Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & -12x+60 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+12x} & = & \color{red}{-12x} +60 \color{blue}{+12x} \color{blue}{-8} \\\Leftrightarrow & 27x& = & 52 \\\Leftrightarrow & \frac{27x}{ \color{red}{27} }& = & \frac{52}{27} \\\Leftrightarrow & x = \frac{52}{27} & & \\ & V = \left\{ \frac{52}{27} \right\} & \\\end{align}\)
2. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{3}{5}x-8 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 30 }{ \color{blue}{165} })& = & (\frac{99}{ \color{blue}{165} }x-\frac{1320}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-30& = & 99x-1320 \\\Leftrightarrow & 55x \color{red}{-30} \color{blue}{+30} \color{blue}{-99x} & = & \color{red}{99x} -1320 \color{blue}{-99x} \color{blue}{+30} \\\Leftrightarrow & -44x& = & -1290 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{-1290}{-44} \\\Leftrightarrow & x = \frac{645}{22} & & \\ & V = \left\{ \frac{645}{22} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & -70x+252 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+70x} & = & \color{red}{-70x} +252 \color{blue}{+70x} \color{blue}{+24} \\\Leftrightarrow & 91x& = & 276 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{276}{91} \\\Leftrightarrow & x = \frac{276}{91} & & \\ & V = \left\{ \frac{276}{91} \right\} & \\\end{align}\)
4. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 45 }{ \color{blue}{105} })& = & (\frac{245}{ \color{blue}{105} }x+\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+45& = & 245x+735 \\\Leftrightarrow & 21x \color{red}{+45} \color{blue}{-45} \color{blue}{-245x} & = & \color{red}{245x} +735 \color{blue}{-245x} \color{blue}{-45} \\\Leftrightarrow & -224x& = & 690 \\\Leftrightarrow & \frac{-224x}{ \color{red}{-224} }& = & \frac{690}{-224} \\\Leftrightarrow & x = \frac{-345}{112} & & \\ & V = \left\{ \frac{-345}{112} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{14}& = & \frac{1}{3}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-18& = & 28x+252 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-28x} & = & \color{red}{28x} +252 \color{blue}{-28x} \color{blue}{+18} \\\Leftrightarrow & -7x& = & 270 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{270}{-7} \\\Leftrightarrow & x = \frac{-270}{7} & & \\ & V = \left\{ \frac{-270}{7} \right\} & \\\end{align}\)
6. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 28 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x-\frac{378}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-28& = & 63x-378 \\\Leftrightarrow & 18x \color{red}{-28} \color{blue}{+28} \color{blue}{-63x} & = & \color{red}{63x} -378 \color{blue}{-63x} \color{blue}{+28} \\\Leftrightarrow & -45x& = & -350 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{-350}{-45} \\\Leftrightarrow & x = \frac{70}{9} & & \\ & V = \left\{ \frac{70}{9} \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x-\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & -112x-140 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{+112x} & = & \color{red}{-112x} -140 \color{blue}{+112x} \color{blue}{-80} \\\Leftrightarrow & 147x& = & -220 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{-220}{147} \\\Leftrightarrow & x = \frac{-220}{147} & & \\ & V = \left\{ \frac{-220}{147} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & 14x+70 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{-14x} & = & \color{red}{14x} +70 \color{blue}{-14x} \color{blue}{+20} \\\Leftrightarrow & 21x& = & 90 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{90}{21} \\\Leftrightarrow & x = \frac{30}{7} & & \\ & V = \left\{ \frac{30}{7} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{2}{5}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{12}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & 12x+180 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-12x} & = & \color{red}{12x} +180 \color{blue}{-12x} \color{blue}{+4} \\\Leftrightarrow & -7x& = & 184 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{184}{-7} \\\Leftrightarrow & x = \frac{-184}{7} & & \\ & V = \left\{ \frac{-184}{7} \right\} & \\\end{align}\)
10. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 80 }{ \color{blue}{180} })& = & (\frac{-252}{ \color{blue}{180} }x-\frac{540}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+80& = & -252x-540 \\\Leftrightarrow & 45x \color{red}{+80} \color{blue}{-80} \color{blue}{+252x} & = & \color{red}{-252x} -540 \color{blue}{+252x} \color{blue}{-80} \\\Leftrightarrow & 297x& = & -620 \\\Leftrightarrow & \frac{297x}{ \color{red}{297} }& = & \frac{-620}{297} \\\Leftrightarrow & x = \frac{-620}{297} & & \\ & V = \left\{ \frac{-620}{297} \right\} & \\\end{align}\)
11. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{3}{2}x-3 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 50 }{ \color{blue}{110} })& = & (\frac{165}{ \color{blue}{110} }x-\frac{330}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+50& = & 165x-330 \\\Leftrightarrow & 22x \color{red}{+50} \color{blue}{-50} \color{blue}{-165x} & = & \color{red}{165x} -330 \color{blue}{-165x} \color{blue}{-50} \\\Leftrightarrow & -143x& = & -380 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-380}{-143} \\\Leftrightarrow & x = \frac{380}{143} & & \\ & V = \left\{ \frac{380}{143} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{7}{3}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{140}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 140x-180 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-140x} & = & \color{red}{140x} -180 \color{blue}{-140x} \color{blue}{+16} \\\Leftrightarrow & -125x& = & -164 \\\Leftrightarrow & \frac{-125x}{ \color{red}{-125} }& = & \frac{-164}{-125} \\\Leftrightarrow & x = \frac{164}{125} & & \\ & V = \left\{ \frac{164}{125} \right\} & \\\end{align}\)