Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{16}& = & \frac{-2}{5}x-5 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-96}{ \color{blue}{240} }x-\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-75& = & -96x-1200 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{+96x} & = & \color{red}{-96x} -1200 \color{blue}{+96x} \color{blue}{+75} \\\Leftrightarrow & 136x& = & -1125 \\\Leftrightarrow & \frac{136x}{ \color{red}{136} }& = & \frac{-1125}{136} \\\Leftrightarrow & x = \frac{-1125}{136} & & \\ & V = \left\{ \frac{-1125}{136} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & -84x-1260 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{+84x} & = & \color{red}{-84x} -1260 \color{blue}{+84x} \color{blue}{-150} \\\Leftrightarrow & 119x& = & -1410 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-1410}{119} \\\Leftrightarrow & x = \frac{-1410}{119} & & \\ & V = \left\{ \frac{-1410}{119} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -168x-630 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+168x} & = & \color{red}{-168x} -630 \color{blue}{+168x} \color{blue}{-120} \\\Leftrightarrow & 203x& = & -750 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-750}{203} \\\Leftrightarrow & x = \frac{-750}{203} & & \\ & V = \left\{ \frac{-750}{203} \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x-\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & -44x-396 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{+44x} & = & \color{red}{-44x} -396 \color{blue}{+44x} \color{blue}{+30} \\\Leftrightarrow & 77x& = & -366 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-366}{77} \\\Leftrightarrow & x = \frac{-366}{77} & & \\ & V = \left\{ \frac{-366}{77} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-84}{ \color{blue}{60} }x+\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & -84x+480 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+84x} & = & \color{red}{-84x} +480 \color{blue}{+84x} \color{blue}{-8} \\\Leftrightarrow & 99x& = & 472 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{472}{99} \\\Leftrightarrow & x = \frac{472}{99} & & \\ & V = \left\{ \frac{472}{99} \right\} & \\\end{align}\)
6. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{7}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 10 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{112}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-10& = & 7x+112 \\\Leftrightarrow & 2x \color{red}{-10} \color{blue}{+10} \color{blue}{-7x} & = & \color{red}{7x} +112 \color{blue}{-7x} \color{blue}{+10} \\\Leftrightarrow & -5x& = & 122 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{122}{-5} \\\Leftrightarrow & x = \frac{-122}{5} & & \\ & V = \left\{ \frac{-122}{5} \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{7}{5}x+5 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{196}{ \color{blue}{140} }x+\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+60& = & 196x+700 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-196x} & = & \color{red}{196x} +700 \color{blue}{-196x} \color{blue}{-60} \\\Leftrightarrow & -161x& = & 640 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{640}{-161} \\\Leftrightarrow & x = \frac{-640}{161} & & \\ & V = \left\{ \frac{-640}{161} \right\} & \\\end{align}\)
8. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{1}{6}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{55}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x-120& = & 55x+2640 \\\Leftrightarrow & 66x \color{red}{-120} \color{blue}{+120} \color{blue}{-55x} & = & \color{red}{55x} +2640 \color{blue}{-55x} \color{blue}{+120} \\\Leftrightarrow & 11x& = & 2760 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{2760}{11} \\\Leftrightarrow & x = \frac{2760}{11} & & \\ & V = \left\{ \frac{2760}{11} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{1}{4}x+2 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{39}{ \color{blue}{156} }x+\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-24& = & 39x+312 \\\Leftrightarrow & 52x \color{red}{-24} \color{blue}{+24} \color{blue}{-39x} & = & \color{red}{39x} +312 \color{blue}{-39x} \color{blue}{+24} \\\Leftrightarrow & 13x& = & 336 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{336}{13} \\\Leftrightarrow & x = \frac{336}{13} & & \\ & V = \left\{ \frac{336}{13} \right\} & \\\end{align}\)
10. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x-\frac{192}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & 24x-192 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{-24x} & = & \color{red}{24x} -192 \color{blue}{-24x} \color{blue}{+9} \\\Leftrightarrow & -8x& = & -183 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-183}{-8} \\\Leftrightarrow & x = \frac{183}{8} & & \\ & V = \left\{ \frac{183}{8} \right\} & \\\end{align}\)
11. \(\text{462 is het kleinste gemene veelvoud van 7, 11 en 6} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{7}{6}x-2 \\\Leftrightarrow & \color{blue}{462.} (\frac{66x}{ \color{blue}{462} }+ \frac{ 210 }{ \color{blue}{462} })& = & (\frac{539}{ \color{blue}{462} }x-\frac{924}{ \color{blue}{462} }) \color{blue}{.462} \\\Leftrightarrow & 66x+210& = & 539x-924 \\\Leftrightarrow & 66x \color{red}{+210} \color{blue}{-210} \color{blue}{-539x} & = & \color{red}{539x} -924 \color{blue}{-539x} \color{blue}{-210} \\\Leftrightarrow & -473x& = & -1134 \\\Leftrightarrow & \frac{-473x}{ \color{red}{-473} }& = & \frac{-1134}{-473} \\\Leftrightarrow & x = \frac{1134}{473} & & \\ & V = \left\{ \frac{1134}{473} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{13}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 30 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x+\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+30& = & 26x+156 \\\Leftrightarrow & 39x \color{red}{+30} \color{blue}{-30} \color{blue}{-26x} & = & \color{red}{26x} +156 \color{blue}{-26x} \color{blue}{-30} \\\Leftrightarrow & 13x& = & 126 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{126}{13} \\\Leftrightarrow & x = \frac{126}{13} & & \\ & V = \left\{ \frac{126}{13} \right\} & \\\end{align}\)