Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x+\frac{234}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-12& = & -130x+234 \\\Leftrightarrow & 39x \color{red}{-12} \color{blue}{+12} \color{blue}{+130x} & = & \color{red}{-130x} +234 \color{blue}{+130x} \color{blue}{+12} \\\Leftrightarrow & 169x& = & 246 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{246}{169} \\\Leftrightarrow & x = \frac{246}{169} & & \\ & V = \left\{ \frac{246}{169} \right\} & \\\end{align}\)
2. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{16}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-96}{ \color{blue}{240} }x-\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-75& = & -96x-1920 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{+96x} & = & \color{red}{-96x} -1920 \color{blue}{+96x} \color{blue}{+75} \\\Leftrightarrow & 136x& = & -1845 \\\Leftrightarrow & \frac{136x}{ \color{red}{136} }& = & \frac{-1845}{136} \\\Leftrightarrow & x = \frac{-1845}{136} & & \\ & V = \left\{ \frac{-1845}{136} \right\} & \\\end{align}\)
3. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{11}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }- \frac{ 60 }{ \color{blue}{220} })& = & (\frac{44}{ \color{blue}{220} }x+\frac{1100}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x-60& = & 44x+1100 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{-44x} & = & \color{red}{44x} +1100 \color{blue}{-44x} \color{blue}{+60} \\\Leftrightarrow & 11x& = & 1160 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{1160}{11} \\\Leftrightarrow & x = \frac{1160}{11} & & \\ & V = \left\{ \frac{1160}{11} \right\} & \\\end{align}\)
4. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-260}{ \color{blue}{156} }x-\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & -260x-156 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{+260x} & = & \color{red}{-260x} -156 \color{blue}{+260x} \color{blue}{+36} \\\Leftrightarrow & 299x& = & -120 \\\Leftrightarrow & \frac{299x}{ \color{red}{299} }& = & \frac{-120}{299} \\\Leftrightarrow & x = \frac{-120}{299} & & \\ & V = \left\{ \frac{-120}{299} \right\} & \\\end{align}\)
5. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{3}{4}x-8 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{99}{ \color{blue}{132} }x-\frac{1056}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x+48& = & 99x-1056 \\\Leftrightarrow & 44x \color{red}{+48} \color{blue}{-48} \color{blue}{-99x} & = & \color{red}{99x} -1056 \color{blue}{-99x} \color{blue}{-48} \\\Leftrightarrow & -55x& = & -1104 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-1104}{-55} \\\Leftrightarrow & x = \frac{1104}{55} & & \\ & V = \left\{ \frac{1104}{55} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{5}{6}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{25}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 25x-120 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-25x} & = & \color{red}{25x} -120 \color{blue}{-25x} \color{blue}{-25} \\\Leftrightarrow & -19x& = & -145 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{-145}{-19} \\\Leftrightarrow & x = \frac{145}{19} & & \\ & V = \left\{ \frac{145}{19} \right\} & \\\end{align}\)
7. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{16}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{-112}{ \color{blue}{80} }x+\frac{480}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x-15& = & -112x+480 \\\Leftrightarrow & 20x \color{red}{-15} \color{blue}{+15} \color{blue}{+112x} & = & \color{red}{-112x} +480 \color{blue}{+112x} \color{blue}{+15} \\\Leftrightarrow & 132x& = & 495 \\\Leftrightarrow & \frac{132x}{ \color{red}{132} }& = & \frac{495}{132} \\\Leftrightarrow & x = \frac{15}{4} & & \\ & V = \left\{ \frac{15}{4} \right\} & \\\end{align}\)
8. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x+\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-24& = & 39x+78 \\\Leftrightarrow & 26x \color{red}{-24} \color{blue}{+24} \color{blue}{-39x} & = & \color{red}{39x} +78 \color{blue}{-39x} \color{blue}{+24} \\\Leftrightarrow & -13x& = & 102 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{102}{-13} \\\Leftrightarrow & x = \frac{-102}{13} & & \\ & V = \left\{ \frac{-102}{13} \right\} & \\\end{align}\)
9. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{11}& = & \frac{-5}{6}x-5 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }- \frac{ 90 }{ \color{blue}{330} })& = & (\frac{-275}{ \color{blue}{330} }x-\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x-90& = & -275x-1650 \\\Leftrightarrow & 66x \color{red}{-90} \color{blue}{+90} \color{blue}{+275x} & = & \color{red}{-275x} -1650 \color{blue}{+275x} \color{blue}{+90} \\\Leftrightarrow & 341x& = & -1560 \\\Leftrightarrow & \frac{341x}{ \color{red}{341} }& = & \frac{-1560}{341} \\\Leftrightarrow & x = \frac{-1560}{341} & & \\ & V = \left\{ \frac{-1560}{341} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{6}{5}x+4 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x+\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & 108x+360 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{-108x} & = & \color{red}{108x} +360 \color{blue}{-108x} \color{blue}{-20} \\\Leftrightarrow & -93x& = & 340 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{340}{-93} \\\Leftrightarrow & x = \frac{-340}{93} & & \\ & V = \left\{ \frac{-340}{93} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{1}{6}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-60& = & 35x-1260 \\\Leftrightarrow & 42x \color{red}{-60} \color{blue}{+60} \color{blue}{-35x} & = & \color{red}{35x} -1260 \color{blue}{-35x} \color{blue}{+60} \\\Leftrightarrow & 7x& = & -1200 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-1200}{7} \\\Leftrightarrow & x = \frac{-1200}{7} & & \\ & V = \left\{ \frac{-1200}{7} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 5, 14 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{14}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 25 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+25& = & 35x+490 \\\Leftrightarrow & 14x \color{red}{+25} \color{blue}{-25} \color{blue}{-35x} & = & \color{red}{35x} +490 \color{blue}{-35x} \color{blue}{-25} \\\Leftrightarrow & -21x& = & 465 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{465}{-21} \\\Leftrightarrow & x = \frac{-155}{7} & & \\ & V = \left\{ \frac{-155}{7} \right\} & \\\end{align}\)