Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{4}{5}x+4 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{208}{ \color{blue}{260} }x+\frac{1040}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-80& = & 208x+1040 \\\Leftrightarrow & 65x \color{red}{-80} \color{blue}{+80} \color{blue}{-208x} & = & \color{red}{208x} +1040 \color{blue}{-208x} \color{blue}{+80} \\\Leftrightarrow & -143x& = & 1120 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{1120}{-143} \\\Leftrightarrow & x = \frac{-1120}{143} & & \\ & V = \left\{ \frac{-1120}{143} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & -546x+3120 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{+546x} & = & \color{red}{-546x} +3120 \color{blue}{+546x} \color{blue}{-90} \\\Leftrightarrow & 611x& = & 3030 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{3030}{611} \\\Leftrightarrow & x = \frac{3030}{611} & & \\ & V = \left\{ \frac{3030}{611} \right\} & \\\end{align}\)
3. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{11}& = & \frac{1}{2}x+5 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 20 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{550}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-20& = & 55x+550 \\\Leftrightarrow & 22x \color{red}{-20} \color{blue}{+20} \color{blue}{-55x} & = & \color{red}{55x} +550 \color{blue}{-55x} \color{blue}{+20} \\\Leftrightarrow & -33x& = & 570 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{570}{-33} \\\Leftrightarrow & x = \frac{-190}{11} & & \\ & V = \left\{ \frac{-190}{11} \right\} & \\\end{align}\)
4. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{-7}{4}x+2 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-231}{ \color{blue}{132} }x+\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x+48& = & -231x+264 \\\Leftrightarrow & 44x \color{red}{+48} \color{blue}{-48} \color{blue}{+231x} & = & \color{red}{-231x} +264 \color{blue}{+231x} \color{blue}{-48} \\\Leftrightarrow & 275x& = & 216 \\\Leftrightarrow & \frac{275x}{ \color{red}{275} }& = & \frac{216}{275} \\\Leftrightarrow & x = \frac{216}{275} & & \\ & V = \left\{ \frac{216}{275} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{12}& = & \frac{-7}{4}x+2 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-105}{ \color{blue}{60} }x+\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+25& = & -105x+120 \\\Leftrightarrow & 12x \color{red}{+25} \color{blue}{-25} \color{blue}{+105x} & = & \color{red}{-105x} +120 \color{blue}{+105x} \color{blue}{-25} \\\Leftrightarrow & 117x& = & 95 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{95}{117} \\\Leftrightarrow & x = \frac{95}{117} & & \\ & V = \left\{ \frac{95}{117} \right\} & \\\end{align}\)
6. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{11}& = & \frac{5}{2}x-5 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{165}{ \color{blue}{66} }x-\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+30& = & 165x-330 \\\Leftrightarrow & 22x \color{red}{+30} \color{blue}{-30} \color{blue}{-165x} & = & \color{red}{165x} -330 \color{blue}{-165x} \color{blue}{-30} \\\Leftrightarrow & -143x& = & -360 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-360}{-143} \\\Leftrightarrow & x = \frac{360}{143} & & \\ & V = \left\{ \frac{360}{143} \right\} & \\\end{align}\)
7. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{4}{3}x+3 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 24 }{ \color{blue}{156} })& = & (\frac{208}{ \color{blue}{156} }x+\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+24& = & 208x+468 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{-208x} & = & \color{red}{208x} +468 \color{blue}{-208x} \color{blue}{-24} \\\Leftrightarrow & -169x& = & 444 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{444}{-169} \\\Leftrightarrow & x = \frac{-444}{169} & & \\ & V = \left\{ \frac{-444}{169} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{15}& = & \frac{1}{4}x+8 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{15}{ \color{blue}{60} }x+\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-16& = & 15x+480 \\\Leftrightarrow & 20x \color{red}{-16} \color{blue}{+16} \color{blue}{-15x} & = & \color{red}{15x} +480 \color{blue}{-15x} \color{blue}{+16} \\\Leftrightarrow & 5x& = & 496 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{496}{5} \\\Leftrightarrow & x = \frac{496}{5} & & \\ & V = \left\{ \frac{496}{5} \right\} & \\\end{align}\)
9. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{3}{4}x-5 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{84}{ \color{blue}{112} }x-\frac{560}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & 84x-560 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{-84x} & = & \color{red}{84x} -560 \color{blue}{-84x} \color{blue}{-35} \\\Leftrightarrow & -68x& = & -595 \\\Leftrightarrow & \frac{-68x}{ \color{red}{-68} }& = & \frac{-595}{-68} \\\Leftrightarrow & x = \frac{35}{4} & & \\ & V = \left\{ \frac{35}{4} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{-7}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & -42x-180 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{+42x} & = & \color{red}{-42x} -180 \color{blue}{+42x} \color{blue}{+9} \\\Leftrightarrow & 47x& = & -171 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{-171}{47} \\\Leftrightarrow & x = \frac{-171}{47} & & \\ & V = \left\{ \frac{-171}{47} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{2}{3}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{40}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-18& = & 40x+240 \\\Leftrightarrow & 15x \color{red}{-18} \color{blue}{+18} \color{blue}{-40x} & = & \color{red}{40x} +240 \color{blue}{-40x} \color{blue}{+18} \\\Leftrightarrow & -25x& = & 258 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{258}{-25} \\\Leftrightarrow & x = \frac{-258}{25} & & \\ & V = \left\{ \frac{-258}{25} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{-4}{5}x+7 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-64}{ \color{blue}{80} }x+\frac{560}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+25& = & -64x+560 \\\Leftrightarrow & 20x \color{red}{+25} \color{blue}{-25} \color{blue}{+64x} & = & \color{red}{-64x} +560 \color{blue}{+64x} \color{blue}{-25} \\\Leftrightarrow & 84x& = & 535 \\\Leftrightarrow & \frac{84x}{ \color{red}{84} }& = & \frac{535}{84} \\\Leftrightarrow & x = \frac{535}{84} & & \\ & V = \left\{ \frac{535}{84} \right\} & \\\end{align}\)