Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+30& = & 33x+528 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-33x} & = & \color{red}{33x} +528 \color{blue}{-33x} \color{blue}{-30} \\\Leftrightarrow & -11x& = & 498 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{498}{-11} \\\Leftrightarrow & x = \frac{-498}{11} & & \\ & V = \left\{ \frac{-498}{11} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x+\frac{624}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+24& = & -130x+624 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{+130x} & = & \color{red}{-130x} +624 \color{blue}{+130x} \color{blue}{-24} \\\Leftrightarrow & 169x& = & 600 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{600}{169} \\\Leftrightarrow & x = \frac{600}{169} & & \\ & V = \left\{ \frac{600}{169} \right\} & \\\end{align}\)
3. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{4}{5}x-2 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{312}{ \color{blue}{390} }x-\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & 312x-780 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{-312x} & = & \color{red}{312x} -780 \color{blue}{-312x} \color{blue}{+120} \\\Leftrightarrow & -247x& = & -660 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-660}{-247} \\\Leftrightarrow & x = \frac{660}{247} & & \\ & V = \left\{ \frac{660}{247} \right\} & \\\end{align}\)
4. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{11}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 20 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{660}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-20& = & 55x+660 \\\Leftrightarrow & 22x \color{red}{-20} \color{blue}{+20} \color{blue}{-55x} & = & \color{red}{55x} +660 \color{blue}{-55x} \color{blue}{+20} \\\Leftrightarrow & -33x& = & 680 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{680}{-33} \\\Leftrightarrow & x = \frac{-680}{33} & & \\ & V = \left\{ \frac{-680}{33} \right\} & \\\end{align}\)
5. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{-2}{3}x+6 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x+\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & -88x+792 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{+88x} & = & \color{red}{-88x} +792 \color{blue}{+88x} \color{blue}{-48} \\\Leftrightarrow & 121x& = & 744 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{744}{121} \\\Leftrightarrow & x = \frac{744}{121} & & \\ & V = \left\{ \frac{744}{121} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{5}{6}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-90& = & 175x+1260 \\\Leftrightarrow & 42x \color{red}{-90} \color{blue}{+90} \color{blue}{-175x} & = & \color{red}{175x} +1260 \color{blue}{-175x} \color{blue}{+90} \\\Leftrightarrow & -133x& = & 1350 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{1350}{-133} \\\Leftrightarrow & x = \frac{-1350}{133} & & \\ & V = \left\{ \frac{-1350}{133} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{4}{3}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & 112x-168 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{-112x} & = & \color{red}{112x} -168 \color{blue}{-112x} \color{blue}{-36} \\\Leftrightarrow & -91x& = & -204 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-204}{-91} \\\Leftrightarrow & x = \frac{204}{91} & & \\ & V = \left\{ \frac{204}{91} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{14}& = & \frac{3}{5}x-1 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 15 }{ \color{blue}{70} })& = & (\frac{42}{ \color{blue}{70} }x-\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+15& = & 42x-70 \\\Leftrightarrow & 35x \color{red}{+15} \color{blue}{-15} \color{blue}{-42x} & = & \color{red}{42x} -70 \color{blue}{-42x} \color{blue}{-15} \\\Leftrightarrow & -7x& = & -85 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-85}{-7} \\\Leftrightarrow & x = \frac{85}{7} & & \\ & V = \left\{ \frac{85}{7} \right\} & \\\end{align}\)
9. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{-104}{ \color{blue}{130} }x+\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-20& = & -104x+260 \\\Leftrightarrow & 65x \color{red}{-20} \color{blue}{+20} \color{blue}{+104x} & = & \color{red}{-104x} +260 \color{blue}{+104x} \color{blue}{+20} \\\Leftrightarrow & 169x& = & 280 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{280}{169} \\\Leftrightarrow & x = \frac{280}{169} & & \\ & V = \left\{ \frac{280}{169} \right\} & \\\end{align}\)
10. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{1}{4}x-7 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 8 }{ \color{blue}{28} })& = & (\frac{7}{ \color{blue}{28} }x-\frac{196}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+8& = & 7x-196 \\\Leftrightarrow & 4x \color{red}{+8} \color{blue}{-8} \color{blue}{-7x} & = & \color{red}{7x} -196 \color{blue}{-7x} \color{blue}{-8} \\\Leftrightarrow & -3x& = & -204 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-204}{-3} \\\Leftrightarrow & x = 68 & & \\ & V = \left\{ 68 \right\} & \\\end{align}\)
11. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x+\frac{1440}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & 288x+1440 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{-288x} & = & \color{red}{288x} +1440 \color{blue}{-288x} \color{blue}{+45} \\\Leftrightarrow & -248x& = & 1485 \\\Leftrightarrow & \frac{-248x}{ \color{red}{-248} }& = & \frac{1485}{-248} \\\Leftrightarrow & x = \frac{-1485}{248} & & \\ & V = \left\{ \frac{-1485}{248} \right\} & \\\end{align}\)
12. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{9}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x+\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+4& = & 6x+36 \\\Leftrightarrow & 9x \color{red}{+4} \color{blue}{-4} \color{blue}{-6x} & = & \color{red}{6x} +36 \color{blue}{-6x} \color{blue}{-4} \\\Leftrightarrow & 3x& = & 32 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{32}{3} \\\Leftrightarrow & x = \frac{32}{3} & & \\ & V = \left\{ \frac{32}{3} \right\} & \\\end{align}\)