Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{7}{2}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{105}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & 105x-240 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{-105x} & = & \color{red}{105x} -240 \color{blue}{-105x} \color{blue}{+9} \\\Leftrightarrow & -95x& = & -231 \\\Leftrightarrow & \frac{-95x}{ \color{red}{-95} }& = & \frac{-231}{-95} \\\Leftrightarrow & x = \frac{231}{95} & & \\ & V = \left\{ \frac{231}{95} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 8 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{28}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-8& = & 7x-28 \\\Leftrightarrow & 2x \color{red}{-8} \color{blue}{+8} \color{blue}{-7x} & = & \color{red}{7x} -28 \color{blue}{-7x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & -20 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-20}{-5} \\\Leftrightarrow & x = 4 & & \\ & V = \left\{ 4 \right\} & \\\end{align}\)
3. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x+\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & 78x+1170 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-78x} & = & \color{red}{78x} +1170 \color{blue}{-78x} \color{blue}{-60} \\\Leftrightarrow & -13x& = & 1110 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{1110}{-13} \\\Leftrightarrow & x = \frac{-1110}{13} & & \\ & V = \left\{ \frac{-1110}{13} \right\} & \\\end{align}\)
4. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{330}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-40& = & 55x-330 \\\Leftrightarrow & 22x \color{red}{-40} \color{blue}{+40} \color{blue}{-55x} & = & \color{red}{55x} -330 \color{blue}{-55x} \color{blue}{+40} \\\Leftrightarrow & -33x& = & -290 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-290}{-33} \\\Leftrightarrow & x = \frac{290}{33} & & \\ & V = \left\{ \frac{290}{33} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x-\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & -98x-420 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{+98x} & = & \color{red}{-98x} -420 \color{blue}{+98x} \color{blue}{+20} \\\Leftrightarrow & 133x& = & -400 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-400}{133} \\\Leftrightarrow & x = \frac{-400}{133} & & \\ & V = \left\{ \frac{-400}{133} \right\} & \\\end{align}\)
6. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{-7}{4}x-1 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{-196}{ \color{blue}{112} }x-\frac{112}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & -196x-112 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{+196x} & = & \color{red}{-196x} -112 \color{blue}{+196x} \color{blue}{-35} \\\Leftrightarrow & 212x& = & -147 \\\Leftrightarrow & \frac{212x}{ \color{red}{212} }& = & \frac{-147}{212} \\\Leftrightarrow & x = \frac{-147}{212} & & \\ & V = \left\{ \frac{-147}{212} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x-\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & -52x-156 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{+52x} & = & \color{red}{-52x} -156 \color{blue}{+52x} \color{blue}{-12} \\\Leftrightarrow & 91x& = & -168 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-168}{91} \\\Leftrightarrow & x = \frac{-24}{13} & & \\ & V = \left\{ \frac{-24}{13} \right\} & \\\end{align}\)
8. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{4}{5}x+4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{312}{ \color{blue}{390} }x+\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & 312x+1560 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{-312x} & = & \color{red}{312x} +1560 \color{blue}{-312x} \color{blue}{-120} \\\Leftrightarrow & -247x& = & 1440 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{1440}{-247} \\\Leftrightarrow & x = \frac{-1440}{247} & & \\ & V = \left\{ \frac{-1440}{247} \right\} & \\\end{align}\)
9. \(\text{45 is het kleinste gemene veelvoud van 5, 9 en 3} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{45.} (\frac{9x}{ \color{blue}{45} }- \frac{ 10 }{ \color{blue}{45} })& = & (\frac{15}{ \color{blue}{45} }x-\frac{315}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 9x-10& = & 15x-315 \\\Leftrightarrow & 9x \color{red}{-10} \color{blue}{+10} \color{blue}{-15x} & = & \color{red}{15x} -315 \color{blue}{-15x} \color{blue}{+10} \\\Leftrightarrow & -6x& = & -305 \\\Leftrightarrow & \frac{-6x}{ \color{red}{-6} }& = & \frac{-305}{-6} \\\Leftrightarrow & x = \frac{305}{6} & & \\ & V = \left\{ \frac{305}{6} \right\} & \\\end{align}\)
10. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{-8}{3}x+1 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-352}{ \color{blue}{132} }x+\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & -352x+132 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{+352x} & = & \color{red}{-352x} +132 \color{blue}{+352x} \color{blue}{-48} \\\Leftrightarrow & 385x& = & 84 \\\Leftrightarrow & \frac{385x}{ \color{red}{385} }& = & \frac{84}{385} \\\Leftrightarrow & x = \frac{12}{55} & & \\ & V = \left\{ \frac{12}{55} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 6x-240 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-6x} & = & \color{red}{6x} -240 \color{blue}{-6x} \color{blue}{-9} \\\Leftrightarrow & -x& = & -249 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-249}{-1} \\\Leftrightarrow & x = 249 & & \\ & V = \left\{ 249 \right\} & \\\end{align}\)
12. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{11}& = & \frac{-2}{5}x+6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{-132}{ \color{blue}{330} }x+\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-60& = & -132x+1980 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{+132x} & = & \color{red}{-132x} +1980 \color{blue}{+132x} \color{blue}{+60} \\\Leftrightarrow & 187x& = & 2040 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{2040}{187} \\\Leftrightarrow & x = \frac{120}{11} & & \\ & V = \left\{ \frac{120}{11} \right\} & \\\end{align}\)