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

---

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x-252 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} -252 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & -228 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-228}{-7} \\\Leftrightarrow & x = \frac{228}{7} & & \\ & V = \left\{ \frac{228}{7} \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & -100x-420 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{+100x} & = & \color{red}{-100x} -420 \color{blue}{+100x} \color{blue}{-16} \\\Leftrightarrow & 115x& = & -436 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{-436}{115} \\\Leftrightarrow & x = \frac{-436}{115} & & \\ & V = \left\{ \frac{-436}{115} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & -24x-60 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{+24x} & = & \color{red}{-24x} -60 \color{blue}{+24x} \color{blue}{-9} \\\Leftrightarrow & 29x& = & -69 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{-69}{29} \\\Leftrightarrow & x = \frac{-69}{29} & & \\ & V = \left\{ \frac{-69}{29} \right\} & \\\end{align}\)
4. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{13}& = & \frac{7}{2}x-5 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }+ \frac{ 70 }{ \color{blue}{182} })& = & (\frac{637}{ \color{blue}{182} }x-\frac{910}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x+70& = & 637x-910 \\\Leftrightarrow & 26x \color{red}{+70} \color{blue}{-70} \color{blue}{-637x} & = & \color{red}{637x} -910 \color{blue}{-637x} \color{blue}{-70} \\\Leftrightarrow & -611x& = & -980 \\\Leftrightarrow & \frac{-611x}{ \color{red}{-611} }& = & \frac{-980}{-611} \\\Leftrightarrow & x = \frac{980}{611} & & \\ & V = \left\{ \frac{980}{611} \right\} & \\\end{align}\)
5. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{5}{2}x+7 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{275}{ \color{blue}{110} }x+\frac{770}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-40& = & 275x+770 \\\Leftrightarrow & 22x \color{red}{-40} \color{blue}{+40} \color{blue}{-275x} & = & \color{red}{275x} +770 \color{blue}{-275x} \color{blue}{+40} \\\Leftrightarrow & -253x& = & 810 \\\Leftrightarrow & \frac{-253x}{ \color{red}{-253} }& = & \frac{810}{-253} \\\Leftrightarrow & x = \frac{-810}{253} & & \\ & V = \left\{ \frac{-810}{253} \right\} & \\\end{align}\)
6. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{2}{5}x+2 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{44}{ \color{blue}{110} }x+\frac{220}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-40& = & 44x+220 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{-44x} & = & \color{red}{44x} +220 \color{blue}{-44x} \color{blue}{+40} \\\Leftrightarrow & 11x& = & 260 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{260}{11} \\\Leftrightarrow & x = \frac{260}{11} & & \\ & V = \left\{ \frac{260}{11} \right\} & \\\end{align}\)
7. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{16}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-80}{ \color{blue}{48} }x+\frac{144}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x-15& = & -80x+144 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{+80x} & = & \color{red}{-80x} +144 \color{blue}{+80x} \color{blue}{+15} \\\Leftrightarrow & 92x& = & 159 \\\Leftrightarrow & \frac{92x}{ \color{red}{92} }& = & \frac{159}{92} \\\Leftrightarrow & x = \frac{159}{92} & & \\ & V = \left\{ \frac{159}{92} \right\} & \\\end{align}\)
8. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }- \frac{ 2 }{ \color{blue}{15} })& = & (\frac{-12}{ \color{blue}{15} }x-\frac{90}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x-2& = & -12x-90 \\\Leftrightarrow & 5x \color{red}{-2} \color{blue}{+2} \color{blue}{+12x} & = & \color{red}{-12x} -90 \color{blue}{+12x} \color{blue}{+2} \\\Leftrightarrow & 17x& = & -88 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-88}{17} \\\Leftrightarrow & x = \frac{-88}{17} & & \\ & V = \left\{ \frac{-88}{17} \right\} & \\\end{align}\)
9. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{16}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{16}{ \color{blue}{80} }x+\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x-15& = & 16x+160 \\\Leftrightarrow & 20x \color{red}{-15} \color{blue}{+15} \color{blue}{-16x} & = & \color{red}{16x} +160 \color{blue}{-16x} \color{blue}{+15} \\\Leftrightarrow & 4x& = & 175 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{175}{4} \\\Leftrightarrow & x = \frac{175}{4} & & \\ & V = \left\{ \frac{175}{4} \right\} & \\\end{align}\)
10. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{13}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x+\frac{1950}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-90& = & -546x+1950 \\\Leftrightarrow & 65x \color{red}{-90} \color{blue}{+90} \color{blue}{+546x} & = & \color{red}{-546x} +1950 \color{blue}{+546x} \color{blue}{+90} \\\Leftrightarrow & 611x& = & 2040 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{2040}{611} \\\Leftrightarrow & x = \frac{2040}{611} & & \\ & V = \left\{ \frac{2040}{611} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 42x+1470 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-42x} & = & \color{red}{42x} +1470 \color{blue}{-42x} \color{blue}{+120} \\\Leftrightarrow & -7x& = & 1590 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{1590}{-7} \\\Leftrightarrow & x = \frac{-1590}{7} & & \\ & V = \left\{ \frac{-1590}{7} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x-\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & -100x-300 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+100x} & = & \color{red}{-100x} -300 \color{blue}{+100x} \color{blue}{-8} \\\Leftrightarrow & 115x& = & -308 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{-308}{115} \\\Leftrightarrow & x = \frac{-308}{115} & & \\ & V = \left\{ \frac{-308}{115} \right\} & \\\end{align}\)