Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{4}-\frac{4}{7}=\frac{-8}{3}x+4\)
2. \(\frac{x}{3}-\frac{4}{11}=\frac{1}{2}x-7\)
3. \(\frac{x}{4}-\frac{4}{15}=\frac{7}{3}x-2\)
4. \(\frac{x}{5}-\frac{2}{13}=\frac{1}{2}x+6\)
5. \(\frac{x}{4}-\frac{4}{11}=\frac{-2}{3}x+5\)
6. \(\frac{x}{6}-\frac{2}{15}=\frac{6}{5}x-2\)
7. \(\frac{x}{6}+\frac{4}{15}=\frac{2}{5}x+5\)
8. \(\frac{x}{6}-\frac{5}{14}=\frac{-7}{5}x-4\)
9. \(\frac{x}{7}+\frac{3}{16}=\frac{1}{2}x-1\)
10. \(\frac{x}{7}-\frac{3}{10}=\frac{1}{2}x-2\)
11. \(\frac{x}{4}+\frac{3}{10}=\frac{-2}{3}x+3\)
12. \(\frac{x}{3}-\frac{4}{7}=\frac{5}{4}x-4\)

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-8}{3}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & -224x+336 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{+224x} & = & \color{red}{-224x} +336 \color{blue}{+224x} \color{blue}{+48} \\\Leftrightarrow & 245x& = & 384 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{384}{245} \\\Leftrightarrow & x = \frac{384}{245} & & \\ & V = \left\{ \frac{384}{245} \right\} & \\\end{align}\)
2. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{462}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-24& = & 33x-462 \\\Leftrightarrow & 22x \color{red}{-24} \color{blue}{+24} \color{blue}{-33x} & = & \color{red}{33x} -462 \color{blue}{-33x} \color{blue}{+24} \\\Leftrightarrow & -11x& = & -438 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-438}{-11} \\\Leftrightarrow & x = \frac{438}{11} & & \\ & V = \left\{ \frac{438}{11} \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{7}{3}x-2 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{140}{ \color{blue}{60} }x-\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 140x-120 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-140x} & = & \color{red}{140x} -120 \color{blue}{-140x} \color{blue}{+16} \\\Leftrightarrow & -125x& = & -104 \\\Leftrightarrow & \frac{-125x}{ \color{red}{-125} }& = & \frac{-104}{-125} \\\Leftrightarrow & x = \frac{104}{125} & & \\ & V = \left\{ \frac{104}{125} \right\} & \\\end{align}\)
4. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x+\frac{780}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-20& = & 65x+780 \\\Leftrightarrow & 26x \color{red}{-20} \color{blue}{+20} \color{blue}{-65x} & = & \color{red}{65x} +780 \color{blue}{-65x} \color{blue}{+20} \\\Leftrightarrow & -39x& = & 800 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{800}{-39} \\\Leftrightarrow & x = \frac{-800}{39} & & \\ & V = \left\{ \frac{-800}{39} \right\} & \\\end{align}\)
5. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{11}& = & \frac{-2}{3}x+5 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x+\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-48& = & -88x+660 \\\Leftrightarrow & 33x \color{red}{-48} \color{blue}{+48} \color{blue}{+88x} & = & \color{red}{-88x} +660 \color{blue}{+88x} \color{blue}{+48} \\\Leftrightarrow & 121x& = & 708 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{708}{121} \\\Leftrightarrow & x = \frac{708}{121} & & \\ & V = \left\{ \frac{708}{121} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & 36x-60 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-36x} & = & \color{red}{36x} -60 \color{blue}{-36x} \color{blue}{+4} \\\Leftrightarrow & -31x& = & -56 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{-56}{-31} \\\Leftrightarrow & x = \frac{56}{31} & & \\ & V = \left\{ \frac{56}{31} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{15}& = & \frac{2}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{12}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+8& = & 12x+150 \\\Leftrightarrow & 5x \color{red}{+8} \color{blue}{-8} \color{blue}{-12x} & = & \color{red}{12x} +150 \color{blue}{-12x} \color{blue}{-8} \\\Leftrightarrow & -7x& = & 142 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{142}{-7} \\\Leftrightarrow & x = \frac{-142}{7} & & \\ & V = \left\{ \frac{-142}{7} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{14}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-75& = & -294x-840 \\\Leftrightarrow & 35x \color{red}{-75} \color{blue}{+75} \color{blue}{+294x} & = & \color{red}{-294x} -840 \color{blue}{+294x} \color{blue}{+75} \\\Leftrightarrow & 329x& = & -765 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-765}{329} \\\Leftrightarrow & x = \frac{-765}{329} & & \\ & V = \left\{ \frac{-765}{329} \right\} & \\\end{align}\)
9. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{16}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 21 }{ \color{blue}{112} })& = & (\frac{56}{ \color{blue}{112} }x-\frac{112}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+21& = & 56x-112 \\\Leftrightarrow & 16x \color{red}{+21} \color{blue}{-21} \color{blue}{-56x} & = & \color{red}{56x} -112 \color{blue}{-56x} \color{blue}{-21} \\\Leftrightarrow & -40x& = & -133 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{-133}{-40} \\\Leftrightarrow & x = \frac{133}{40} & & \\ & V = \left\{ \frac{133}{40} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 21 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-21& = & 35x-140 \\\Leftrightarrow & 10x \color{red}{-21} \color{blue}{+21} \color{blue}{-35x} & = & \color{red}{35x} -140 \color{blue}{-35x} \color{blue}{+21} \\\Leftrightarrow & -25x& = & -119 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{-119}{-25} \\\Leftrightarrow & x = \frac{119}{25} & & \\ & V = \left\{ \frac{119}{25} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+18& = & -40x+180 \\\Leftrightarrow & 15x \color{red}{+18} \color{blue}{-18} \color{blue}{+40x} & = & \color{red}{-40x} +180 \color{blue}{+40x} \color{blue}{-18} \\\Leftrightarrow & 55x& = & 162 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{162}{55} \\\Leftrightarrow & x = \frac{162}{55} & & \\ & V = \left\{ \frac{162}{55} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{5}{4}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{105}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-48& = & 105x-336 \\\Leftrightarrow & 28x \color{red}{-48} \color{blue}{+48} \color{blue}{-105x} & = & \color{red}{105x} -336 \color{blue}{-105x} \color{blue}{+48} \\\Leftrightarrow & -77x& = & -288 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-288}{-77} \\\Leftrightarrow & x = \frac{288}{77} & & \\ & V = \left\{ \frac{288}{77} \right\} & \\\end{align}\)