Vgln. eerste graad (reeks 4)

Hoofdmenu Eentje per keer  🖨

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

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

---

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

Verbetersleutel

1. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 60 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x+\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+60& = & 44x+660 \\\Leftrightarrow & 33x \color{red}{+60} \color{blue}{-60} \color{blue}{-44x} & = & \color{red}{44x} +660 \color{blue}{-44x} \color{blue}{-60} \\\Leftrightarrow & -11x& = & 600 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{600}{-11} \\\Leftrightarrow & x = \frac{-600}{11} & & \\ & V = \left\{ \frac{-600}{11} \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{72}{ \color{blue}{60} }x+\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 72x+480 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-72x} & = & \color{red}{72x} +480 \color{blue}{-72x} \color{blue}{+16} \\\Leftrightarrow & -57x& = & 496 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{496}{-57} \\\Leftrightarrow & x = \frac{-496}{57} & & \\ & V = \left\{ \frac{-496}{57} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{3}{4}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{63}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-48& = & 63x-588 \\\Leftrightarrow & 28x \color{red}{-48} \color{blue}{+48} \color{blue}{-63x} & = & \color{red}{63x} -588 \color{blue}{-63x} \color{blue}{+48} \\\Leftrightarrow & -35x& = & -540 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-540}{-35} \\\Leftrightarrow & x = \frac{108}{7} & & \\ & V = \left\{ \frac{108}{7} \right\} & \\\end{align}\)
4. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{9}& = & \frac{1}{6}x+5 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{15}{ \color{blue}{90} }x+\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-50& = & 15x+450 \\\Leftrightarrow & 18x \color{red}{-50} \color{blue}{+50} \color{blue}{-15x} & = & \color{red}{15x} +450 \color{blue}{-15x} \color{blue}{+50} \\\Leftrightarrow & 3x& = & 500 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{500}{3} \\\Leftrightarrow & x = \frac{500}{3} & & \\ & V = \left\{ \frac{500}{3} \right\} & \\\end{align}\)
5. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{13}& = & \frac{3}{5}x-1 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{234}{ \color{blue}{390} }x-\frac{390}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-90& = & 234x-390 \\\Leftrightarrow & 65x \color{red}{-90} \color{blue}{+90} \color{blue}{-234x} & = & \color{red}{234x} -390 \color{blue}{-234x} \color{blue}{+90} \\\Leftrightarrow & -169x& = & -300 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-300}{-169} \\\Leftrightarrow & x = \frac{300}{169} & & \\ & V = \left\{ \frac{300}{169} \right\} & \\\end{align}\)
6. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{16}& = & \frac{7}{2}x+6 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }- \frac{ 35 }{ \color{blue}{112} })& = & (\frac{392}{ \color{blue}{112} }x+\frac{672}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x-35& = & 392x+672 \\\Leftrightarrow & 16x \color{red}{-35} \color{blue}{+35} \color{blue}{-392x} & = & \color{red}{392x} +672 \color{blue}{-392x} \color{blue}{+35} \\\Leftrightarrow & -376x& = & 707 \\\Leftrightarrow & \frac{-376x}{ \color{red}{-376} }& = & \frac{707}{-376} \\\Leftrightarrow & x = \frac{-707}{376} & & \\ & V = \left\{ \frac{-707}{376} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-20& = & 18x-630 \\\Leftrightarrow & 45x \color{red}{-20} \color{blue}{+20} \color{blue}{-18x} & = & \color{red}{18x} -630 \color{blue}{-18x} \color{blue}{+20} \\\Leftrightarrow & 27x& = & -610 \\\Leftrightarrow & \frac{27x}{ \color{red}{27} }& = & \frac{-610}{27} \\\Leftrightarrow & x = \frac{-610}{27} & & \\ & V = \left\{ \frac{-610}{27} \right\} & \\\end{align}\)
8. \(\text{10 is het kleinste gemene veelvoud van 5, 10 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{10.} (\frac{2x}{ \color{blue}{10} }- \frac{ 3 }{ \color{blue}{10} })& = & (\frac{5}{ \color{blue}{10} }x+\frac{80}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 2x-3& = & 5x+80 \\\Leftrightarrow & 2x \color{red}{-3} \color{blue}{+3} \color{blue}{-5x} & = & \color{red}{5x} +80 \color{blue}{-5x} \color{blue}{+3} \\\Leftrightarrow & -3x& = & 83 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{83}{-3} \\\Leftrightarrow & x = \frac{-83}{3} & & \\ & V = \left\{ \frac{-83}{3} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 7, 14 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{14}& = & \frac{-5}{6}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 15 }{ \color{blue}{42} })& = & (\frac{-35}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-15& = & -35x-336 \\\Leftrightarrow & 6x \color{red}{-15} \color{blue}{+15} \color{blue}{+35x} & = & \color{red}{-35x} -336 \color{blue}{+35x} \color{blue}{+15} \\\Leftrightarrow & 41x& = & -321 \\\Leftrightarrow & \frac{41x}{ \color{red}{41} }& = & \frac{-321}{41} \\\Leftrightarrow & x = \frac{-321}{41} & & \\ & V = \left\{ \frac{-321}{41} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{3}{5}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & 18x+30 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-18x} & = & \color{red}{18x} +30 \color{blue}{-18x} \color{blue}{-8} \\\Leftrightarrow & -3x& = & 22 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{22}{-3} \\\Leftrightarrow & x = \frac{-22}{3} & & \\ & V = \left\{ \frac{-22}{3} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & 196x+420 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{-196x} & = & \color{red}{196x} +420 \color{blue}{-196x} \color{blue}{-36} \\\Leftrightarrow & -175x& = & 384 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{384}{-175} \\\Leftrightarrow & x = \frac{-384}{175} & & \\ & V = \left\{ \frac{-384}{175} \right\} & \\\end{align}\)
12. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{4}{5}x+8 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }+ \frac{ 2 }{ \color{blue}{15} })& = & (\frac{12}{ \color{blue}{15} }x+\frac{120}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x+2& = & 12x+120 \\\Leftrightarrow & 5x \color{red}{+2} \color{blue}{-2} \color{blue}{-12x} & = & \color{red}{12x} +120 \color{blue}{-12x} \color{blue}{-2} \\\Leftrightarrow & -7x& = & 118 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{118}{-7} \\\Leftrightarrow & x = \frac{-118}{7} & & \\ & V = \left\{ \frac{-118}{7} \right\} & \\\end{align}\)