Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{7}{5}x+8 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }+ \frac{ 10 }{ \color{blue}{45} })& = & (\frac{63}{ \color{blue}{45} }x+\frac{360}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x+10& = & 63x+360 \\\Leftrightarrow & 15x \color{red}{+10} \color{blue}{-10} \color{blue}{-63x} & = & \color{red}{63x} +360 \color{blue}{-63x} \color{blue}{-10} \\\Leftrightarrow & -48x& = & 350 \\\Leftrightarrow & \frac{-48x}{ \color{red}{-48} }& = & \frac{350}{-48} \\\Leftrightarrow & x = \frac{-175}{24} & & \\ & V = \left\{ \frac{-175}{24} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-40& = & 35x-350 \\\Leftrightarrow & 14x \color{red}{-40} \color{blue}{+40} \color{blue}{-35x} & = & \color{red}{35x} -350 \color{blue}{-35x} \color{blue}{+40} \\\Leftrightarrow & -21x& = & -310 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-310}{-21} \\\Leftrightarrow & x = \frac{310}{21} & & \\ & V = \left\{ \frac{310}{21} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 21x-294 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-21x} & = & \color{red}{21x} -294 \color{blue}{-21x} \color{blue}{-18} \\\Leftrightarrow & -7x& = & -312 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-312}{-7} \\\Leftrightarrow & x = \frac{312}{7} & & \\ & V = \left\{ \frac{312}{7} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{11}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x-\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-120& = & 396x-1650 \\\Leftrightarrow & 55x \color{red}{-120} \color{blue}{+120} \color{blue}{-396x} & = & \color{red}{396x} -1650 \color{blue}{-396x} \color{blue}{+120} \\\Leftrightarrow & -341x& = & -1530 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{-1530}{-341} \\\Leftrightarrow & x = \frac{1530}{341} & & \\ & V = \left\{ \frac{1530}{341} \right\} & \\\end{align}\)
5. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{13}& = & \frac{2}{3}x+2 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{104}{ \color{blue}{156} }x+\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-60& = & 104x+312 \\\Leftrightarrow & 39x \color{red}{-60} \color{blue}{+60} \color{blue}{-104x} & = & \color{red}{104x} +312 \color{blue}{-104x} \color{blue}{+60} \\\Leftrightarrow & -65x& = & 372 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{372}{-65} \\\Leftrightarrow & x = \frac{-372}{65} & & \\ & V = \left\{ \frac{-372}{65} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-24& = & -63x-504 \\\Leftrightarrow & 28x \color{red}{-24} \color{blue}{+24} \color{blue}{+63x} & = & \color{red}{-63x} -504 \color{blue}{+63x} \color{blue}{+24} \\\Leftrightarrow & 91x& = & -480 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-480}{91} \\\Leftrightarrow & x = \frac{-480}{91} & & \\ & V = \left\{ \frac{-480}{91} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 7, 12 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{12}& = & \frac{-3}{4}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }- \frac{ 35 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x-35& = & -63x+84 \\\Leftrightarrow & 12x \color{red}{-35} \color{blue}{+35} \color{blue}{+63x} & = & \color{red}{-63x} +84 \color{blue}{+63x} \color{blue}{+35} \\\Leftrightarrow & 75x& = & 119 \\\Leftrightarrow & \frac{75x}{ \color{red}{75} }& = & \frac{119}{75} \\\Leftrightarrow & x = \frac{119}{75} & & \\ & V = \left\{ \frac{119}{75} \right\} & \\\end{align}\)
8. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{8}& = & \frac{7}{4}x-7 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }- \frac{ 25 }{ \color{blue}{40} })& = & (\frac{70}{ \color{blue}{40} }x-\frac{280}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x-25& = & 70x-280 \\\Leftrightarrow & 8x \color{red}{-25} \color{blue}{+25} \color{blue}{-70x} & = & \color{red}{70x} -280 \color{blue}{-70x} \color{blue}{+25} \\\Leftrightarrow & -62x& = & -255 \\\Leftrightarrow & \frac{-62x}{ \color{red}{-62} }& = & \frac{-255}{-62} \\\Leftrightarrow & x = \frac{255}{62} & & \\ & V = \left\{ \frac{255}{62} \right\} & \\\end{align}\)
9. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{8}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{24}{ \color{blue}{120} }x-\frac{480}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-75& = & 24x-480 \\\Leftrightarrow & 20x \color{red}{-75} \color{blue}{+75} \color{blue}{-24x} & = & \color{red}{24x} -480 \color{blue}{-24x} \color{blue}{+75} \\\Leftrightarrow & -4x& = & -405 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-405}{-4} \\\Leftrightarrow & x = \frac{405}{4} & & \\ & V = \left\{ \frac{405}{4} \right\} & \\\end{align}\)
10. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 35 }{ \color{blue}{112} })& = & (\frac{56}{ \color{blue}{112} }x-\frac{448}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-35& = & 56x-448 \\\Leftrightarrow & 16x \color{red}{-35} \color{blue}{+35} \color{blue}{-56x} & = & \color{red}{56x} -448 \color{blue}{-56x} \color{blue}{+35} \\\Leftrightarrow & -40x& = & -413 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{-413}{-40} \\\Leftrightarrow & x = \frac{413}{40} & & \\ & V = \left\{ \frac{413}{40} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{3}{5}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 126x+1260 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-126x} & = & \color{red}{126x} +1260 \color{blue}{-126x} \color{blue}{+120} \\\Leftrightarrow & -91x& = & 1380 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{1380}{-91} \\\Leftrightarrow & x = \frac{-1380}{91} & & \\ & V = \left\{ \frac{-1380}{91} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x+\frac{390}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & -312x+390 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{+312x} & = & \color{red}{-312x} +390 \color{blue}{+312x} \color{blue}{-60} \\\Leftrightarrow & 377x& = & 330 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{330}{377} \\\Leftrightarrow & x = \frac{330}{377} & & \\ & V = \left\{ \frac{330}{377} \right\} & \\\end{align}\)