Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{4}{7}=\frac{-3}{4}x-4\)
2. \(\frac{x}{2}-\frac{4}{7}=\frac{-8}{3}x-2\)
3. \(\frac{x}{2}-\frac{3}{11}=\frac{7}{3}x+5\)
4. \(\frac{x}{6}+\frac{3}{11}=\frac{6}{5}x-1\)
5. \(\frac{x}{3}-\frac{4}{11}=\frac{5}{4}x-5\)
6. \(\frac{x}{6}+\frac{3}{16}=\frac{3}{5}x-5\)
7. \(\frac{x}{6}+\frac{2}{7}=\frac{7}{5}x-5\)
8. \(\frac{x}{6}+\frac{4}{15}=\frac{1}{5}x-3\)
9. \(\frac{x}{3}+\frac{3}{7}=\frac{3}{4}x-7\)
10. \(\frac{x}{2}+\frac{5}{14}=\frac{-7}{5}x+7\)
11. \(\frac{x}{3}-\frac{2}{15}=\frac{7}{4}x-8\)
12. \(\frac{x}{6}+\frac{4}{7}=\frac{-4}{5}x+3\)

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

Verbetersleutel

1. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{-3}{4}x-4 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 16 }{ \color{blue}{28} })& = & (\frac{-21}{ \color{blue}{28} }x-\frac{112}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+16& = & -21x-112 \\\Leftrightarrow & 4x \color{red}{+16} \color{blue}{-16} \color{blue}{+21x} & = & \color{red}{-21x} -112 \color{blue}{+21x} \color{blue}{-16} \\\Leftrightarrow & 25x& = & -128 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{-128}{25} \\\Leftrightarrow & x = \frac{-128}{25} & & \\ & V = \left\{ \frac{-128}{25} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-8}{3}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & -112x-84 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+112x} & = & \color{red}{-112x} -84 \color{blue}{+112x} \color{blue}{+24} \\\Leftrightarrow & 133x& = & -60 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-60}{133} \\\Leftrightarrow & x = \frac{-60}{133} & & \\ & V = \left\{ \frac{-60}{133} \right\} & \\\end{align}\)
3. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{11}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 18 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x+\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-18& = & 154x+330 \\\Leftrightarrow & 33x \color{red}{-18} \color{blue}{+18} \color{blue}{-154x} & = & \color{red}{154x} +330 \color{blue}{-154x} \color{blue}{+18} \\\Leftrightarrow & -121x& = & 348 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{348}{-121} \\\Leftrightarrow & x = \frac{-348}{121} & & \\ & V = \left\{ \frac{-348}{121} \right\} & \\\end{align}\)
4. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{11}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 90 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x-\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+90& = & 396x-330 \\\Leftrightarrow & 55x \color{red}{+90} \color{blue}{-90} \color{blue}{-396x} & = & \color{red}{396x} -330 \color{blue}{-396x} \color{blue}{-90} \\\Leftrightarrow & -341x& = & -420 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{-420}{-341} \\\Leftrightarrow & x = \frac{420}{341} & & \\ & V = \left\{ \frac{420}{341} \right\} & \\\end{align}\)
5. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{5}{4}x-5 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{165}{ \color{blue}{132} }x-\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-48& = & 165x-660 \\\Leftrightarrow & 44x \color{red}{-48} \color{blue}{+48} \color{blue}{-165x} & = & \color{red}{165x} -660 \color{blue}{-165x} \color{blue}{+48} \\\Leftrightarrow & -121x& = & -612 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{-612}{-121} \\\Leftrightarrow & x = \frac{612}{121} & & \\ & V = \left\{ \frac{612}{121} \right\} & \\\end{align}\)
6. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{16}& = & \frac{3}{5}x-5 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{144}{ \color{blue}{240} }x-\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+45& = & 144x-1200 \\\Leftrightarrow & 40x \color{red}{+45} \color{blue}{-45} \color{blue}{-144x} & = & \color{red}{144x} -1200 \color{blue}{-144x} \color{blue}{-45} \\\Leftrightarrow & -104x& = & -1245 \\\Leftrightarrow & \frac{-104x}{ \color{red}{-104} }& = & \frac{-1245}{-104} \\\Leftrightarrow & x = \frac{1245}{104} & & \\ & V = \left\{ \frac{1245}{104} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{7}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 294x-1050 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-294x} & = & \color{red}{294x} -1050 \color{blue}{-294x} \color{blue}{-60} \\\Leftrightarrow & -259x& = & -1110 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{-1110}{-259} \\\Leftrightarrow & x = \frac{30}{7} & & \\ & V = \left\{ \frac{30}{7} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{15}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+8& = & 6x-90 \\\Leftrightarrow & 5x \color{red}{+8} \color{blue}{-8} \color{blue}{-6x} & = & \color{red}{6x} -90 \color{blue}{-6x} \color{blue}{-8} \\\Leftrightarrow & -x& = & -98 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-98}{-1} \\\Leftrightarrow & x = 98 & & \\ & V = \left\{ 98 \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{3}{4}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{63}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+36& = & 63x-588 \\\Leftrightarrow & 28x \color{red}{+36} \color{blue}{-36} \color{blue}{-63x} & = & \color{red}{63x} -588 \color{blue}{-63x} \color{blue}{-36} \\\Leftrightarrow & -35x& = & -624 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-624}{-35} \\\Leftrightarrow & x = \frac{624}{35} & & \\ & V = \left\{ \frac{624}{35} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{14}& = & \frac{-7}{5}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 25 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+25& = & -98x+490 \\\Leftrightarrow & 35x \color{red}{+25} \color{blue}{-25} \color{blue}{+98x} & = & \color{red}{-98x} +490 \color{blue}{+98x} \color{blue}{-25} \\\Leftrightarrow & 133x& = & 465 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{465}{133} \\\Leftrightarrow & x = \frac{465}{133} & & \\ & V = \left\{ \frac{465}{133} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{7}{4}x-8 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{105}{ \color{blue}{60} }x-\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-8& = & 105x-480 \\\Leftrightarrow & 20x \color{red}{-8} \color{blue}{+8} \color{blue}{-105x} & = & \color{red}{105x} -480 \color{blue}{-105x} \color{blue}{+8} \\\Leftrightarrow & -85x& = & -472 \\\Leftrightarrow & \frac{-85x}{ \color{red}{-85} }& = & \frac{-472}{-85} \\\Leftrightarrow & x = \frac{472}{85} & & \\ & V = \left\{ \frac{472}{85} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -168x+630 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+168x} & = & \color{red}{-168x} +630 \color{blue}{+168x} \color{blue}{-120} \\\Leftrightarrow & 203x& = & 510 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{510}{203} \\\Leftrightarrow & x = \frac{510}{203} & & \\ & V = \left\{ \frac{510}{203} \right\} & \\\end{align}\)