Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{16}& = & \frac{7}{6}x+4 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }- \frac{ 63 }{ \color{blue}{336} })& = & (\frac{392}{ \color{blue}{336} }x+\frac{1344}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x-63& = & 392x+1344 \\\Leftrightarrow & 48x \color{red}{-63} \color{blue}{+63} \color{blue}{-392x} & = & \color{red}{392x} +1344 \color{blue}{-392x} \color{blue}{+63} \\\Leftrightarrow & -344x& = & 1407 \\\Leftrightarrow & \frac{-344x}{ \color{red}{-344} }& = & \frac{1407}{-344} \\\Leftrightarrow & x = \frac{-1407}{344} & & \\ & V = \left\{ \frac{-1407}{344} \right\} & \\\end{align}\)
2. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{-8}{3}x+2 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 48 }{ \color{blue}{156} })& = & (\frac{-416}{ \color{blue}{156} }x+\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-48& = & -416x+312 \\\Leftrightarrow & 39x \color{red}{-48} \color{blue}{+48} \color{blue}{+416x} & = & \color{red}{-416x} +312 \color{blue}{+416x} \color{blue}{+48} \\\Leftrightarrow & 455x& = & 360 \\\Leftrightarrow & \frac{455x}{ \color{red}{455} }& = & \frac{360}{455} \\\Leftrightarrow & x = \frac{72}{91} & & \\ & V = \left\{ \frac{72}{91} \right\} & \\\end{align}\)
3. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{11}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x+\frac{1320}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+150& = & -462x+1320 \\\Leftrightarrow & 55x \color{red}{+150} \color{blue}{-150} \color{blue}{+462x} & = & \color{red}{-462x} +1320 \color{blue}{+462x} \color{blue}{-150} \\\Leftrightarrow & 517x& = & 1170 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{1170}{517} \\\Leftrightarrow & x = \frac{1170}{517} & & \\ & V = \left\{ \frac{1170}{517} \right\} & \\\end{align}\)
4. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{-5}{3}x-3 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{-110}{ \color{blue}{66} }x-\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+30& = & -110x-198 \\\Leftrightarrow & 33x \color{red}{+30} \color{blue}{-30} \color{blue}{+110x} & = & \color{red}{-110x} -198 \color{blue}{+110x} \color{blue}{-30} \\\Leftrightarrow & 143x& = & -228 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-228}{143} \\\Leftrightarrow & x = \frac{-228}{143} & & \\ & V = \left\{ \frac{-228}{143} \right\} & \\\end{align}\)
5. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{4}{5}x-4 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 10 }{ \color{blue}{35} })& = & (\frac{28}{ \color{blue}{35} }x-\frac{140}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+10& = & 28x-140 \\\Leftrightarrow & 5x \color{red}{+10} \color{blue}{-10} \color{blue}{-28x} & = & \color{red}{28x} -140 \color{blue}{-28x} \color{blue}{-10} \\\Leftrightarrow & -23x& = & -150 \\\Leftrightarrow & \frac{-23x}{ \color{red}{-23} }& = & \frac{-150}{-23} \\\Leftrightarrow & x = \frac{150}{23} & & \\ & V = \left\{ \frac{150}{23} \right\} & \\\end{align}\)
6. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-30& = & 33x+396 \\\Leftrightarrow & 22x \color{red}{-30} \color{blue}{+30} \color{blue}{-33x} & = & \color{red}{33x} +396 \color{blue}{-33x} \color{blue}{+30} \\\Leftrightarrow & -11x& = & 426 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{426}{-11} \\\Leftrightarrow & x = \frac{-426}{11} & & \\ & V = \left\{ \frac{-426}{11} \right\} & \\\end{align}\)
7. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{13}& = & \frac{7}{5}x+5 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 60 }{ \color{blue}{260} })& = & (\frac{364}{ \color{blue}{260} }x+\frac{1300}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+60& = & 364x+1300 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-364x} & = & \color{red}{364x} +1300 \color{blue}{-364x} \color{blue}{-60} \\\Leftrightarrow & -299x& = & 1240 \\\Leftrightarrow & \frac{-299x}{ \color{red}{-299} }& = & \frac{1240}{-299} \\\Leftrightarrow & x = \frac{-1240}{299} & & \\ & V = \left\{ \frac{-1240}{299} \right\} & \\\end{align}\)
8. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 8 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{98}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-8& = & 7x-98 \\\Leftrightarrow & 2x \color{red}{-8} \color{blue}{+8} \color{blue}{-7x} & = & \color{red}{7x} -98 \color{blue}{-7x} \color{blue}{+8} \\\Leftrightarrow & -5x& = & -90 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-90}{-5} \\\Leftrightarrow & x = 18 & & \\ & V = \left\{ 18 \right\} & \\\end{align}\)
9. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{3}{5}x-5 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 28 }{ \color{blue}{105} })& = & (\frac{63}{ \color{blue}{105} }x-\frac{525}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+28& = & 63x-525 \\\Leftrightarrow & 15x \color{red}{+28} \color{blue}{-28} \color{blue}{-63x} & = & \color{red}{63x} -525 \color{blue}{-63x} \color{blue}{-28} \\\Leftrightarrow & -48x& = & -553 \\\Leftrightarrow & \frac{-48x}{ \color{red}{-48} }& = & \frac{-553}{-48} \\\Leftrightarrow & x = \frac{553}{48} & & \\ & V = \left\{ \frac{553}{48} \right\} & \\\end{align}\)
10. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & 28x-252 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{-28x} & = & \color{red}{28x} -252 \color{blue}{-28x} \color{blue}{+48} \\\Leftrightarrow & -7x& = & -204 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-204}{-7} \\\Leftrightarrow & x = \frac{204}{7} & & \\ & V = \left\{ \frac{204}{7} \right\} & \\\end{align}\)
11. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{13}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x-\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+150& = & -156x-1170 \\\Leftrightarrow & 65x \color{red}{+150} \color{blue}{-150} \color{blue}{+156x} & = & \color{red}{-156x} -1170 \color{blue}{+156x} \color{blue}{-150} \\\Leftrightarrow & 221x& = & -1320 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{-1320}{221} \\\Leftrightarrow & x = \frac{-1320}{221} & & \\ & V = \left\{ \frac{-1320}{221} \right\} & \\\end{align}\)
12. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }- \frac{ 75 }{ \color{blue}{165} })& = & (\frac{-110}{ \color{blue}{165} }x+\frac{1155}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x-75& = & -110x+1155 \\\Leftrightarrow & 33x \color{red}{-75} \color{blue}{+75} \color{blue}{+110x} & = & \color{red}{-110x} +1155 \color{blue}{+110x} \color{blue}{+75} \\\Leftrightarrow & 143x& = & 1230 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{1230}{143} \\\Leftrightarrow & x = \frac{1230}{143} & & \\ & V = \left\{ \frac{1230}{143} \right\} & \\\end{align}\)