Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{11}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 40 }{ \color{blue}{110} })& = & (\frac{22}{ \color{blue}{110} }x-\frac{440}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+40& = & 22x-440 \\\Leftrightarrow & 55x \color{red}{+40} \color{blue}{-40} \color{blue}{-22x} & = & \color{red}{22x} -440 \color{blue}{-22x} \color{blue}{-40} \\\Leftrightarrow & 33x& = & -480 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-480}{33} \\\Leftrightarrow & x = \frac{-160}{11} & & \\ & V = \left\{ \frac{-160}{11} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{3}{2}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{45}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 45x-210 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-45x} & = & \color{red}{45x} -210 \color{blue}{-45x} \color{blue}{-25} \\\Leftrightarrow & -39x& = & -235 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-235}{-39} \\\Leftrightarrow & x = \frac{235}{39} & & \\ & V = \left\{ \frac{235}{39} \right\} & \\\end{align}\)
3. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{-3}{4}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-36}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & -36x+336 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{+36x} & = & \color{red}{-36x} +336 \color{blue}{+36x} \color{blue}{+9} \\\Leftrightarrow & 52x& = & 345 \\\Leftrightarrow & \frac{52x}{ \color{red}{52} }& = & \frac{345}{52} \\\Leftrightarrow & x = \frac{345}{52} & & \\ & V = \left\{ \frac{345}{52} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{10}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & 10x+240 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-10x} & = & \color{red}{10x} +240 \color{blue}{-10x} \color{blue}{+8} \\\Leftrightarrow & 5x& = & 248 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{248}{5} \\\Leftrightarrow & x = \frac{248}{5} & & \\ & V = \left\{ \frac{248}{5} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{5}{2}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{75}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+4& = & 75x-180 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-75x} & = & \color{red}{75x} -180 \color{blue}{-75x} \color{blue}{-4} \\\Leftrightarrow & -69x& = & -184 \\\Leftrightarrow & \frac{-69x}{ \color{red}{-69} }& = & \frac{-184}{-69} \\\Leftrightarrow & x = \frac{8}{3} & & \\ & V = \left\{ \frac{8}{3} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{5}{6}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{25}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-9& = & 25x+30 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{-25x} & = & \color{red}{25x} +30 \color{blue}{-25x} \color{blue}{+9} \\\Leftrightarrow & -19x& = & 39 \\\Leftrightarrow & \frac{-19x}{ \color{red}{-19} }& = & \frac{39}{-19} \\\Leftrightarrow & x = \frac{-39}{19} & & \\ & V = \left\{ \frac{-39}{19} \right\} & \\\end{align}\)
7. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{-2}{3}x+2 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 60 }{ \color{blue}{195} })& = & (\frac{-130}{ \color{blue}{195} }x+\frac{390}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+60& = & -130x+390 \\\Leftrightarrow & 39x \color{red}{+60} \color{blue}{-60} \color{blue}{+130x} & = & \color{red}{-130x} +390 \color{blue}{+130x} \color{blue}{-60} \\\Leftrightarrow & 169x& = & 330 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{330}{169} \\\Leftrightarrow & x = \frac{330}{169} & & \\ & V = \left\{ \frac{330}{169} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{7}& = & \frac{1}{6}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 150 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-150& = & 35x-1050 \\\Leftrightarrow & 42x \color{red}{-150} \color{blue}{+150} \color{blue}{-35x} & = & \color{red}{35x} -1050 \color{blue}{-35x} \color{blue}{+150} \\\Leftrightarrow & 7x& = & -900 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-900}{7} \\\Leftrightarrow & x = \frac{-900}{7} & & \\ & V = \left\{ \frac{-900}{7} \right\} & \\\end{align}\)
9. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{13}& = & \frac{1}{5}x+4 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{26}{ \color{blue}{130} }x+\frac{520}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+40& = & 26x+520 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{-26x} & = & \color{red}{26x} +520 \color{blue}{-26x} \color{blue}{-40} \\\Leftrightarrow & 39x& = & 480 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{480}{39} \\\Leftrightarrow & x = \frac{160}{13} & & \\ & V = \left\{ \frac{160}{13} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-20}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-9& = & -20x+210 \\\Leftrightarrow & 15x \color{red}{-9} \color{blue}{+9} \color{blue}{+20x} & = & \color{red}{-20x} +210 \color{blue}{+20x} \color{blue}{+9} \\\Leftrightarrow & 35x& = & 219 \\\Leftrightarrow & \frac{35x}{ \color{red}{35} }& = & \frac{219}{35} \\\Leftrightarrow & x = \frac{219}{35} & & \\ & V = \left\{ \frac{219}{35} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -168x-210 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+168x} & = & \color{red}{-168x} -210 \color{blue}{+168x} \color{blue}{-120} \\\Leftrightarrow & 203x& = & -330 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-330}{203} \\\Leftrightarrow & x = \frac{-330}{203} & & \\ & V = \left\{ \frac{-330}{203} \right\} & \\\end{align}\)
12. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-30& = & 21x-210 \\\Leftrightarrow & 14x \color{red}{-30} \color{blue}{+30} \color{blue}{-21x} & = & \color{red}{21x} -210 \color{blue}{-21x} \color{blue}{+30} \\\Leftrightarrow & -7x& = & -180 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-180}{-7} \\\Leftrightarrow & x = \frac{180}{7} & & \\ & V = \left\{ \frac{180}{7} \right\} & \\\end{align}\)