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

---

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

Verbetersleutel

1. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 9 }{ \color{blue}{21} })& = & (\frac{-35}{ \color{blue}{21} }x+\frac{105}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-9& = & -35x+105 \\\Leftrightarrow & 3x \color{red}{-9} \color{blue}{+9} \color{blue}{+35x} & = & \color{red}{-35x} +105 \color{blue}{+35x} \color{blue}{+9} \\\Leftrightarrow & 38x& = & 114 \\\Leftrightarrow & \frac{38x}{ \color{red}{38} }& = & \frac{114}{38} \\\Leftrightarrow & x = 3 & & \\ & V = \left\{ 3 \right\} & \\\end{align}\)
2. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x+\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & 66x+330 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{-66x} & = & \color{red}{66x} +330 \color{blue}{-66x} \color{blue}{-120} \\\Leftrightarrow & -11x& = & 210 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{210}{-11} \\\Leftrightarrow & x = \frac{-210}{11} & & \\ & V = \left\{ \frac{-210}{11} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 7, 14 en 5} \\ \begin{align} & \frac{x}{7}+\frac{3}{14}& = & \frac{2}{5}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }+ \frac{ 15 }{ \color{blue}{70} })& = & (\frac{28}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x+15& = & 28x+490 \\\Leftrightarrow & 10x \color{red}{+15} \color{blue}{-15} \color{blue}{-28x} & = & \color{red}{28x} +490 \color{blue}{-28x} \color{blue}{-15} \\\Leftrightarrow & -18x& = & 475 \\\Leftrightarrow & \frac{-18x}{ \color{red}{-18} }& = & \frac{475}{-18} \\\Leftrightarrow & x = \frac{-475}{18} & & \\ & V = \left\{ \frac{-475}{18} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }- \frac{ 28 }{ \color{blue}{210} })& = & (\frac{105}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x-28& = & 105x-1470 \\\Leftrightarrow & 30x \color{red}{-28} \color{blue}{+28} \color{blue}{-105x} & = & \color{red}{105x} -1470 \color{blue}{-105x} \color{blue}{+28} \\\Leftrightarrow & -75x& = & -1442 \\\Leftrightarrow & \frac{-75x}{ \color{red}{-75} }& = & \frac{-1442}{-75} \\\Leftrightarrow & x = \frac{1442}{75} & & \\ & V = \left\{ \frac{1442}{75} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{1}{3}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-9& = & 14x-294 \\\Leftrightarrow & 21x \color{red}{-9} \color{blue}{+9} \color{blue}{-14x} & = & \color{red}{14x} -294 \color{blue}{-14x} \color{blue}{+9} \\\Leftrightarrow & 7x& = & -285 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-285}{7} \\\Leftrightarrow & x = \frac{-285}{7} & & \\ & V = \left\{ \frac{-285}{7} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{-2}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & -12x+240 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{+12x} & = & \color{red}{-12x} +240 \color{blue}{+12x} \color{blue}{-9} \\\Leftrightarrow & 17x& = & 231 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{231}{17} \\\Leftrightarrow & x = \frac{231}{17} & & \\ & V = \left\{ \frac{231}{17} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{14}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+45& = & 42x-630 \\\Leftrightarrow & 35x \color{red}{+45} \color{blue}{-45} \color{blue}{-42x} & = & \color{red}{42x} -630 \color{blue}{-42x} \color{blue}{-45} \\\Leftrightarrow & -7x& = & -675 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-675}{-7} \\\Leftrightarrow & x = \frac{675}{7} & & \\ & V = \left\{ \frac{675}{7} \right\} & \\\end{align}\)
8. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 28 }{ \color{blue}{105} })& = & (\frac{-84}{ \color{blue}{105} }x+\frac{840}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+28& = & -84x+840 \\\Leftrightarrow & 15x \color{red}{+28} \color{blue}{-28} \color{blue}{+84x} & = & \color{red}{-84x} +840 \color{blue}{+84x} \color{blue}{-28} \\\Leftrightarrow & 99x& = & 812 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{812}{99} \\\Leftrightarrow & x = \frac{812}{99} & & \\ & V = \left\{ \frac{812}{99} \right\} & \\\end{align}\)
9. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{5}{6}x+3 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{75}{ \color{blue}{90} }x+\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-20& = & 75x+270 \\\Leftrightarrow & 18x \color{red}{-20} \color{blue}{+20} \color{blue}{-75x} & = & \color{red}{75x} +270 \color{blue}{-75x} \color{blue}{+20} \\\Leftrightarrow & -57x& = & 290 \\\Leftrightarrow & \frac{-57x}{ \color{red}{-57} }& = & \frac{290}{-57} \\\Leftrightarrow & x = \frac{-290}{57} & & \\ & V = \left\{ \frac{-290}{57} \right\} & \\\end{align}\)
10. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{8}& = & \frac{7}{4}x-1 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }- \frac{ 35 }{ \color{blue}{56} })& = & (\frac{98}{ \color{blue}{56} }x-\frac{56}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x-35& = & 98x-56 \\\Leftrightarrow & 8x \color{red}{-35} \color{blue}{+35} \color{blue}{-98x} & = & \color{red}{98x} -56 \color{blue}{-98x} \color{blue}{+35} \\\Leftrightarrow & -90x& = & -21 \\\Leftrightarrow & \frac{-90x}{ \color{red}{-90} }& = & \frac{-21}{-90} \\\Leftrightarrow & x = \frac{7}{30} & & \\ & V = \left\{ \frac{7}{30} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x-210 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} -210 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & -234 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-234}{-7} \\\Leftrightarrow & x = \frac{234}{7} & & \\ & V = \left\{ \frac{234}{7} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-20& = & 35x-280 \\\Leftrightarrow & 14x \color{red}{-20} \color{blue}{+20} \color{blue}{-35x} & = & \color{red}{35x} -280 \color{blue}{-35x} \color{blue}{+20} \\\Leftrightarrow & -21x& = & -260 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-260}{-21} \\\Leftrightarrow & x = \frac{260}{21} & & \\ & V = \left\{ \frac{260}{21} \right\} & \\\end{align}\)