Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{120 is het kleinste gemene veelvoud van 3, 8 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{8}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{120.} (\frac{40x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{24}{ \color{blue}{120} }x+\frac{600}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 40x-75& = & 24x+600 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{-24x} & = & \color{red}{24x} +600 \color{blue}{-24x} \color{blue}{+75} \\\Leftrightarrow & 16x& = & 675 \\\Leftrightarrow & \frac{16x}{ \color{red}{16} }& = & \frac{675}{16} \\\Leftrightarrow & x = \frac{675}{16} & & \\ & V = \left\{ \frac{675}{16} \right\} & \\\end{align}\)
2. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{-3}{4}x+6 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-9}{ \color{blue}{12} }x+\frac{72}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+5& = & -9x+72 \\\Leftrightarrow & 4x \color{red}{+5} \color{blue}{-5} \color{blue}{+9x} & = & \color{red}{-9x} +72 \color{blue}{+9x} \color{blue}{-5} \\\Leftrightarrow & 13x& = & 67 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{67}{13} \\\Leftrightarrow & x = \frac{67}{13} & & \\ & V = \left\{ \frac{67}{13} \right\} & \\\end{align}\)
3. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{3}{2}x-7 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 28 }{ \color{blue}{154} })& = & (\frac{231}{ \color{blue}{154} }x-\frac{1078}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-28& = & 231x-1078 \\\Leftrightarrow & 22x \color{red}{-28} \color{blue}{+28} \color{blue}{-231x} & = & \color{red}{231x} -1078 \color{blue}{-231x} \color{blue}{+28} \\\Leftrightarrow & -209x& = & -1050 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-1050}{-209} \\\Leftrightarrow & x = \frac{1050}{209} & & \\ & V = \left\{ \frac{1050}{209} \right\} & \\\end{align}\)
4. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 42 }{ \color{blue}{273} })& = & (\frac{364}{ \color{blue}{273} }x+\frac{2184}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-42& = & 364x+2184 \\\Leftrightarrow & 39x \color{red}{-42} \color{blue}{+42} \color{blue}{-364x} & = & \color{red}{364x} +2184 \color{blue}{-364x} \color{blue}{+42} \\\Leftrightarrow & -325x& = & 2226 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{2226}{-325} \\\Leftrightarrow & x = \frac{-2226}{325} & & \\ & V = \left\{ \frac{-2226}{325} \right\} & \\\end{align}\)
5. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{5}{2}x+7 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 28 }{ \color{blue}{126} })& = & (\frac{315}{ \color{blue}{126} }x+\frac{882}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-28& = & 315x+882 \\\Leftrightarrow & 18x \color{red}{-28} \color{blue}{+28} \color{blue}{-315x} & = & \color{red}{315x} +882 \color{blue}{-315x} \color{blue}{+28} \\\Leftrightarrow & -297x& = & 910 \\\Leftrightarrow & \frac{-297x}{ \color{red}{-297} }& = & \frac{910}{-297} \\\Leftrightarrow & x = \frac{-910}{297} & & \\ & V = \left\{ \frac{-910}{297} \right\} & \\\end{align}\)
6. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{1}{3}x-8 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 9 }{ \color{blue}{21} })& = & (\frac{7}{ \color{blue}{21} }x-\frac{168}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+9& = & 7x-168 \\\Leftrightarrow & 3x \color{red}{+9} \color{blue}{-9} \color{blue}{-7x} & = & \color{red}{7x} -168 \color{blue}{-7x} \color{blue}{-9} \\\Leftrightarrow & -4x& = & -177 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-177}{-4} \\\Leftrightarrow & x = \frac{177}{4} & & \\ & V = \left\{ \frac{177}{4} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{1}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 42x-1260 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-42x} & = & \color{red}{42x} -1260 \color{blue}{-42x} \color{blue}{+120} \\\Leftrightarrow & -7x& = & -1140 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-1140}{-7} \\\Leftrightarrow & x = \frac{1140}{7} & & \\ & V = \left\{ \frac{1140}{7} \right\} & \\\end{align}\)
8. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{7}{3}x-4 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{182}{ \color{blue}{78} }x-\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+24& = & 182x-312 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{-182x} & = & \color{red}{182x} -312 \color{blue}{-182x} \color{blue}{-24} \\\Leftrightarrow & -143x& = & -336 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-336}{-143} \\\Leftrightarrow & x = \frac{336}{143} & & \\ & V = \left\{ \frac{336}{143} \right\} & \\\end{align}\)
9. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-56}{ \color{blue}{140} }x+\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & -56x+420 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{+56x} & = & \color{red}{-56x} +420 \color{blue}{+56x} \color{blue}{-40} \\\Leftrightarrow & 91x& = & 380 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{380}{91} \\\Leftrightarrow & x = \frac{380}{91} & & \\ & V = \left\{ \frac{380}{91} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{11}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x+\frac{1320}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+150& = & -462x+1320 \\\Leftrightarrow & 55x \color{red}{+150} \color{blue}{-150} \color{blue}{+462x} & = & \color{red}{-462x} +1320 \color{blue}{+462x} \color{blue}{-150} \\\Leftrightarrow & 517x& = & 1170 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{1170}{517} \\\Leftrightarrow & x = \frac{1170}{517} & & \\ & V = \left\{ \frac{1170}{517} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{10}& = & \frac{-5}{6}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-25}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+9& = & -25x+240 \\\Leftrightarrow & 6x \color{red}{+9} \color{blue}{-9} \color{blue}{+25x} & = & \color{red}{-25x} +240 \color{blue}{+25x} \color{blue}{-9} \\\Leftrightarrow & 31x& = & 231 \\\Leftrightarrow & \frac{31x}{ \color{red}{31} }& = & \frac{231}{31} \\\Leftrightarrow & x = \frac{231}{31} & & \\ & V = \left\{ \frac{231}{31} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+20& = & -56x-490 \\\Leftrightarrow & 35x \color{red}{+20} \color{blue}{-20} \color{blue}{+56x} & = & \color{red}{-56x} -490 \color{blue}{+56x} \color{blue}{-20} \\\Leftrightarrow & 91x& = & -510 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-510}{91} \\\Leftrightarrow & x = \frac{-510}{91} & & \\ & V = \left\{ \frac{-510}{91} \right\} & \\\end{align}\)