrow & -75x& = & -266 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{-266}{-75} \\\Leftrightarrow & x = \frac{266}{75} & & \\ & V = \left\{ \frac{266}{75} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{14}& = & \frac{4}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+75& = & 168x+840 \\\Leftrightarrow & 35x \color{red}{+75} \color{blue}{-75} \color{blue}{-168x} & = & \color{red}{168x} +840 \color{blue}{-168x} \color{blue}{-75} \\\Leftrightarrow & -133x& = & 765 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{765}{-133} \\\Leftrightarrow & x = \frac{-765}{133} & & \\ & V = \left\{ \frac{-765}{133} \right\} & \\\end{align}\)
8. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-24}{ \color{blue}{36} }x-\frac{72}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-16& = & -24x-72 \\\Leftrightarrow & 9x \color{red}{-16} \color{blue}{+16} \color{blue}{+24x} & = & \color{red}{-24x} -72 \color{blue}{+24x} \color{blue}{+16} \\\Leftrightarrow & 33x& = & -56 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-56}{33} \\\Leftrightarrow & x = \frac{-56}{33} & & \\ & V = \left\{ \frac{-56}{33} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{4}{3}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{80}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 80x+360 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-80x} & = & \color{red}{80x} +360 \color{blue}{-80x} \color{blue}{+16} \\\Leftrightarrow & -65x& = & 376 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{376}{-65} \\\Leftrightarrow & x = \frac{-376}{65} & & \\ & V = \left\{ \frac{-376}{65} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{11}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-132}{ \color{blue}{330} }x+\frac{1320}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-120& = & -132x+1320 \\\Leftrightarrow & 55x \color{red}{-120} \color{blue}{+120} \color{blue}{+132x} & = & \color{red}{-132x} +1320 \color{blue}{+132x} \color{blue}{+120} \\\Leftrightarrow & 187x& = & 1440 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{1440}{187} \\\Leftrightarrow & x = \frac{1440}{187} & & \\ & V = \left\{ \frac{1440}{187} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{7}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 294x-1260 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-294x} & = & \color{red}{294x} -1260 \color{blue}{-294x} \color{blue}{+60} \\\Leftrightarrow & -259x& = & -1200 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{-1200}{-259} \\\Leftrightarrow & x = \frac{1200}{259} & & \\ & V = \left\{ \frac{1200}{259} \right\} & \\\end{align}\)
12. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{-4}{5}x+7 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 14 }{ \color{blue}{105} })& = & (\frac{-84}{ \color{blue}{105} }x+\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-14& = & -84x+735 \\\Leftrightarrow & 15x \color{red}{-14} \color{blue}{+14} \color{blue}{+84x} & = & \color{red}{-84x} +735 \color{blue}{+84x} \color{blue}{+14} \\\Leftrightarrow & 99x& = & 749 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{749}{99} \\\Leftrightarrow & x = \frac{749}{99} & & \\ & V = \left\{ \frac{749}{99} \right\} & \\\end{align}\)