Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{9}& = & \frac{4}{5}x+5 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{72}{ \color{blue}{90} }x+\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-50& = & 72x+450 \\\Leftrightarrow & 45x \color{red}{-50} \color{blue}{+50} \color{blue}{-72x} & = & \color{red}{72x} +450 \color{blue}{-72x} \color{blue}{+50} \\\Leftrightarrow & -27x& = & 500 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{500}{-27} \\\Leftrightarrow & x = \frac{-500}{27} & & \\ & V = \left\{ \frac{-500}{27} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{10}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }+ \frac{ 21 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x+21& = & 35x+70 \\\Leftrightarrow & 10x \color{red}{+21} \color{blue}{-21} \color{blue}{-35x} & = & \color{red}{35x} +70 \color{blue}{-35x} \color{blue}{-21} \\\Leftrightarrow & -25x& = & 49 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{49}{-25} \\\Leftrightarrow & x = \frac{-49}{25} & & \\ & V = \left\{ \frac{-49}{25} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x-\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & -196x-560 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{+196x} & = & \color{red}{-196x} -560 \color{blue}{+196x} \color{blue}{-80} \\\Leftrightarrow & 231x& = & -640 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{-640}{231} \\\Leftrightarrow & x = \frac{-640}{231} & & \\ & V = \left\{ \frac{-640}{231} \right\} & \\\end{align}\)
4. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{-4}{5}x+7 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{-104}{ \color{blue}{130} }x+\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-20& = & -104x+910 \\\Leftrightarrow & 65x \color{red}{-20} \color{blue}{+20} \color{blue}{+104x} & = & \color{red}{-104x} +910 \color{blue}{+104x} \color{blue}{+20} \\\Leftrightarrow & 169x& = & 930 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{930}{169} \\\Leftrightarrow & x = \frac{930}{169} & & \\ & V = \left\{ \frac{930}{169} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{-3}{4}x-5 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x-\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-24& = & -63x-420 \\\Leftrightarrow & 28x \color{red}{-24} \color{blue}{+24} \color{blue}{+63x} & = & \color{red}{-63x} -420 \color{blue}{+63x} \color{blue}{+24} \\\Leftrightarrow & 91x& = & -396 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-396}{91} \\\Leftrightarrow & x = \frac{-396}{91} & & \\ & V = \left\{ \frac{-396}{91} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{-7}{5}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-84}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & -84x+120 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+84x} & = & \color{red}{-84x} +120 \color{blue}{+84x} \color{blue}{-8} \\\Leftrightarrow & 99x& = & 112 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{112}{99} \\\Leftrightarrow & x = \frac{112}{99} & & \\ & V = \left\{ \frac{112}{99} \right\} & \\\end{align}\)
7. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{11}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 42 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x+\frac{154}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-42& = & 77x+154 \\\Leftrightarrow & 22x \color{red}{-42} \color{blue}{+42} \color{blue}{-77x} & = & \color{red}{77x} +154 \color{blue}{-77x} \color{blue}{+42} \\\Leftrightarrow & -55x& = & 196 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{196}{-55} \\\Leftrightarrow & x = \frac{-196}{55} & & \\ & V = \left\{ \frac{-196}{55} \right\} & \\\end{align}\)
8. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{12}& = & \frac{7}{4}x+6 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{21}{ \color{blue}{12} }x+\frac{72}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x-5& = & 21x+72 \\\Leftrightarrow & 4x \color{red}{-5} \color{blue}{+5} \color{blue}{-21x} & = & \color{red}{21x} +72 \color{blue}{-21x} \color{blue}{+5} \\\Leftrightarrow & -17x& = & 77 \\\Leftrightarrow & \frac{-17x}{ \color{red}{-17} }& = & \frac{77}{-17} \\\Leftrightarrow & x = \frac{-77}{17} & & \\ & V = \left\{ \frac{-77}{17} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+9& = & 15x+120 \\\Leftrightarrow & 10x \color{red}{+9} \color{blue}{-9} \color{blue}{-15x} & = & \color{red}{15x} +120 \color{blue}{-15x} \color{blue}{-9} \\\Leftrightarrow & -5x& = & 111 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{111}{-5} \\\Leftrightarrow & x = \frac{-111}{5} & & \\ & V = \left\{ \frac{-111}{5} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x-252 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} -252 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & -228 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-228}{-7} \\\Leftrightarrow & x = \frac{228}{7} & & \\ & V = \left\{ \frac{228}{7} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & 42x-630 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{-42x} & = & \color{red}{42x} -630 \color{blue}{-42x} \color{blue}{+90} \\\Leftrightarrow & -7x& = & -540 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-540}{-7} \\\Leftrightarrow & x = \frac{540}{7} & & \\ & V = \left\{ \frac{540}{7} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & -24x-150 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{+24x} & = & \color{red}{-24x} -150 \color{blue}{+24x} \color{blue}{+9} \\\Leftrightarrow & 29x& = & -141 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-141}{29} \\\Leftrightarrow & x = \frac{-141}{29} & & \\ & V = \left\{ \frac{-141}{29} \right\} & \\\end{align}\)