Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{6}+\frac{4}{7}=\frac{6}{5}x+3\)
2. \(\frac{x}{2}+\frac{4}{13}=\frac{-4}{5}x-8\)
3. \(\frac{x}{2}-\frac{3}{14}=\frac{3}{5}x+8\)
4. \(\frac{x}{5}-\frac{5}{12}=\frac{1}{2}x+6\)
5. \(\frac{x}{4}+\frac{5}{16}=\frac{3}{5}x-3\)
6. \(\frac{x}{4}+\frac{3}{14}=\frac{-8}{3}x+5\)
7. \(\frac{x}{3}+\frac{3}{11}=\frac{-3}{4}x-1\)
8. \(\frac{x}{4}+\frac{4}{15}=\frac{-4}{5}x-1\)
9. \(\frac{x}{4}+\frac{4}{7}=\frac{-5}{3}x-1\)
10. \(\frac{x}{5}+\frac{3}{16}=\frac{7}{2}x-3\)
11. \(\frac{x}{5}-\frac{2}{11}=\frac{3}{4}x+8\)
12. \(\frac{x}{5}+\frac{3}{11}=\frac{1}{2}x-2\)

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{6}{5}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 252x+630 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-252x} & = & \color{red}{252x} +630 \color{blue}{-252x} \color{blue}{-120} \\\Leftrightarrow & -217x& = & 510 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{510}{-217} \\\Leftrightarrow & x = \frac{-510}{217} & & \\ & V = \left\{ \frac{-510}{217} \right\} & \\\end{align}\)
2. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{-104}{ \color{blue}{130} }x-\frac{1040}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+40& = & -104x-1040 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{+104x} & = & \color{red}{-104x} -1040 \color{blue}{+104x} \color{blue}{-40} \\\Leftrightarrow & 169x& = & -1080 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-1080}{169} \\\Leftrightarrow & x = \frac{-1080}{169} & & \\ & V = \left\{ \frac{-1080}{169} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{3}{5}x+8 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{42}{ \color{blue}{70} }x+\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-15& = & 42x+560 \\\Leftrightarrow & 35x \color{red}{-15} \color{blue}{+15} \color{blue}{-42x} & = & \color{red}{42x} +560 \color{blue}{-42x} \color{blue}{+15} \\\Leftrightarrow & -7x& = & 575 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{575}{-7} \\\Leftrightarrow & x = \frac{-575}{7} & & \\ & V = \left\{ \frac{-575}{7} \right\} & \\\end{align}\)
4. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{12}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{30}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-25& = & 30x+360 \\\Leftrightarrow & 12x \color{red}{-25} \color{blue}{+25} \color{blue}{-30x} & = & \color{red}{30x} +360 \color{blue}{-30x} \color{blue}{+25} \\\Leftrightarrow & -18x& = & 385 \\\Leftrightarrow & \frac{-18x}{ \color{red}{-18} }& = & \frac{385}{-18} \\\Leftrightarrow & x = \frac{-385}{18} & & \\ & V = \left\{ \frac{-385}{18} \right\} & \\\end{align}\)
5. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{3}{5}x-3 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{48}{ \color{blue}{80} }x-\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+25& = & 48x-240 \\\Leftrightarrow & 20x \color{red}{+25} \color{blue}{-25} \color{blue}{-48x} & = & \color{red}{48x} -240 \color{blue}{-48x} \color{blue}{-25} \\\Leftrightarrow & -28x& = & -265 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{-265}{-28} \\\Leftrightarrow & x = \frac{265}{28} & & \\ & V = \left\{ \frac{265}{28} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{14}& = & \frac{-8}{3}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 18 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+18& = & -224x+420 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{+224x} & = & \color{red}{-224x} +420 \color{blue}{+224x} \color{blue}{-18} \\\Leftrightarrow & 245x& = & 402 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{402}{245} \\\Leftrightarrow & x = \frac{402}{245} & & \\ & V = \left\{ \frac{402}{245} \right\} & \\\end{align}\)
7. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{11}& = & \frac{-3}{4}x-1 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }+ \frac{ 36 }{ \color{blue}{132} })& = & (\frac{-99}{ \color{blue}{132} }x-\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x+36& = & -99x-132 \\\Leftrightarrow & 44x \color{red}{+36} \color{blue}{-36} \color{blue}{+99x} & = & \color{red}{-99x} -132 \color{blue}{+99x} \color{blue}{-36} \\\Leftrightarrow & 143x& = & -168 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-168}{143} \\\Leftrightarrow & x = \frac{-168}{143} & & \\ & V = \left\{ \frac{-168}{143} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-48}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & -48x-60 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{+48x} & = & \color{red}{-48x} -60 \color{blue}{+48x} \color{blue}{-16} \\\Leftrightarrow & 63x& = & -76 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{-76}{63} \\\Leftrightarrow & x = \frac{-76}{63} & & \\ & V = \left\{ \frac{-76}{63} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -140x-84 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+140x} & = & \color{red}{-140x} -84 \color{blue}{+140x} \color{blue}{-48} \\\Leftrightarrow & 161x& = & -132 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-132}{161} \\\Leftrightarrow & x = \frac{-132}{161} & & \\ & V = \left\{ \frac{-132}{161} \right\} & \\\end{align}\)
10. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{7}{2}x-3 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{280}{ \color{blue}{80} }x-\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+15& = & 280x-240 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-280x} & = & \color{red}{280x} -240 \color{blue}{-280x} \color{blue}{-15} \\\Leftrightarrow & -264x& = & -255 \\\Leftrightarrow & \frac{-264x}{ \color{red}{-264} }& = & \frac{-255}{-264} \\\Leftrightarrow & x = \frac{85}{88} & & \\ & V = \left\{ \frac{85}{88} \right\} & \\\end{align}\)
11. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{11}& = & \frac{3}{4}x+8 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }- \frac{ 40 }{ \color{blue}{220} })& = & (\frac{165}{ \color{blue}{220} }x+\frac{1760}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x-40& = & 165x+1760 \\\Leftrightarrow & 44x \color{red}{-40} \color{blue}{+40} \color{blue}{-165x} & = & \color{red}{165x} +1760 \color{blue}{-165x} \color{blue}{+40} \\\Leftrightarrow & -121x& = & 1800 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{1800}{-121} \\\Leftrightarrow & x = \frac{-1800}{121} & & \\ & V = \left\{ \frac{-1800}{121} \right\} & \\\end{align}\)
12. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{11}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 30 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{220}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+30& = & 55x-220 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-55x} & = & \color{red}{55x} -220 \color{blue}{-55x} \color{blue}{-30} \\\Leftrightarrow & -33x& = & -250 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-250}{-33} \\\Leftrightarrow & x = \frac{250}{33} & & \\ & V = \left\{ \frac{250}{33} \right\} & \\\end{align}\)