Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{2}{9}=\frac{1}{4}x-3\)
2. \(\frac{x}{2}-\frac{2}{7}=\frac{-2}{3}x-7\)
3. \(\frac{x}{3}-\frac{5}{8}=\frac{-3}{4}x-6\)
4. \(\frac{x}{5}+\frac{4}{11}=\frac{1}{2}x+7\)
5. \(\frac{x}{7}-\frac{2}{7}=\frac{-4}{5}x-2\)
6. \(\frac{x}{7}+\frac{5}{12}=\frac{1}{2}x+3\)
7. \(\frac{x}{2}-\frac{5}{16}=\frac{2}{3}x-8\)
8. \(\frac{x}{3}-\frac{2}{15}=\frac{1}{2}x+6\)
9. \(\frac{x}{6}-\frac{3}{10}=\frac{6}{5}x-3\)
10. \(\frac{x}{4}+\frac{3}{13}=\frac{6}{5}x+6\)
11. \(\frac{x}{6}+\frac{5}{16}=\frac{-2}{5}x-1\)
12. \(\frac{x}{4}+\frac{2}{7}=\frac{-5}{3}x-1\)

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

Verbetersleutel

1. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{1}{4}x-3 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }+ \frac{ 56 }{ \color{blue}{252} })& = & (\frac{63}{ \color{blue}{252} }x-\frac{756}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x+56& = & 63x-756 \\\Leftrightarrow & 36x \color{red}{+56} \color{blue}{-56} \color{blue}{-63x} & = & \color{red}{63x} -756 \color{blue}{-63x} \color{blue}{-56} \\\Leftrightarrow & -27x& = & -812 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-812}{-27} \\\Leftrightarrow & x = \frac{812}{27} & & \\ & V = \left\{ \frac{812}{27} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & -28x-294 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{+28x} & = & \color{red}{-28x} -294 \color{blue}{+28x} \color{blue}{+12} \\\Leftrightarrow & 49x& = & -282 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-282}{49} \\\Leftrightarrow & x = \frac{-282}{49} & & \\ & V = \left\{ \frac{-282}{49} \right\} & \\\end{align}\)
3. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{8}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-18}{ \color{blue}{24} }x-\frac{144}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-15& = & -18x-144 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{+18x} & = & \color{red}{-18x} -144 \color{blue}{+18x} \color{blue}{+15} \\\Leftrightarrow & 26x& = & -129 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{-129}{26} \\\Leftrightarrow & x = \frac{-129}{26} & & \\ & V = \left\{ \frac{-129}{26} \right\} & \\\end{align}\)
4. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 40 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{770}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+40& = & 55x+770 \\\Leftrightarrow & 22x \color{red}{+40} \color{blue}{-40} \color{blue}{-55x} & = & \color{red}{55x} +770 \color{blue}{-55x} \color{blue}{-40} \\\Leftrightarrow & -33x& = & 730 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{730}{-33} \\\Leftrightarrow & x = \frac{-730}{33} & & \\ & V = \left\{ \frac{-730}{33} \right\} & \\\end{align}\)
5. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 10 }{ \color{blue}{35} })& = & (\frac{-28}{ \color{blue}{35} }x-\frac{70}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-10& = & -28x-70 \\\Leftrightarrow & 5x \color{red}{-10} \color{blue}{+10} \color{blue}{+28x} & = & \color{red}{-28x} -70 \color{blue}{+28x} \color{blue}{+10} \\\Leftrightarrow & 33x& = & -60 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-60}{33} \\\Leftrightarrow & x = \frac{-20}{11} & & \\ & V = \left\{ \frac{-20}{11} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{12}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }+ \frac{ 35 }{ \color{blue}{84} })& = & (\frac{42}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x+35& = & 42x+252 \\\Leftrightarrow & 12x \color{red}{+35} \color{blue}{-35} \color{blue}{-42x} & = & \color{red}{42x} +252 \color{blue}{-42x} \color{blue}{-35} \\\Leftrightarrow & -30x& = & 217 \\\Leftrightarrow & \frac{-30x}{ \color{red}{-30} }& = & \frac{217}{-30} \\\Leftrightarrow & x = \frac{-217}{30} & & \\ & V = \left\{ \frac{-217}{30} \right\} & \\\end{align}\)
7. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{2}{3}x-8 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{32}{ \color{blue}{48} }x-\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & 32x-384 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{-32x} & = & \color{red}{32x} -384 \color{blue}{-32x} \color{blue}{+15} \\\Leftrightarrow & -8x& = & -369 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-369}{-8} \\\Leftrightarrow & x = \frac{369}{8} & & \\ & V = \left\{ \frac{369}{8} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-4& = & 15x+180 \\\Leftrightarrow & 10x \color{red}{-4} \color{blue}{+4} \color{blue}{-15x} & = & \color{red}{15x} +180 \color{blue}{-15x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & 184 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{184}{-5} \\\Leftrightarrow & x = \frac{-184}{5} & & \\ & V = \left\{ \frac{-184}{5} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{6}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & 36x-90 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{-36x} & = & \color{red}{36x} -90 \color{blue}{-36x} \color{blue}{+9} \\\Leftrightarrow & -31x& = & -81 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{-81}{-31} \\\Leftrightarrow & x = \frac{81}{31} & & \\ & V = \left\{ \frac{81}{31} \right\} & \\\end{align}\)
10. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{13}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 60 }{ \color{blue}{260} })& = & (\frac{312}{ \color{blue}{260} }x+\frac{1560}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+60& = & 312x+1560 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-312x} & = & \color{red}{312x} +1560 \color{blue}{-312x} \color{blue}{-60} \\\Leftrightarrow & -247x& = & 1500 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{1500}{-247} \\\Leftrightarrow & x = \frac{-1500}{247} & & \\ & V = \left\{ \frac{-1500}{247} \right\} & \\\end{align}\)
11. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{-2}{5}x-1 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-96}{ \color{blue}{240} }x-\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & -96x-240 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{+96x} & = & \color{red}{-96x} -240 \color{blue}{+96x} \color{blue}{-75} \\\Leftrightarrow & 136x& = & -315 \\\Leftrightarrow & \frac{136x}{ \color{red}{136} }& = & \frac{-315}{136} \\\Leftrightarrow & x = \frac{-315}{136} & & \\ & V = \left\{ \frac{-315}{136} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & -140x-84 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+140x} & = & \color{red}{-140x} -84 \color{blue}{+140x} \color{blue}{-24} \\\Leftrightarrow & 161x& = & -108 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-108}{161} \\\Leftrightarrow & x = \frac{-108}{161} & & \\ & V = \left\{ \frac{-108}{161} \right\} & \\\end{align}\)