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

---

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

Verbetersleutel

1. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x+\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & 22x+264 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{-22x} & = & \color{red}{22x} +264 \color{blue}{-22x} \color{blue}{+12} \\\Leftrightarrow & 11x& = & 276 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{276}{11} \\\Leftrightarrow & x = \frac{276}{11} & & \\ & V = \left\{ \frac{276}{11} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 42x-630 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-42x} & = & \color{red}{42x} -630 \color{blue}{-42x} \color{blue}{+120} \\\Leftrightarrow & -7x& = & -510 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-510}{-7} \\\Leftrightarrow & x = \frac{510}{7} & & \\ & V = \left\{ \frac{510}{7} \right\} & \\\end{align}\)
3. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{8}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }+ \frac{ 21 }{ \color{blue}{56} })& = & (\frac{28}{ \color{blue}{56} }x+\frac{336}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x+21& = & 28x+336 \\\Leftrightarrow & 8x \color{red}{+21} \color{blue}{-21} \color{blue}{-28x} & = & \color{red}{28x} +336 \color{blue}{-28x} \color{blue}{-21} \\\Leftrightarrow & -20x& = & 315 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{315}{-20} \\\Leftrightarrow & x = \frac{-63}{4} & & \\ & V = \left\{ \frac{-63}{4} \right\} & \\\end{align}\)
4. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{-3}{4}x-2 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 8 }{ \color{blue}{28} })& = & (\frac{-21}{ \color{blue}{28} }x-\frac{56}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+8& = & -21x-56 \\\Leftrightarrow & 4x \color{red}{+8} \color{blue}{-8} \color{blue}{+21x} & = & \color{red}{-21x} -56 \color{blue}{+21x} \color{blue}{-8} \\\Leftrightarrow & 25x& = & -64 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{-64}{25} \\\Leftrightarrow & x = \frac{-64}{25} & & \\ & V = \left\{ \frac{-64}{25} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-4& = & 15x-120 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-15x} & = & \color{red}{15x} -120 \color{blue}{-15x} \color{blue}{+4} \\\Leftrightarrow & -9x& = & -116 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-116}{-9} \\\Leftrightarrow & x = \frac{116}{9} & & \\ & V = \left\{ \frac{116}{9} \right\} & \\\end{align}\)
6. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{11}& = & \frac{-2}{5}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-132}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-120& = & -132x+2640 \\\Leftrightarrow & 55x \color{red}{-120} \color{blue}{+120} \color{blue}{+132x} & = & \color{red}{-132x} +2640 \color{blue}{+132x} \color{blue}{+120} \\\Leftrightarrow & 187x& = & 2760 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{2760}{187} \\\Leftrightarrow & x = \frac{2760}{187} & & \\ & V = \left\{ \frac{2760}{187} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 7, 10 en 3} \\ \begin{align} & \frac{x}{7}+\frac{3}{10}& = & \frac{-5}{3}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 63 }{ \color{blue}{210} })& = & (\frac{-350}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+63& = & -350x+840 \\\Leftrightarrow & 30x \color{red}{+63} \color{blue}{-63} \color{blue}{+350x} & = & \color{red}{-350x} +840 \color{blue}{+350x} \color{blue}{-63} \\\Leftrightarrow & 380x& = & 777 \\\Leftrightarrow & \frac{380x}{ \color{red}{380} }& = & \frac{777}{380} \\\Leftrightarrow & x = \frac{777}{380} & & \\ & V = \left\{ \frac{777}{380} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 4, 6 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{2}{5}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 50 }{ \color{blue}{60} })& = & (\frac{24}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+50& = & 24x+420 \\\Leftrightarrow & 15x \color{red}{+50} \color{blue}{-50} \color{blue}{-24x} & = & \color{red}{24x} +420 \color{blue}{-24x} \color{blue}{-50} \\\Leftrightarrow & -9x& = & 370 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{370}{-9} \\\Leftrightarrow & x = \frac{-370}{9} & & \\ & V = \left\{ \frac{-370}{9} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{1}{6}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-60& = & 35x-1050 \\\Leftrightarrow & 42x \color{red}{-60} \color{blue}{+60} \color{blue}{-35x} & = & \color{red}{35x} -1050 \color{blue}{-35x} \color{blue}{+60} \\\Leftrightarrow & 7x& = & -990 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-990}{7} \\\Leftrightarrow & x = \frac{-990}{7} & & \\ & V = \left\{ \frac{-990}{7} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{10}& = & \frac{2}{3}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{20}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+9& = & 20x+90 \\\Leftrightarrow & 15x \color{red}{+9} \color{blue}{-9} \color{blue}{-20x} & = & \color{red}{20x} +90 \color{blue}{-20x} \color{blue}{-9} \\\Leftrightarrow & -5x& = & 81 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{81}{-5} \\\Leftrightarrow & x = \frac{-81}{5} & & \\ & V = \left\{ \frac{-81}{5} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{3}{4}x-4 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{63}{ \color{blue}{84} }x-\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-24& = & 63x-336 \\\Leftrightarrow & 28x \color{red}{-24} \color{blue}{+24} \color{blue}{-63x} & = & \color{red}{63x} -336 \color{blue}{-63x} \color{blue}{+24} \\\Leftrightarrow & -35x& = & -312 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-312}{-35} \\\Leftrightarrow & x = \frac{312}{35} & & \\ & V = \left\{ \frac{312}{35} \right\} & \\\end{align}\)
12. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{16}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-96}{ \color{blue}{240} }x-\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-75& = & -96x-1920 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{+96x} & = & \color{red}{-96x} -1920 \color{blue}{+96x} \color{blue}{+75} \\\Leftrightarrow & 136x& = & -1845 \\\Leftrightarrow & \frac{136x}{ \color{red}{136} }& = & \frac{-1845}{136} \\\Leftrightarrow & x = \frac{-1845}{136} & & \\ & V = \left\{ \frac{-1845}{136} \right\} & \\\end{align}\)