Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{10}& = & \frac{-8}{3}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-80}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+9& = & -80x+30 \\\Leftrightarrow & 15x \color{red}{+9} \color{blue}{-9} \color{blue}{+80x} & = & \color{red}{-80x} +30 \color{blue}{+80x} \color{blue}{-9} \\\Leftrightarrow & 95x& = & 21 \\\Leftrightarrow & \frac{95x}{ \color{red}{95} }& = & \frac{21}{95} \\\Leftrightarrow & x = \frac{21}{95} & & \\ & V = \left\{ \frac{21}{95} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & -168x+210 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{+168x} & = & \color{red}{-168x} +210 \color{blue}{+168x} \color{blue}{+90} \\\Leftrightarrow & 203x& = & 300 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{300}{203} \\\Leftrightarrow & x = \frac{300}{203} & & \\ & V = \left\{ \frac{300}{203} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{2}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{28}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+30& = & 28x-490 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{-28x} & = & \color{red}{28x} -490 \color{blue}{-28x} \color{blue}{-30} \\\Leftrightarrow & 7x& = & -520 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-520}{7} \\\Leftrightarrow & x = \frac{-520}{7} & & \\ & V = \left\{ \frac{-520}{7} \right\} & \\\end{align}\)
4. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{-7}{5}x-5 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 100 }{ \color{blue}{220} })& = & (\frac{-308}{ \color{blue}{220} }x-\frac{1100}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+100& = & -308x-1100 \\\Leftrightarrow & 55x \color{red}{+100} \color{blue}{-100} \color{blue}{+308x} & = & \color{red}{-308x} -1100 \color{blue}{+308x} \color{blue}{-100} \\\Leftrightarrow & 363x& = & -1200 \\\Leftrightarrow & \frac{363x}{ \color{red}{363} }& = & \frac{-1200}{363} \\\Leftrightarrow & x = \frac{-400}{121} & & \\ & V = \left\{ \frac{-400}{121} \right\} & \\\end{align}\)
5. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{6}{ \color{blue}{12} }x+\frac{48}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+5& = & 6x+48 \\\Leftrightarrow & 4x \color{red}{+5} \color{blue}{-5} \color{blue}{-6x} & = & \color{red}{6x} +48 \color{blue}{-6x} \color{blue}{-5} \\\Leftrightarrow & -2x& = & 43 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{43}{-2} \\\Leftrightarrow & x = \frac{-43}{2} & & \\ & V = \left\{ \frac{-43}{2} \right\} & \\\end{align}\)
6. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{5}{7}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 25 }{ \color{blue}{35} })& = & (\frac{-49}{ \color{blue}{35} }x-\frac{140}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-25& = & -49x-140 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{+49x} & = & \color{red}{-49x} -140 \color{blue}{+49x} \color{blue}{+25} \\\Leftrightarrow & 54x& = & -115 \\\Leftrightarrow & \frac{54x}{ \color{red}{54} }& = & \frac{-115}{54} \\\Leftrightarrow & x = \frac{-115}{54} & & \\ & V = \left\{ \frac{-115}{54} \right\} & \\\end{align}\)
7. \(\text{12 is het kleinste gemene veelvoud van 3, 6 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }- \frac{ 10 }{ \color{blue}{12} })& = & (\frac{3}{ \color{blue}{12} }x+\frac{48}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x-10& = & 3x+48 \\\Leftrightarrow & 4x \color{red}{-10} \color{blue}{+10} \color{blue}{-3x} & = & \color{red}{3x} +48 \color{blue}{-3x} \color{blue}{+10} \\\Leftrightarrow & x& = & 58 \\ & V = \left\{ 58 \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{9}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x-\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-50& = & 108x-450 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{-108x} & = & \color{red}{108x} -450 \color{blue}{-108x} \color{blue}{+50} \\\Leftrightarrow & -93x& = & -400 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{-400}{-93} \\\Leftrightarrow & x = \frac{400}{93} & & \\ & V = \left\{ \frac{400}{93} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{15}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+8& = & -12x-240 \\\Leftrightarrow & 5x \color{red}{+8} \color{blue}{-8} \color{blue}{+12x} & = & \color{red}{-12x} -240 \color{blue}{+12x} \color{blue}{-8} \\\Leftrightarrow & 17x& = & -248 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-248}{17} \\\Leftrightarrow & x = \frac{-248}{17} & & \\ & V = \left\{ \frac{-248}{17} \right\} & \\\end{align}\)
10. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 16 }{ \color{blue}{28} })& = & (\frac{-21}{ \color{blue}{28} }x+\frac{84}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-16& = & -21x+84 \\\Leftrightarrow & 4x \color{red}{-16} \color{blue}{+16} \color{blue}{+21x} & = & \color{red}{-21x} +84 \color{blue}{+21x} \color{blue}{+16} \\\Leftrightarrow & 25x& = & 100 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{100}{25} \\\Leftrightarrow & x = 4 & & \\ & V = \left\{ 4 \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{14}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-75& = & -168x-1680 \\\Leftrightarrow & 35x \color{red}{-75} \color{blue}{+75} \color{blue}{+168x} & = & \color{red}{-168x} -1680 \color{blue}{+168x} \color{blue}{+75} \\\Leftrightarrow & 203x& = & -1605 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1605}{203} \\\Leftrightarrow & x = \frac{-1605}{203} & & \\ & V = \left\{ \frac{-1605}{203} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{3}{2}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-18& = & 63x+42 \\\Leftrightarrow & 14x \color{red}{-18} \color{blue}{+18} \color{blue}{-63x} & = & \color{red}{63x} +42 \color{blue}{-63x} \color{blue}{+18} \\\Leftrightarrow & -49x& = & 60 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{60}{-49} \\\Leftrightarrow & x = \frac{-60}{49} & & \\ & V = \left\{ \frac{-60}{49} \right\} & \\\end{align}\)