Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{-32}{ \color{blue}{80} }x-\frac{640}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+15& = & -32x-640 \\\Leftrightarrow & 40x \color{red}{+15} \color{blue}{-15} \color{blue}{+32x} & = & \color{red}{-32x} -640 \color{blue}{+32x} \color{blue}{-15} \\\Leftrightarrow & 72x& = & -655 \\\Leftrightarrow & \frac{72x}{ \color{red}{72} }& = & \frac{-655}{72} \\\Leftrightarrow & x = \frac{-655}{72} & & \\ & V = \left\{ \frac{-655}{72} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 5, 14 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{14}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-15& = & 35x+70 \\\Leftrightarrow & 14x \color{red}{-15} \color{blue}{+15} \color{blue}{-35x} & = & \color{red}{35x} +70 \color{blue}{-35x} \color{blue}{+15} \\\Leftrightarrow & -21x& = & 85 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{85}{-21} \\\Leftrightarrow & x = \frac{-85}{21} & & \\ & V = \left\{ \frac{-85}{21} \right\} & \\\end{align}\)
3. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x-\frac{80}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+25& = & 40x-80 \\\Leftrightarrow & 16x \color{red}{+25} \color{blue}{-25} \color{blue}{-40x} & = & \color{red}{40x} -80 \color{blue}{-40x} \color{blue}{-25} \\\Leftrightarrow & -24x& = & -105 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{-105}{-24} \\\Leftrightarrow & x = \frac{35}{8} & & \\ & V = \left\{ \frac{35}{8} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+120& = & 35x-1470 \\\Leftrightarrow & 42x \color{red}{+120} \color{blue}{-120} \color{blue}{-35x} & = & \color{red}{35x} -1470 \color{blue}{-35x} \color{blue}{-120} \\\Leftrightarrow & 7x& = & -1590 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-1590}{7} \\\Leftrightarrow & x = \frac{-1590}{7} & & \\ & V = \left\{ \frac{-1590}{7} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{1}{6}x-3 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x-\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+60& = & 65x-1170 \\\Leftrightarrow & 78x \color{red}{+60} \color{blue}{-60} \color{blue}{-65x} & = & \color{red}{65x} -1170 \color{blue}{-65x} \color{blue}{-60} \\\Leftrightarrow & 13x& = & -1230 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-1230}{13} \\\Leftrightarrow & x = \frac{-1230}{13} & & \\ & V = \left\{ \frac{-1230}{13} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{2}{5}x-6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{156}{ \color{blue}{390} }x-\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & 156x-2340 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{-156x} & = & \color{red}{156x} -2340 \color{blue}{-156x} \color{blue}{-90} \\\Leftrightarrow & -91x& = & -2430 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-2430}{-91} \\\Leftrightarrow & x = \frac{2430}{91} & & \\ & V = \left\{ \frac{2430}{91} \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{7}{5}x-7 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{196}{ \color{blue}{140} }x-\frac{980}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & 196x-980 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-196x} & = & \color{red}{196x} -980 \color{blue}{-196x} \color{blue}{-40} \\\Leftrightarrow & -161x& = & -1020 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{-1020}{-161} \\\Leftrightarrow & x = \frac{1020}{161} & & \\ & V = \left\{ \frac{1020}{161} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{6}& = & \frac{2}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{12}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+25& = & 12x+240 \\\Leftrightarrow & 5x \color{red}{+25} \color{blue}{-25} \color{blue}{-12x} & = & \color{red}{12x} +240 \color{blue}{-12x} \color{blue}{-25} \\\Leftrightarrow & -7x& = & 215 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{215}{-7} \\\Leftrightarrow & x = \frac{-215}{7} & & \\ & V = \left\{ \frac{-215}{7} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+9& = & 15x-210 \\\Leftrightarrow & 10x \color{red}{+9} \color{blue}{-9} \color{blue}{-15x} & = & \color{red}{15x} -210 \color{blue}{-15x} \color{blue}{-9} \\\Leftrightarrow & -5x& = & -219 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-219}{-5} \\\Leftrightarrow & x = \frac{219}{5} & & \\ & V = \left\{ \frac{219}{5} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{5}{6}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{35}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+18& = & 35x-210 \\\Leftrightarrow & 6x \color{red}{+18} \color{blue}{-18} \color{blue}{-35x} & = & \color{red}{35x} -210 \color{blue}{-35x} \color{blue}{-18} \\\Leftrightarrow & -29x& = & -228 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{-228}{-29} \\\Leftrightarrow & x = \frac{228}{29} & & \\ & V = \left\{ \frac{228}{29} \right\} & \\\end{align}\)
11. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{-182}{ \color{blue}{130} }x+\frac{1040}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-30& = & -182x+1040 \\\Leftrightarrow & 65x \color{red}{-30} \color{blue}{+30} \color{blue}{+182x} & = & \color{red}{-182x} +1040 \color{blue}{+182x} \color{blue}{+30} \\\Leftrightarrow & 247x& = & 1070 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{1070}{247} \\\Leftrightarrow & x = \frac{1070}{247} & & \\ & V = \left\{ \frac{1070}{247} \right\} & \\\end{align}\)
12. \(\text{6 is het kleinste gemene veelvoud van 3, 6 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{6}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{6.} (\frac{2x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{3}{ \color{blue}{6} }x-\frac{30}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 2x+5& = & 3x-30 \\\Leftrightarrow & 2x \color{red}{+5} \color{blue}{-5} \color{blue}{-3x} & = & \color{red}{3x} -30 \color{blue}{-3x} \color{blue}{-5} \\\Leftrightarrow & -x& = & -35 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-35}{-1} \\\Leftrightarrow & x = 35 & & \\ & V = \left\{ 35 \right\} & \\\end{align}\)