Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -70x-210 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+70x} & = & \color{red}{-70x} -210 \color{blue}{+70x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & -234 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-234}{91} \\\Leftrightarrow & x = \frac{-18}{7} & & \\ & V = \left\{ \frac{-18}{7} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{7}{2}x+2 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{245}{ \color{blue}{70} }x+\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-40& = & 245x+140 \\\Leftrightarrow & 14x \color{red}{-40} \color{blue}{+40} \color{blue}{-245x} & = & \color{red}{245x} +140 \color{blue}{-245x} \color{blue}{+40} \\\Leftrightarrow & -231x& = & 180 \\\Leftrightarrow & \frac{-231x}{ \color{red}{-231} }& = & \frac{180}{-231} \\\Leftrightarrow & x = \frac{-60}{77} & & \\ & V = \left\{ \frac{-60}{77} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{-7}{5}x-1 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x-\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-60& = & -196x-140 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+196x} & = & \color{red}{-196x} -140 \color{blue}{+196x} \color{blue}{+60} \\\Leftrightarrow & 231x& = & -80 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{-80}{231} \\\Leftrightarrow & x = \frac{-80}{231} & & \\ & V = \left\{ \frac{-80}{231} \right\} & \\\end{align}\)
4. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{176}{ \color{blue}{132} }x+\frac{924}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & 176x+924 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{-176x} & = & \color{red}{176x} +924 \color{blue}{-176x} \color{blue}{-24} \\\Leftrightarrow & -143x& = & 900 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{900}{-143} \\\Leftrightarrow & x = \frac{-900}{143} & & \\ & V = \left\{ \frac{-900}{143} \right\} & \\\end{align}\)
5. \(\text{10 is het kleinste gemene veelvoud van 5, 10 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{10}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{10.} (\frac{2x}{ \color{blue}{10} }+ \frac{ 3 }{ \color{blue}{10} })& = & (\frac{5}{ \color{blue}{10} }x-\frac{20}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 2x+3& = & 5x-20 \\\Leftrightarrow & 2x \color{red}{+3} \color{blue}{-3} \color{blue}{-5x} & = & \color{red}{5x} -20 \color{blue}{-5x} \color{blue}{-3} \\\Leftrightarrow & -3x& = & -23 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-23}{-3} \\\Leftrightarrow & x = \frac{23}{3} & & \\ & V = \left\{ \frac{23}{3} \right\} & \\\end{align}\)
6. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+4& = & 9x-18 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-9x} & = & \color{red}{9x} -18 \color{blue}{-9x} \color{blue}{-4} \\\Leftrightarrow & -3x& = & -22 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-22}{-3} \\\Leftrightarrow & x = \frac{22}{3} & & \\ & V = \left\{ \frac{22}{3} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{4}{3}x+6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x+\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & 112x+504 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-112x} & = & \color{red}{112x} +504 \color{blue}{-112x} \color{blue}{+24} \\\Leftrightarrow & -91x& = & 528 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{528}{-91} \\\Leftrightarrow & x = \frac{-528}{91} & & \\ & V = \left\{ \frac{-528}{91} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{12}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-48}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x+25& = & -48x+120 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{+48x} & = & \color{red}{-48x} +120 \color{blue}{+48x} \color{blue}{-25} \\\Leftrightarrow & 58x& = & 95 \\\Leftrightarrow & \frac{58x}{ \color{red}{58} }& = & \frac{95}{58} \\\Leftrightarrow & x = \frac{95}{58} & & \\ & V = \left\{ \frac{95}{58} \right\} & \\\end{align}\)
9. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{2}{5}x+7 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{36}{ \color{blue}{90} }x+\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & 36x+630 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{-36x} & = & \color{red}{36x} +630 \color{blue}{-36x} \color{blue}{+40} \\\Leftrightarrow & -21x& = & 670 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{670}{-21} \\\Leftrightarrow & x = \frac{-670}{21} & & \\ & V = \left\{ \frac{-670}{21} \right\} & \\\end{align}\)
10. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 28 }{ \color{blue}{105} })& = & (\frac{-175}{ \color{blue}{105} }x-\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-28& = & -175x-210 \\\Leftrightarrow & 15x \color{red}{-28} \color{blue}{+28} \color{blue}{+175x} & = & \color{red}{-175x} -210 \color{blue}{+175x} \color{blue}{+28} \\\Leftrightarrow & 190x& = & -182 \\\Leftrightarrow & \frac{190x}{ \color{red}{190} }& = & \frac{-182}{190} \\\Leftrightarrow & x = \frac{-91}{95} & & \\ & V = \left\{ \frac{-91}{95} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{72}{ \color{blue}{60} }x+\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & 72x+60 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-72x} & = & \color{red}{72x} +60 \color{blue}{-72x} \color{blue}{+8} \\\Leftrightarrow & -57x& = & 68 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{68}{-57} \\\Leftrightarrow & x = \frac{-68}{57} & & \\ & V = \left\{ \frac{-68}{57} \right\} & \\\end{align}\)
12. \(\text{28 is het kleinste gemene veelvoud van 7, 14 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{-7}{4}x+1 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 6 }{ \color{blue}{28} })& = & (\frac{-49}{ \color{blue}{28} }x+\frac{28}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-6& = & -49x+28 \\\Leftrightarrow & 4x \color{red}{-6} \color{blue}{+6} \color{blue}{+49x} & = & \color{red}{-49x} +28 \color{blue}{+49x} \color{blue}{+6} \\\Leftrightarrow & 53x& = & 34 \\\Leftrightarrow & \frac{53x}{ \color{red}{53} }& = & \frac{34}{53} \\\Leftrightarrow & x = \frac{34}{53} & & \\ & V = \left\{ \frac{34}{53} \right\} & \\\end{align}\)