Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{1}{6}x+5 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{15}{ \color{blue}{90} }x+\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 15x+450 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-15x} & = & \color{red}{15x} +450 \color{blue}{-15x} \color{blue}{+40} \\\Leftrightarrow & 3x& = & 490 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{490}{3} \\\Leftrightarrow & x = \frac{490}{3} & & \\ & V = \left\{ \frac{490}{3} \right\} & \\\end{align}\)
2. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }+ \frac{ 112 }{ \color{blue}{420} })& = & (\frac{-315}{ \color{blue}{420} }x+\frac{1260}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x+112& = & -315x+1260 \\\Leftrightarrow & 60x \color{red}{+112} \color{blue}{-112} \color{blue}{+315x} & = & \color{red}{-315x} +1260 \color{blue}{+315x} \color{blue}{-112} \\\Leftrightarrow & 375x& = & 1148 \\\Leftrightarrow & \frac{375x}{ \color{red}{375} }& = & \frac{1148}{375} \\\Leftrightarrow & x = \frac{1148}{375} & & \\ & V = \left\{ \frac{1148}{375} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 5, 14 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{14}& = & \frac{-3}{4}x-1 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 50 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x-\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+50& = & -105x-140 \\\Leftrightarrow & 28x \color{red}{+50} \color{blue}{-50} \color{blue}{+105x} & = & \color{red}{-105x} -140 \color{blue}{+105x} \color{blue}{-50} \\\Leftrightarrow & 133x& = & -190 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-190}{133} \\\Leftrightarrow & x = \frac{-10}{7} & & \\ & V = \left\{ \frac{-10}{7} \right\} & \\\end{align}\)
4. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{2}{5}x-5 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 45 }{ \color{blue}{195} })& = & (\frac{78}{ \color{blue}{195} }x-\frac{975}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+45& = & 78x-975 \\\Leftrightarrow & 65x \color{red}{+45} \color{blue}{-45} \color{blue}{-78x} & = & \color{red}{78x} -975 \color{blue}{-78x} \color{blue}{-45} \\\Leftrightarrow & -13x& = & -1020 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-1020}{-13} \\\Leftrightarrow & x = \frac{1020}{13} & & \\ & V = \left\{ \frac{1020}{13} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & -28x-42 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{+28x} & = & \color{red}{-28x} -42 \color{blue}{+28x} \color{blue}{+12} \\\Leftrightarrow & 49x& = & -30 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-30}{49} \\\Leftrightarrow & x = \frac{-30}{49} & & \\ & V = \left\{ \frac{-30}{49} \right\} & \\\end{align}\)
6. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{11}& = & \frac{5}{4}x+5 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }- \frac{ 40 }{ \color{blue}{220} })& = & (\frac{275}{ \color{blue}{220} }x+\frac{1100}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x-40& = & 275x+1100 \\\Leftrightarrow & 44x \color{red}{-40} \color{blue}{+40} \color{blue}{-275x} & = & \color{red}{275x} +1100 \color{blue}{-275x} \color{blue}{+40} \\\Leftrightarrow & -231x& = & 1140 \\\Leftrightarrow & \frac{-231x}{ \color{red}{-231} }& = & \frac{1140}{-231} \\\Leftrightarrow & x = \frac{-380}{77} & & \\ & V = \left\{ \frac{-380}{77} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{5}{3}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & 140x-336 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-140x} & = & \color{red}{140x} -336 \color{blue}{-140x} \color{blue}{-24} \\\Leftrightarrow & -119x& = & -360 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{-360}{-119} \\\Leftrightarrow & x = \frac{360}{119} & & \\ & V = \left\{ \frac{360}{119} \right\} & \\\end{align}\)
8. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{9}& = & \frac{-7}{5}x+7 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }+ \frac{ 25 }{ \color{blue}{45} })& = & (\frac{-63}{ \color{blue}{45} }x+\frac{315}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x+25& = & -63x+315 \\\Leftrightarrow & 15x \color{red}{+25} \color{blue}{-25} \color{blue}{+63x} & = & \color{red}{-63x} +315 \color{blue}{+63x} \color{blue}{-25} \\\Leftrightarrow & 78x& = & 290 \\\Leftrightarrow & \frac{78x}{ \color{red}{78} }& = & \frac{290}{78} \\\Leftrightarrow & x = \frac{145}{39} & & \\ & V = \left\{ \frac{145}{39} \right\} & \\\end{align}\)
9. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{4}{5}x-4 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 20 }{ \color{blue}{130} })& = & (\frac{104}{ \color{blue}{130} }x-\frac{520}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+20& = & 104x-520 \\\Leftrightarrow & 65x \color{red}{+20} \color{blue}{-20} \color{blue}{-104x} & = & \color{red}{104x} -520 \color{blue}{-104x} \color{blue}{-20} \\\Leftrightarrow & -39x& = & -540 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-540}{-39} \\\Leftrightarrow & x = \frac{180}{13} & & \\ & V = \left\{ \frac{180}{13} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{1}{6}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-18& = & 7x-42 \\\Leftrightarrow & 6x \color{red}{-18} \color{blue}{+18} \color{blue}{-7x} & = & \color{red}{7x} -42 \color{blue}{-7x} \color{blue}{+18} \\\Leftrightarrow & -x& = & -24 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-24}{-1} \\\Leftrightarrow & x = 24 & & \\ & V = \left\{ 24 \right\} & \\\end{align}\)
11. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{8}& = & \frac{-7}{4}x+7 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{-42}{ \color{blue}{24} }x+\frac{168}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-9& = & -42x+168 \\\Leftrightarrow & 8x \color{red}{-9} \color{blue}{+9} \color{blue}{+42x} & = & \color{red}{-42x} +168 \color{blue}{+42x} \color{blue}{+9} \\\Leftrightarrow & 50x& = & 177 \\\Leftrightarrow & \frac{50x}{ \color{red}{50} }& = & \frac{177}{50} \\\Leftrightarrow & x = \frac{177}{50} & & \\ & V = \left\{ \frac{177}{50} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{6}& = & \frac{-8}{3}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-80}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-25& = & -80x-240 \\\Leftrightarrow & 6x \color{red}{-25} \color{blue}{+25} \color{blue}{+80x} & = & \color{red}{-80x} -240 \color{blue}{+80x} \color{blue}{+25} \\\Leftrightarrow & 86x& = & -215 \\\Leftrightarrow & \frac{86x}{ \color{red}{86} }& = & \frac{-215}{86} \\\Leftrightarrow & x = \frac{-5}{2} & & \\ & V = \left\{ \frac{-5}{2} \right\} & \\\end{align}\)