Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 7, 14 en 5} \\ \begin{align} & \frac{x}{7}+\frac{5}{14}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }+ \frac{ 25 }{ \color{blue}{70} })& = & (\frac{84}{ \color{blue}{70} }x+\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x+25& = & 84x+140 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{-84x} & = & \color{red}{84x} +140 \color{blue}{-84x} \color{blue}{-25} \\\Leftrightarrow & -74x& = & 115 \\\Leftrightarrow & \frac{-74x}{ \color{red}{-74} }& = & \frac{115}{-74} \\\Leftrightarrow & x = \frac{-115}{74} & & \\ & V = \left\{ \frac{-115}{74} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-325}{ \color{blue}{390} }x-\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+90& = & -325x-2730 \\\Leftrightarrow & 78x \color{red}{+90} \color{blue}{-90} \color{blue}{+325x} & = & \color{red}{-325x} -2730 \color{blue}{+325x} \color{blue}{-90} \\\Leftrightarrow & 403x& = & -2820 \\\Leftrightarrow & \frac{403x}{ \color{red}{403} }& = & \frac{-2820}{403} \\\Leftrightarrow & x = \frac{-2820}{403} & & \\ & V = \left\{ \frac{-2820}{403} \right\} & \\\end{align}\)
3. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{3}{2}x+4 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{36}{ \color{blue}{24} }x+\frac{96}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+9& = & 36x+96 \\\Leftrightarrow & 8x \color{red}{+9} \color{blue}{-9} \color{blue}{-36x} & = & \color{red}{36x} +96 \color{blue}{-36x} \color{blue}{-9} \\\Leftrightarrow & -28x& = & 87 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{87}{-28} \\\Leftrightarrow & x = \frac{-87}{28} & & \\ & V = \left\{ \frac{-87}{28} \right\} & \\\end{align}\)
4. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{-8}{3}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-160}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+18& = & -160x+180 \\\Leftrightarrow & 15x \color{red}{+18} \color{blue}{-18} \color{blue}{+160x} & = & \color{red}{-160x} +180 \color{blue}{+160x} \color{blue}{-18} \\\Leftrightarrow & 175x& = & 162 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{162}{175} \\\Leftrightarrow & x = \frac{162}{175} & & \\ & V = \left\{ \frac{162}{175} \right\} & \\\end{align}\)
5. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x+\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+40& = & 45x+720 \\\Leftrightarrow & 18x \color{red}{+40} \color{blue}{-40} \color{blue}{-45x} & = & \color{red}{45x} +720 \color{blue}{-45x} \color{blue}{-40} \\\Leftrightarrow & -27x& = & 680 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{680}{-27} \\\Leftrightarrow & x = \frac{-680}{27} & & \\ & V = \left\{ \frac{-680}{27} \right\} & \\\end{align}\)
6. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{-3}{4}x-3 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-99}{ \color{blue}{132} }x-\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-24& = & -99x-396 \\\Leftrightarrow & 44x \color{red}{-24} \color{blue}{+24} \color{blue}{+99x} & = & \color{red}{-99x} -396 \color{blue}{+99x} \color{blue}{+24} \\\Leftrightarrow & 143x& = & -372 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-372}{143} \\\Leftrightarrow & x = \frac{-372}{143} & & \\ & V = \left\{ \frac{-372}{143} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x+\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+20& = & -98x+420 \\\Leftrightarrow & 35x \color{red}{+20} \color{blue}{-20} \color{blue}{+98x} & = & \color{red}{-98x} +420 \color{blue}{+98x} \color{blue}{-20} \\\Leftrightarrow & 133x& = & 400 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{400}{133} \\\Leftrightarrow & x = \frac{400}{133} & & \\ & V = \left\{ \frac{400}{133} \right\} & \\\end{align}\)
8. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{7}{3}x-4 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 28 }{ \color{blue}{105} })& = & (\frac{245}{ \color{blue}{105} }x-\frac{420}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+28& = & 245x-420 \\\Leftrightarrow & 15x \color{red}{+28} \color{blue}{-28} \color{blue}{-245x} & = & \color{red}{245x} -420 \color{blue}{-245x} \color{blue}{-28} \\\Leftrightarrow & -230x& = & -448 \\\Leftrightarrow & \frac{-230x}{ \color{red}{-230} }& = & \frac{-448}{-230} \\\Leftrightarrow & x = \frac{224}{115} & & \\ & V = \left\{ \frac{224}{115} \right\} & \\\end{align}\)
9. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & -462x+2640 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{+462x} & = & \color{red}{-462x} +2640 \color{blue}{+462x} \color{blue}{+150} \\\Leftrightarrow & 517x& = & 2790 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{2790}{517} \\\Leftrightarrow & x = \frac{2790}{517} & & \\ & V = \left\{ \frac{2790}{517} \right\} & \\\end{align}\)
10. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{11}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 30 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{220}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-30& = & 55x-220 \\\Leftrightarrow & 22x \color{red}{-30} \color{blue}{+30} \color{blue}{-55x} & = & \color{red}{55x} -220 \color{blue}{-55x} \color{blue}{+30} \\\Leftrightarrow & -33x& = & -190 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-190}{-33} \\\Leftrightarrow & x = \frac{190}{33} & & \\ & V = \left\{ \frac{190}{33} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{1}{6}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+90& = & 35x-1050 \\\Leftrightarrow & 42x \color{red}{+90} \color{blue}{-90} \color{blue}{-35x} & = & \color{red}{35x} -1050 \color{blue}{-35x} \color{blue}{-90} \\\Leftrightarrow & 7x& = & -1140 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-1140}{7} \\\Leftrightarrow & x = \frac{-1140}{7} & & \\ & V = \left\{ \frac{-1140}{7} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{14}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-75& = & 42x-840 \\\Leftrightarrow & 35x \color{red}{-75} \color{blue}{+75} \color{blue}{-42x} & = & \color{red}{42x} -840 \color{blue}{-42x} \color{blue}{+75} \\\Leftrightarrow & -7x& = & -765 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-765}{-7} \\\Leftrightarrow & x = \frac{765}{7} & & \\ & V = \left\{ \frac{765}{7} \right\} & \\\end{align}\)