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

---

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

Verbetersleutel

1. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{-5}{3}x+3 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x+\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & -220x+396 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{+220x} & = & \color{red}{-220x} +396 \color{blue}{+220x} \color{blue}{-24} \\\Leftrightarrow & 253x& = & 372 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{372}{253} \\\Leftrightarrow & x = \frac{372}{253} & & \\ & V = \left\{ \frac{372}{253} \right\} & \\\end{align}\)
2. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{11}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+12& = & 33x-198 \\\Leftrightarrow & 22x \color{red}{+12} \color{blue}{-12} \color{blue}{-33x} & = & \color{red}{33x} -198 \color{blue}{-33x} \color{blue}{-12} \\\Leftrightarrow & -11x& = & -210 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-210}{-11} \\\Leftrightarrow & x = \frac{210}{11} & & \\ & V = \left\{ \frac{210}{11} \right\} & \\\end{align}\)
3. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{7}{2}x+3 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{392}{ \color{blue}{112} }x+\frac{336}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & 392x+336 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{-392x} & = & \color{red}{392x} +336 \color{blue}{-392x} \color{blue}{-35} \\\Leftrightarrow & -376x& = & 301 \\\Leftrightarrow & \frac{-376x}{ \color{red}{-376} }& = & \frac{301}{-376} \\\Leftrightarrow & x = \frac{-301}{376} & & \\ & V = \left\{ \frac{-301}{376} \right\} & \\\end{align}\)
4. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{7}{5}x+1 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 20 }{ \color{blue}{35} })& = & (\frac{49}{ \color{blue}{35} }x+\frac{35}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+20& = & 49x+35 \\\Leftrightarrow & 5x \color{red}{+20} \color{blue}{-20} \color{blue}{-49x} & = & \color{red}{49x} +35 \color{blue}{-49x} \color{blue}{-20} \\\Leftrightarrow & -44x& = & 15 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{15}{-44} \\\Leftrightarrow & x = \frac{-15}{44} & & \\ & V = \left\{ \frac{-15}{44} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{1}{6}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 56 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+56& = & 35x+1050 \\\Leftrightarrow & 30x \color{red}{+56} \color{blue}{-56} \color{blue}{-35x} & = & \color{red}{35x} +1050 \color{blue}{-35x} \color{blue}{-56} \\\Leftrightarrow & -5x& = & 994 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{994}{-5} \\\Leftrightarrow & x = \frac{-994}{5} & & \\ & V = \left\{ \frac{-994}{5} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{7}{3}x-6 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{140}{ \color{blue}{60} }x-\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+18& = & 140x-360 \\\Leftrightarrow & 15x \color{red}{+18} \color{blue}{-18} \color{blue}{-140x} & = & \color{red}{140x} -360 \color{blue}{-140x} \color{blue}{-18} \\\Leftrightarrow & -125x& = & -378 \\\Leftrightarrow & \frac{-125x}{ \color{red}{-125} }& = & \frac{-378}{-125} \\\Leftrightarrow & x = \frac{378}{125} & & \\ & V = \left\{ \frac{378}{125} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x-\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+24& = & -52x-546 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{+52x} & = & \color{red}{-52x} -546 \color{blue}{+52x} \color{blue}{-24} \\\Leftrightarrow & 91x& = & -570 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-570}{91} \\\Leftrightarrow & x = \frac{-570}{91} & & \\ & V = \left\{ \frac{-570}{91} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{2}{3}x-5 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{40}{ \color{blue}{60} }x-\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & 40x-300 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{-40x} & = & \color{red}{40x} -300 \color{blue}{-40x} \color{blue}{-16} \\\Leftrightarrow & -25x& = & -316 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{-316}{-25} \\\Leftrightarrow & x = \frac{316}{25} & & \\ & V = \left\{ \frac{316}{25} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{7}{3}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{70}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & 70x+240 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-70x} & = & \color{red}{70x} +240 \color{blue}{-70x} \color{blue}{+8} \\\Leftrightarrow & -55x& = & 248 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{248}{-55} \\\Leftrightarrow & x = \frac{-248}{55} & & \\ & V = \left\{ \frac{-248}{55} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{11}& = & \frac{2}{5}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 90 }{ \color{blue}{330} })& = & (\frac{132}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+90& = & 132x+2640 \\\Leftrightarrow & 55x \color{red}{+90} \color{blue}{-90} \color{blue}{-132x} & = & \color{red}{132x} +2640 \color{blue}{-132x} \color{blue}{-90} \\\Leftrightarrow & -77x& = & 2550 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{2550}{-77} \\\Leftrightarrow & x = \frac{-2550}{77} & & \\ & V = \left\{ \frac{-2550}{77} \right\} & \\\end{align}\)
11. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{4}{5}x-5 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 60 }{ \color{blue}{105} })& = & (\frac{84}{ \color{blue}{105} }x-\frac{525}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-60& = & 84x-525 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-84x} & = & \color{red}{84x} -525 \color{blue}{-84x} \color{blue}{+60} \\\Leftrightarrow & -49x& = & -465 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-465}{-49} \\\Leftrightarrow & x = \frac{465}{49} & & \\ & V = \left\{ \frac{465}{49} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{14}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 15 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+15& = & 21x+84 \\\Leftrightarrow & 14x \color{red}{+15} \color{blue}{-15} \color{blue}{-21x} & = & \color{red}{21x} +84 \color{blue}{-21x} \color{blue}{-15} \\\Leftrightarrow & -7x& = & 69 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{69}{-7} \\\Leftrightarrow & x = \frac{-69}{7} & & \\ & V = \left\{ \frac{-69}{7} \right\} & \\\end{align}\)