Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{3}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & 18x+240 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{-18x} & = & \color{red}{18x} +240 \color{blue}{-18x} \color{blue}{+25} \\\Leftrightarrow & -13x& = & 265 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{265}{-13} \\\Leftrightarrow & x = \frac{-265}{13} & & \\ & V = \left\{ \frac{-265}{13} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -294x+210 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+294x} & = & \color{red}{-294x} +210 \color{blue}{+294x} \color{blue}{-120} \\\Leftrightarrow & 329x& = & 90 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{90}{329} \\\Leftrightarrow & x = \frac{90}{329} & & \\ & V = \left\{ \frac{90}{329} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 6x+240 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-6x} & = & \color{red}{6x} +240 \color{blue}{-6x} \color{blue}{-9} \\\Leftrightarrow & -x& = & 231 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{231}{-1} \\\Leftrightarrow & x = -231 & & \\ & V = \left\{ -231 \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & -28x-168 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{+28x} & = & \color{red}{-28x} -168 \color{blue}{+28x} \color{blue}{-12} \\\Leftrightarrow & 49x& = & -180 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-180}{49} \\\Leftrightarrow & x = \frac{-180}{49} & & \\ & V = \left\{ \frac{-180}{49} \right\} & \\\end{align}\)
5. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{4}{5}x+3 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{72}{ \color{blue}{90} }x+\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+20& = & 72x+270 \\\Leftrightarrow & 45x \color{red}{+20} \color{blue}{-20} \color{blue}{-72x} & = & \color{red}{72x} +270 \color{blue}{-72x} \color{blue}{-20} \\\Leftrightarrow & -27x& = & 250 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{250}{-27} \\\Leftrightarrow & x = \frac{-250}{27} & & \\ & V = \left\{ \frac{-250}{27} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{7}{2}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{105}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-4& = & 105x+240 \\\Leftrightarrow & 10x \color{red}{-4} \color{blue}{+4} \color{blue}{-105x} & = & \color{red}{105x} +240 \color{blue}{-105x} \color{blue}{+4} \\\Leftrightarrow & -95x& = & 244 \\\Leftrightarrow & \frac{-95x}{ \color{red}{-95} }& = & \frac{244}{-95} \\\Leftrightarrow & x = \frac{-244}{95} & & \\ & V = \left\{ \frac{-244}{95} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{-3}{4}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+24& = & -63x+420 \\\Leftrightarrow & 28x \color{red}{+24} \color{blue}{-24} \color{blue}{+63x} & = & \color{red}{-63x} +420 \color{blue}{+63x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & 396 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{396}{91} \\\Leftrightarrow & x = \frac{396}{91} & & \\ & V = \left\{ \frac{396}{91} \right\} & \\\end{align}\)
8. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{1}{6}x+1 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }+ \frac{ 168 }{ \color{blue}{462} })& = & (\frac{77}{ \color{blue}{462} }x+\frac{462}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x+168& = & 77x+462 \\\Leftrightarrow & 66x \color{red}{+168} \color{blue}{-168} \color{blue}{-77x} & = & \color{red}{77x} +462 \color{blue}{-77x} \color{blue}{-168} \\\Leftrightarrow & -11x& = & 294 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{294}{-11} \\\Leftrightarrow & x = \frac{-294}{11} & & \\ & V = \left\{ \frac{-294}{11} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & -40x+360 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+40x} & = & \color{red}{-40x} +360 \color{blue}{+40x} \color{blue}{-8} \\\Leftrightarrow & 55x& = & 352 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{352}{55} \\\Leftrightarrow & x = \frac{32}{5} & & \\ & V = \left\{ \frac{32}{5} \right\} & \\\end{align}\)
10. \(\text{560 is het kleinste gemene veelvoud van 7, 16 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{560.} (\frac{80x}{ \color{blue}{560} }- \frac{ 105 }{ \color{blue}{560} })& = & (\frac{-448}{ \color{blue}{560} }x-\frac{4480}{ \color{blue}{560} }) \color{blue}{.560} \\\Leftrightarrow & 80x-105& = & -448x-4480 \\\Leftrightarrow & 80x \color{red}{-105} \color{blue}{+105} \color{blue}{+448x} & = & \color{red}{-448x} -4480 \color{blue}{+448x} \color{blue}{+105} \\\Leftrightarrow & 528x& = & -4375 \\\Leftrightarrow & \frac{528x}{ \color{red}{528} }& = & \frac{-4375}{528} \\\Leftrightarrow & x = \frac{-4375}{528} & & \\ & V = \left\{ \frac{-4375}{528} \right\} & \\\end{align}\)
11. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{5}{2}x+1 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 40 }{ \color{blue}{130} })& = & (\frac{325}{ \color{blue}{130} }x+\frac{130}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-40& = & 325x+130 \\\Leftrightarrow & 26x \color{red}{-40} \color{blue}{+40} \color{blue}{-325x} & = & \color{red}{325x} +130 \color{blue}{-325x} \color{blue}{+40} \\\Leftrightarrow & -299x& = & 170 \\\Leftrightarrow & \frac{-299x}{ \color{red}{-299} }& = & \frac{170}{-299} \\\Leftrightarrow & x = \frac{-170}{299} & & \\ & V = \left\{ \frac{-170}{299} \right\} & \\\end{align}\)
12. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{5}{2}x+8 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{60}{ \color{blue}{24} }x+\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+9& = & 60x+192 \\\Leftrightarrow & 8x \color{red}{+9} \color{blue}{-9} \color{blue}{-60x} & = & \color{red}{60x} +192 \color{blue}{-60x} \color{blue}{-9} \\\Leftrightarrow & -52x& = & 183 \\\Leftrightarrow & \frac{-52x}{ \color{red}{-52} }& = & \frac{183}{-52} \\\Leftrightarrow & x = \frac{-183}{52} & & \\ & V = \left\{ \frac{-183}{52} \right\} & \\\end{align}\)