Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{-2}{3}x+8 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }+ \frac{ 105 }{ \color{blue}{231} })& = & (\frac{-154}{ \color{blue}{231} }x+\frac{1848}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x+105& = & -154x+1848 \\\Leftrightarrow & 33x \color{red}{+105} \color{blue}{-105} \color{blue}{+154x} & = & \color{red}{-154x} +1848 \color{blue}{+154x} \color{blue}{-105} \\\Leftrightarrow & 187x& = & 1743 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{1743}{187} \\\Leftrightarrow & x = \frac{1743}{187} & & \\ & V = \left\{ \frac{1743}{187} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{6}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 252x-1470 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-252x} & = & \color{red}{252x} -1470 \color{blue}{-252x} \color{blue}{+120} \\\Leftrightarrow & -217x& = & -1350 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-1350}{-217} \\\Leftrightarrow & x = \frac{1350}{217} & & \\ & V = \left\{ \frac{1350}{217} \right\} & \\\end{align}\)
3. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{9}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 10 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x+\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-10& = & 9x+126 \\\Leftrightarrow & 6x \color{red}{-10} \color{blue}{+10} \color{blue}{-9x} & = & \color{red}{9x} +126 \color{blue}{-9x} \color{blue}{+10} \\\Leftrightarrow & -3x& = & 136 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{136}{-3} \\\Leftrightarrow & x = \frac{-136}{3} & & \\ & V = \left\{ \frac{-136}{3} \right\} & \\\end{align}\)
4. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+40& = & 45x+540 \\\Leftrightarrow & 18x \color{red}{+40} \color{blue}{-40} \color{blue}{-45x} & = & \color{red}{45x} +540 \color{blue}{-45x} \color{blue}{-40} \\\Leftrightarrow & -27x& = & 500 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{500}{-27} \\\Leftrightarrow & x = \frac{-500}{27} & & \\ & V = \left\{ \frac{-500}{27} \right\} & \\\end{align}\)
5. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{35}{ \color{blue}{105} }x+\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+60& = & 35x+210 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{-35x} & = & \color{red}{35x} +210 \color{blue}{-35x} \color{blue}{-60} \\\Leftrightarrow & -14x& = & 150 \\\Leftrightarrow & \frac{-14x}{ \color{red}{-14} }& = & \frac{150}{-14} \\\Leftrightarrow & x = \frac{-75}{7} & & \\ & V = \left\{ \frac{-75}{7} \right\} & \\\end{align}\)
6. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x+\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-40& = & -196x+700 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{+196x} & = & \color{red}{-196x} +700 \color{blue}{+196x} \color{blue}{+40} \\\Leftrightarrow & 231x& = & 740 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{740}{231} \\\Leftrightarrow & x = \frac{740}{231} & & \\ & V = \left\{ \frac{740}{231} \right\} & \\\end{align}\)
7. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{3}{2}x+5 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 8 }{ \color{blue}{14} })& = & (\frac{21}{ \color{blue}{14} }x+\frac{70}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-8& = & 21x+70 \\\Leftrightarrow & 2x \color{red}{-8} \color{blue}{+8} \color{blue}{-21x} & = & \color{red}{21x} +70 \color{blue}{-21x} \color{blue}{+8} \\\Leftrightarrow & -19x& = & 78 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{78}{-19} \\\Leftrightarrow & x = \frac{-78}{19} & & \\ & V = \left\{ \frac{-78}{19} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & 42x-1470 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{-42x} & = & \color{red}{42x} -1470 \color{blue}{-42x} \color{blue}{+90} \\\Leftrightarrow & -7x& = & -1380 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-1380}{-7} \\\Leftrightarrow & x = \frac{1380}{7} & & \\ & V = \left\{ \frac{1380}{7} \right\} & \\\end{align}\)
9. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 40 }{ \color{blue}{180} })& = & (\frac{36}{ \color{blue}{180} }x+\frac{360}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+40& = & 36x+360 \\\Leftrightarrow & 45x \color{red}{+40} \color{blue}{-40} \color{blue}{-36x} & = & \color{red}{36x} +360 \color{blue}{-36x} \color{blue}{-40} \\\Leftrightarrow & 9x& = & 320 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{320}{9} \\\Leftrightarrow & x = \frac{320}{9} & & \\ & V = \left\{ \frac{320}{9} \right\} & \\\end{align}\)
10. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{8}& = & \frac{7}{3}x+3 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{56}{ \color{blue}{24} }x+\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x+15& = & 56x+72 \\\Leftrightarrow & 6x \color{red}{+15} \color{blue}{-15} \color{blue}{-56x} & = & \color{red}{56x} +72 \color{blue}{-56x} \color{blue}{-15} \\\Leftrightarrow & -50x& = & 57 \\\Leftrightarrow & \frac{-50x}{ \color{red}{-50} }& = & \frac{57}{-50} \\\Leftrightarrow & x = \frac{-57}{50} & & \\ & V = \left\{ \frac{-57}{50} \right\} & \\\end{align}\)
11. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{7}{3}x-6 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 28 }{ \color{blue}{105} })& = & (\frac{245}{ \color{blue}{105} }x-\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+28& = & 245x-630 \\\Leftrightarrow & 15x \color{red}{+28} \color{blue}{-28} \color{blue}{-245x} & = & \color{red}{245x} -630 \color{blue}{-245x} \color{blue}{-28} \\\Leftrightarrow & -230x& = & -658 \\\Leftrightarrow & \frac{-230x}{ \color{red}{-230} }& = & \frac{-658}{-230} \\\Leftrightarrow & x = \frac{329}{115} & & \\ & V = \left\{ \frac{329}{115} \right\} & \\\end{align}\)
12. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{8}& = & \frac{7}{5}x-4 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 75 }{ \color{blue}{120} })& = & (\frac{168}{ \color{blue}{120} }x-\frac{480}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-75& = & 168x-480 \\\Leftrightarrow & 20x \color{red}{-75} \color{blue}{+75} \color{blue}{-168x} & = & \color{red}{168x} -480 \color{blue}{-168x} \color{blue}{+75} \\\Leftrightarrow & -148x& = & -405 \\\Leftrightarrow & \frac{-148x}{ \color{red}{-148} }& = & \frac{-405}{-148} \\\Leftrightarrow & x = \frac{405}{148} & & \\ & V = \left\{ \frac{405}{148} \right\} & \\\end{align}\)