Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{4}+\frac{2}{7}=\frac{6}{5}x-1\)
2. \(\frac{x}{2}+\frac{5}{16}=\frac{-5}{3}x+8\)
3. \(\frac{x}{5}+\frac{5}{16}=\frac{1}{6}x-4\)
4. \(\frac{x}{5}-\frac{3}{16}=\frac{1}{2}x+8\)
5. \(\frac{x}{4}+\frac{5}{7}=\frac{-8}{3}x-7\)
6. \(\frac{x}{3}-\frac{5}{11}=\frac{1}{4}x+8\)
7. \(\frac{x}{5}+\frac{5}{14}=\frac{5}{6}x+3\)
8. \(\frac{x}{2}-\frac{4}{9}=\frac{7}{3}x-8\)
9. \(\frac{x}{2}-\frac{2}{7}=\frac{-2}{3}x-7\)
10. \(\frac{x}{6}+\frac{3}{7}=\frac{6}{5}x+7\)
11. \(\frac{x}{4}+\frac{4}{9}=\frac{-2}{5}x-3\)
12. \(\frac{x}{5}-\frac{5}{6}=\frac{5}{4}x+3\)

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x-\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & 168x-140 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-168x} & = & \color{red}{168x} -140 \color{blue}{-168x} \color{blue}{-40} \\\Leftrightarrow & -133x& = & -180 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-180}{-133} \\\Leftrightarrow & x = \frac{180}{133} & & \\ & V = \left\{ \frac{180}{133} \right\} & \\\end{align}\)
2. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x+\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & -80x+384 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{+80x} & = & \color{red}{-80x} +384 \color{blue}{+80x} \color{blue}{-15} \\\Leftrightarrow & 104x& = & 369 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{369}{104} \\\Leftrightarrow & x = \frac{369}{104} & & \\ & V = \left\{ \frac{369}{104} \right\} & \\\end{align}\)
3. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{1}{6}x-4 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{40}{ \color{blue}{240} }x-\frac{960}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+75& = & 40x-960 \\\Leftrightarrow & 48x \color{red}{+75} \color{blue}{-75} \color{blue}{-40x} & = & \color{red}{40x} -960 \color{blue}{-40x} \color{blue}{-75} \\\Leftrightarrow & 8x& = & -1035 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{-1035}{8} \\\Leftrightarrow & x = \frac{-1035}{8} & & \\ & V = \left\{ \frac{-1035}{8} \right\} & \\\end{align}\)
4. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x+\frac{640}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-15& = & 40x+640 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-40x} & = & \color{red}{40x} +640 \color{blue}{-40x} \color{blue}{+15} \\\Leftrightarrow & -24x& = & 655 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{655}{-24} \\\Leftrightarrow & x = \frac{-655}{24} & & \\ & V = \left\{ \frac{-655}{24} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{7}& = & \frac{-8}{3}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 60 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+60& = & -224x-588 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{+224x} & = & \color{red}{-224x} -588 \color{blue}{+224x} \color{blue}{-60} \\\Leftrightarrow & 245x& = & -648 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{-648}{245} \\\Leftrightarrow & x = \frac{-648}{245} & & \\ & V = \left\{ \frac{-648}{245} \right\} & \\\end{align}\)
6. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{4}x+8 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{33}{ \color{blue}{132} }x+\frac{1056}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-60& = & 33x+1056 \\\Leftrightarrow & 44x \color{red}{-60} \color{blue}{+60} \color{blue}{-33x} & = & \color{red}{33x} +1056 \color{blue}{-33x} \color{blue}{+60} \\\Leftrightarrow & 11x& = & 1116 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{1116}{11} \\\Leftrightarrow & x = \frac{1116}{11} & & \\ & V = \left\{ \frac{1116}{11} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{14}& = & \frac{5}{6}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+75& = & 175x+630 \\\Leftrightarrow & 42x \color{red}{+75} \color{blue}{-75} \color{blue}{-175x} & = & \color{red}{175x} +630 \color{blue}{-175x} \color{blue}{-75} \\\Leftrightarrow & -133x& = & 555 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{555}{-133} \\\Leftrightarrow & x = \frac{-555}{133} & & \\ & V = \left\{ \frac{-555}{133} \right\} & \\\end{align}\)
8. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{42}{ \color{blue}{18} }x-\frac{144}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-8& = & 42x-144 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{-42x} & = & \color{red}{42x} -144 \color{blue}{-42x} \color{blue}{+8} \\\Leftrightarrow & -33x& = & -136 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-136}{-33} \\\Leftrightarrow & x = \frac{136}{33} & & \\ & V = \left\{ \frac{136}{33} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & -28x-294 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{+28x} & = & \color{red}{-28x} -294 \color{blue}{+28x} \color{blue}{+12} \\\Leftrightarrow & 49x& = & -282 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-282}{49} \\\Leftrightarrow & x = \frac{-282}{49} & & \\ & V = \left\{ \frac{-282}{49} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{6}{5}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & 252x+1470 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{-252x} & = & \color{red}{252x} +1470 \color{blue}{-252x} \color{blue}{-90} \\\Leftrightarrow & -217x& = & 1380 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{1380}{-217} \\\Leftrightarrow & x = \frac{-1380}{217} & & \\ & V = \left\{ \frac{-1380}{217} \right\} & \\\end{align}\)
11. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 80 }{ \color{blue}{180} })& = & (\frac{-72}{ \color{blue}{180} }x-\frac{540}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+80& = & -72x-540 \\\Leftrightarrow & 45x \color{red}{+80} \color{blue}{-80} \color{blue}{+72x} & = & \color{red}{-72x} -540 \color{blue}{+72x} \color{blue}{-80} \\\Leftrightarrow & 117x& = & -620 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-620}{117} \\\Leftrightarrow & x = \frac{-620}{117} & & \\ & V = \left\{ \frac{-620}{117} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 5, 6 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{6}& = & \frac{5}{4}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 50 }{ \color{blue}{60} })& = & (\frac{75}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-50& = & 75x+180 \\\Leftrightarrow & 12x \color{red}{-50} \color{blue}{+50} \color{blue}{-75x} & = & \color{red}{75x} +180 \color{blue}{-75x} \color{blue}{+50} \\\Leftrightarrow & -63x& = & 230 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{230}{-63} \\\Leftrightarrow & x = \frac{-230}{63} & & \\ & V = \left\{ \frac{-230}{63} \right\} & \\\end{align}\)