Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+30& = & 65x-650 \\\Leftrightarrow & 26x \color{red}{+30} \color{blue}{-30} \color{blue}{-65x} & = & \color{red}{65x} -650 \color{blue}{-65x} \color{blue}{-30} \\\Leftrightarrow & -39x& = & -680 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-680}{-39} \\\Leftrightarrow & x = \frac{680}{39} & & \\ & V = \left\{ \frac{680}{39} \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 2, 12 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{12}& = & \frac{7}{5}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{30x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{84}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 30x+25& = & 84x+240 \\\Leftrightarrow & 30x \color{red}{+25} \color{blue}{-25} \color{blue}{-84x} & = & \color{red}{84x} +240 \color{blue}{-84x} \color{blue}{-25} \\\Leftrightarrow & -54x& = & 215 \\\Leftrightarrow & \frac{-54x}{ \color{red}{-54} }& = & \frac{215}{-54} \\\Leftrightarrow & x = \frac{-215}{54} & & \\ & V = \left\{ \frac{-215}{54} \right\} & \\\end{align}\)
3. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 28 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x+\frac{924}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+28& = & 77x+924 \\\Leftrightarrow & 22x \color{red}{+28} \color{blue}{-28} \color{blue}{-77x} & = & \color{red}{77x} +924 \color{blue}{-77x} \color{blue}{-28} \\\Leftrightarrow & -55x& = & 896 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{896}{-55} \\\Leftrightarrow & x = \frac{-896}{55} & & \\ & V = \left\{ \frac{-896}{55} \right\} & \\\end{align}\)
4. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{3}{2}x-8 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{27}{ \color{blue}{18} }x-\frac{144}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-8& = & 27x-144 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-27x} & = & \color{red}{27x} -144 \color{blue}{-27x} \color{blue}{+8} \\\Leftrightarrow & -21x& = & -136 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-136}{-21} \\\Leftrightarrow & x = \frac{136}{21} & & \\ & V = \left\{ \frac{136}{21} \right\} & \\\end{align}\)
5. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{12}& = & \frac{-8}{3}x+3 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{-32}{ \color{blue}{12} }x+\frac{36}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x-5& = & -32x+36 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{+32x} & = & \color{red}{-32x} +36 \color{blue}{+32x} \color{blue}{+5} \\\Leftrightarrow & 35x& = & 41 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{41}{35} \\\Leftrightarrow & x = \frac{41}{35} & & \\ & V = \left\{ \frac{41}{35} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{1}{6}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+90& = & 35x+1470 \\\Leftrightarrow & 42x \color{red}{+90} \color{blue}{-90} \color{blue}{-35x} & = & \color{red}{35x} +1470 \color{blue}{-35x} \color{blue}{-90} \\\Leftrightarrow & 7x& = & 1380 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{1380}{7} \\\Leftrightarrow & x = \frac{1380}{7} & & \\ & V = \left\{ \frac{1380}{7} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 252x-210 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-252x} & = & \color{red}{252x} -210 \color{blue}{-252x} \color{blue}{+120} \\\Leftrightarrow & -217x& = & -90 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-90}{-217} \\\Leftrightarrow & x = \frac{90}{217} & & \\ & V = \left\{ \frac{90}{217} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{5}{6}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{35}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-12& = & 35x+126 \\\Leftrightarrow & 6x \color{red}{-12} \color{blue}{+12} \color{blue}{-35x} & = & \color{red}{35x} +126 \color{blue}{-35x} \color{blue}{+12} \\\Leftrightarrow & -29x& = & 138 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{138}{-29} \\\Leftrightarrow & x = \frac{-138}{29} & & \\ & V = \left\{ \frac{-138}{29} \right\} & \\\end{align}\)
9. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{9}& = & \frac{7}{2}x+8 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 70 }{ \color{blue}{126} })& = & (\frac{441}{ \color{blue}{126} }x+\frac{1008}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+70& = & 441x+1008 \\\Leftrightarrow & 18x \color{red}{+70} \color{blue}{-70} \color{blue}{-441x} & = & \color{red}{441x} +1008 \color{blue}{-441x} \color{blue}{-70} \\\Leftrightarrow & -423x& = & 938 \\\Leftrightarrow & \frac{-423x}{ \color{red}{-423} }& = & \frac{938}{-423} \\\Leftrightarrow & x = \frac{-938}{423} & & \\ & V = \left\{ \frac{-938}{423} \right\} & \\\end{align}\)
10. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{5}{3}x-1 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }- \frac{ 75 }{ \color{blue}{165} })& = & (\frac{275}{ \color{blue}{165} }x-\frac{165}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x-75& = & 275x-165 \\\Leftrightarrow & 33x \color{red}{-75} \color{blue}{+75} \color{blue}{-275x} & = & \color{red}{275x} -165 \color{blue}{-275x} \color{blue}{+75} \\\Leftrightarrow & -242x& = & -90 \\\Leftrightarrow & \frac{-242x}{ \color{red}{-242} }& = & \frac{-90}{-242} \\\Leftrightarrow & x = \frac{45}{121} & & \\ & V = \left\{ \frac{45}{121} \right\} & \\\end{align}\)
11. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{16}& = & \frac{5}{2}x+5 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{120}{ \color{blue}{48} }x+\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+15& = & 120x+240 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-120x} & = & \color{red}{120x} +240 \color{blue}{-120x} \color{blue}{-15} \\\Leftrightarrow & -104x& = & 225 \\\Leftrightarrow & \frac{-104x}{ \color{red}{-104} }& = & \frac{225}{-104} \\\Leftrightarrow & x = \frac{-225}{104} & & \\ & V = \left\{ \frac{-225}{104} \right\} & \\\end{align}\)
12. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 8 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{14}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-8& = & 7x-14 \\\Leftrightarrow & 2x \color{red}{-8} \color{blue}{+8} \color{blue}{-7x} & = & \color{red}{7x} -14 \color{blue}{-7x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & -6 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-6}{-5} \\\Leftrightarrow & x = \frac{6}{5} & & \\ & V = \left\{ \frac{6}{5} \right\} & \\\end{align}\)