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

---

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{9}& = & \frac{6}{5}x+8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x+\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-50& = & 108x+720 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{-108x} & = & \color{red}{108x} +720 \color{blue}{-108x} \color{blue}{+50} \\\Leftrightarrow & -93x& = & 770 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{770}{-93} \\\Leftrightarrow & x = \frac{-770}{93} & & \\ & V = \left\{ \frac{-770}{93} \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 5, 15 en 4} \\ \begin{align} & \frac{x}{5}+\frac{4}{15}& = & \frac{7}{4}x+5 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{105}{ \color{blue}{60} }x+\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x+16& = & 105x+300 \\\Leftrightarrow & 12x \color{red}{+16} \color{blue}{-16} \color{blue}{-105x} & = & \color{red}{105x} +300 \color{blue}{-105x} \color{blue}{-16} \\\Leftrightarrow & -93x& = & 284 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{284}{-93} \\\Leftrightarrow & x = \frac{-284}{93} & & \\ & V = \left\{ \frac{-284}{93} \right\} & \\\end{align}\)
3. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x+\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-15& = & 24x+48 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-24x} & = & \color{red}{24x} +48 \color{blue}{-24x} \color{blue}{+15} \\\Leftrightarrow & -8x& = & 63 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{63}{-8} \\\Leftrightarrow & x = \frac{-63}{8} & & \\ & V = \left\{ \frac{-63}{8} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & -12x-180 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{+12x} & = & \color{red}{-12x} -180 \color{blue}{+12x} \color{blue}{+9} \\\Leftrightarrow & 22x& = & -171 \\\Leftrightarrow & \frac{22x}{ \color{red}{22} }& = & \frac{-171}{22} \\\Leftrightarrow & x = \frac{-171}{22} & & \\ & V = \left\{ \frac{-171}{22} \right\} & \\\end{align}\)
5. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{1}{5}x-6 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 10 }{ \color{blue}{35} })& = & (\frac{7}{ \color{blue}{35} }x-\frac{210}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-10& = & 7x-210 \\\Leftrightarrow & 5x \color{red}{-10} \color{blue}{+10} \color{blue}{-7x} & = & \color{red}{7x} -210 \color{blue}{-7x} \color{blue}{+10} \\\Leftrightarrow & -2x& = & -200 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-200}{-2} \\\Leftrightarrow & x = 100 & & \\ & V = \left\{ 100 \right\} & \\\end{align}\)
6. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 70 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x+\frac{1078}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+70& = & 77x+1078 \\\Leftrightarrow & 22x \color{red}{+70} \color{blue}{-70} \color{blue}{-77x} & = & \color{red}{77x} +1078 \color{blue}{-77x} \color{blue}{-70} \\\Leftrightarrow & -55x& = & 1008 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{1008}{-55} \\\Leftrightarrow & x = \frac{-1008}{55} & & \\ & V = \left\{ \frac{-1008}{55} \right\} & \\\end{align}\)
7. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }+ \frac{ 30 }{ \color{blue}{105} })& = & (\frac{35}{ \color{blue}{105} }x-\frac{420}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x+30& = & 35x-420 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{-35x} & = & \color{red}{35x} -420 \color{blue}{-35x} \color{blue}{-30} \\\Leftrightarrow & -14x& = & -450 \\\Leftrightarrow & \frac{-14x}{ \color{red}{-14} }& = & \frac{-450}{-14} \\\Leftrightarrow & x = \frac{225}{7} & & \\ & V = \left\{ \frac{225}{7} \right\} & \\\end{align}\)
8. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{13}& = & \frac{3}{5}x-6 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 60 }{ \color{blue}{260} })& = & (\frac{156}{ \color{blue}{260} }x-\frac{1560}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+60& = & 156x-1560 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-156x} & = & \color{red}{156x} -1560 \color{blue}{-156x} \color{blue}{-60} \\\Leftrightarrow & -91x& = & -1620 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-1620}{-91} \\\Leftrightarrow & x = \frac{1620}{91} & & \\ & V = \left\{ \frac{1620}{91} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{-5}{6}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+90& = & -175x+1470 \\\Leftrightarrow & 42x \color{red}{+90} \color{blue}{-90} \color{blue}{+175x} & = & \color{red}{-175x} +1470 \color{blue}{+175x} \color{blue}{-90} \\\Leftrightarrow & 217x& = & 1380 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{1380}{217} \\\Leftrightarrow & x = \frac{1380}{217} & & \\ & V = \left\{ \frac{1380}{217} \right\} & \\\end{align}\)
10. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{5}{2}x-6 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 28 }{ \color{blue}{154} })& = & (\frac{385}{ \color{blue}{154} }x-\frac{924}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+28& = & 385x-924 \\\Leftrightarrow & 22x \color{red}{+28} \color{blue}{-28} \color{blue}{-385x} & = & \color{red}{385x} -924 \color{blue}{-385x} \color{blue}{-28} \\\Leftrightarrow & -363x& = & -952 \\\Leftrightarrow & \frac{-363x}{ \color{red}{-363} }& = & \frac{-952}{-363} \\\Leftrightarrow & x = \frac{952}{363} & & \\ & V = \left\{ \frac{952}{363} \right\} & \\\end{align}\)
11. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{4}{5}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{56}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & 56x-350 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{-56x} & = & \color{red}{56x} -350 \color{blue}{-56x} \color{blue}{+20} \\\Leftrightarrow & -21x& = & -330 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-330}{-21} \\\Leftrightarrow & x = \frac{110}{7} & & \\ & V = \left\{ \frac{110}{7} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{15}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+8& = & -24x-90 \\\Leftrightarrow & 5x \color{red}{+8} \color{blue}{-8} \color{blue}{+24x} & = & \color{red}{-24x} -90 \color{blue}{+24x} \color{blue}{-8} \\\Leftrightarrow & 29x& = & -98 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-98}{29} \\\Leftrightarrow & x = \frac{-98}{29} & & \\ & V = \left\{ \frac{-98}{29} \right\} & \\\end{align}\)