Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{4}{13}=\frac{-4}{5}x+8\)
2. \(\frac{x}{7}-\frac{5}{9}=\frac{-5}{3}x-2\)
3. \(\frac{x}{2}+\frac{2}{7}=\frac{-2}{3}x-8\)
4. \(\frac{x}{3}-\frac{5}{16}=\frac{-4}{5}x+1\)
5. \(\frac{x}{2}-\frac{5}{13}=\frac{-5}{3}x-2\)
6. \(\frac{x}{5}+\frac{3}{7}=\frac{4}{3}x-4\)
7. \(\frac{x}{5}-\frac{2}{15}=\frac{5}{2}x-8\)
8. \(\frac{x}{7}+\frac{2}{13}=\frac{4}{3}x-7\)
9. \(\frac{x}{7}+\frac{4}{11}=\frac{7}{5}x+8\)
10. \(\frac{x}{2}+\frac{4}{7}=\frac{2}{5}x+3\)
11. \(\frac{x}{2}+\frac{2}{11}=\frac{7}{3}x+5\)
12. \(\frac{x}{4}+\frac{3}{16}=\frac{-2}{3}x+7\)

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

Verbetersleutel

1. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }+ \frac{ 140 }{ \color{blue}{455} })& = & (\frac{-364}{ \color{blue}{455} }x+\frac{3640}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x+140& = & -364x+3640 \\\Leftrightarrow & 65x \color{red}{+140} \color{blue}{-140} \color{blue}{+364x} & = & \color{red}{-364x} +3640 \color{blue}{+364x} \color{blue}{-140} \\\Leftrightarrow & 429x& = & 3500 \\\Leftrightarrow & \frac{429x}{ \color{red}{429} }& = & \frac{3500}{429} \\\Leftrightarrow & x = \frac{3500}{429} & & \\ & V = \left\{ \frac{3500}{429} \right\} & \\\end{align}\)
2. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}-\frac{5}{9}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }- \frac{ 35 }{ \color{blue}{63} })& = & (\frac{-105}{ \color{blue}{63} }x-\frac{126}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x-35& = & -105x-126 \\\Leftrightarrow & 9x \color{red}{-35} \color{blue}{+35} \color{blue}{+105x} & = & \color{red}{-105x} -126 \color{blue}{+105x} \color{blue}{+35} \\\Leftrightarrow & 114x& = & -91 \\\Leftrightarrow & \frac{114x}{ \color{red}{114} }& = & \frac{-91}{114} \\\Leftrightarrow & x = \frac{-91}{114} & & \\ & V = \left\{ \frac{-91}{114} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-2}{3}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & -28x-336 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{+28x} & = & \color{red}{-28x} -336 \color{blue}{+28x} \color{blue}{-12} \\\Leftrightarrow & 49x& = & -348 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-348}{49} \\\Leftrightarrow & x = \frac{-348}{49} & & \\ & V = \left\{ \frac{-348}{49} \right\} & \\\end{align}\)
4. \(\text{240 is het kleinste gemene veelvoud van 3, 16 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{240.} (\frac{80x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-192}{ \color{blue}{240} }x+\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 80x-75& = & -192x+240 \\\Leftrightarrow & 80x \color{red}{-75} \color{blue}{+75} \color{blue}{+192x} & = & \color{red}{-192x} +240 \color{blue}{+192x} \color{blue}{+75} \\\Leftrightarrow & 272x& = & 315 \\\Leftrightarrow & \frac{272x}{ \color{red}{272} }& = & \frac{315}{272} \\\Leftrightarrow & x = \frac{315}{272} & & \\ & V = \left\{ \frac{315}{272} \right\} & \\\end{align}\)
5. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{13}& = & \frac{-5}{3}x-2 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 30 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x-\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-30& = & -130x-156 \\\Leftrightarrow & 39x \color{red}{-30} \color{blue}{+30} \color{blue}{+130x} & = & \color{red}{-130x} -156 \color{blue}{+130x} \color{blue}{+30} \\\Leftrightarrow & 169x& = & -126 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-126}{169} \\\Leftrightarrow & x = \frac{-126}{169} & & \\ & V = \left\{ \frac{-126}{169} \right\} & \\\end{align}\)
6. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{4}{3}x-4 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 45 }{ \color{blue}{105} })& = & (\frac{140}{ \color{blue}{105} }x-\frac{420}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+45& = & 140x-420 \\\Leftrightarrow & 21x \color{red}{+45} \color{blue}{-45} \color{blue}{-140x} & = & \color{red}{140x} -420 \color{blue}{-140x} \color{blue}{-45} \\\Leftrightarrow & -119x& = & -465 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{-465}{-119} \\\Leftrightarrow & x = \frac{465}{119} & & \\ & V = \left\{ \frac{465}{119} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{5}{2}x-8 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{75}{ \color{blue}{30} }x-\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-4& = & 75x-240 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-75x} & = & \color{red}{75x} -240 \color{blue}{-75x} \color{blue}{+4} \\\Leftrightarrow & -69x& = & -236 \\\Leftrightarrow & \frac{-69x}{ \color{red}{-69} }& = & \frac{-236}{-69} \\\Leftrightarrow & x = \frac{236}{69} & & \\ & V = \left\{ \frac{236}{69} \right\} & \\\end{align}\)
8. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{4}{3}x-7 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }+ \frac{ 42 }{ \color{blue}{273} })& = & (\frac{364}{ \color{blue}{273} }x-\frac{1911}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x+42& = & 364x-1911 \\\Leftrightarrow & 39x \color{red}{+42} \color{blue}{-42} \color{blue}{-364x} & = & \color{red}{364x} -1911 \color{blue}{-364x} \color{blue}{-42} \\\Leftrightarrow & -325x& = & -1953 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{-1953}{-325} \\\Leftrightarrow & x = \frac{1953}{325} & & \\ & V = \left\{ \frac{1953}{325} \right\} & \\\end{align}\)
9. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{7}{5}x+8 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }+ \frac{ 140 }{ \color{blue}{385} })& = & (\frac{539}{ \color{blue}{385} }x+\frac{3080}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x+140& = & 539x+3080 \\\Leftrightarrow & 55x \color{red}{+140} \color{blue}{-140} \color{blue}{-539x} & = & \color{red}{539x} +3080 \color{blue}{-539x} \color{blue}{-140} \\\Leftrightarrow & -484x& = & 2940 \\\Leftrightarrow & \frac{-484x}{ \color{red}{-484} }& = & \frac{2940}{-484} \\\Leftrightarrow & x = \frac{-735}{121} & & \\ & V = \left\{ \frac{-735}{121} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{2}{5}x+3 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{28}{ \color{blue}{70} }x+\frac{210}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+40& = & 28x+210 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{-28x} & = & \color{red}{28x} +210 \color{blue}{-28x} \color{blue}{-40} \\\Leftrightarrow & 7x& = & 170 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{170}{7} \\\Leftrightarrow & x = \frac{170}{7} & & \\ & V = \left\{ \frac{170}{7} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x+\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+12& = & 154x+330 \\\Leftrightarrow & 33x \color{red}{+12} \color{blue}{-12} \color{blue}{-154x} & = & \color{red}{154x} +330 \color{blue}{-154x} \color{blue}{-12} \\\Leftrightarrow & -121x& = & 318 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{318}{-121} \\\Leftrightarrow & x = \frac{-318}{121} & & \\ & V = \left\{ \frac{-318}{121} \right\} & \\\end{align}\)
12. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-32}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+9& = & -32x+336 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{+32x} & = & \color{red}{-32x} +336 \color{blue}{+32x} \color{blue}{-9} \\\Leftrightarrow & 44x& = & 327 \\\Leftrightarrow & \frac{44x}{ \color{red}{44} }& = & \frac{327}{44} \\\Leftrightarrow & x = \frac{327}{44} & & \\ & V = \left\{ \frac{327}{44} \right\} & \\\end{align}\)