Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}-\frac{5}{13}& = & \frac{1}{4}x+5 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }- \frac{ 140 }{ \color{blue}{364} })& = & (\frac{91}{ \color{blue}{364} }x+\frac{1820}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x-140& = & 91x+1820 \\\Leftrightarrow & 52x \color{red}{-140} \color{blue}{+140} \color{blue}{-91x} & = & \color{red}{91x} +1820 \color{blue}{-91x} \color{blue}{+140} \\\Leftrightarrow & -39x& = & 1960 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{1960}{-39} \\\Leftrightarrow & x = \frac{-1960}{39} & & \\ & V = \left\{ \frac{-1960}{39} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{7}& = & \frac{7}{3}x+8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 60 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+60& = & 196x+672 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{-196x} & = & \color{red}{196x} +672 \color{blue}{-196x} \color{blue}{-60} \\\Leftrightarrow & -175x& = & 612 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{612}{-175} \\\Leftrightarrow & x = \frac{-612}{175} & & \\ & V = \left\{ \frac{-612}{175} \right\} & \\\end{align}\)
3. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 60 }{ \color{blue}{195} })& = & (\frac{234}{ \color{blue}{195} }x+\frac{975}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+60& = & 234x+975 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-234x} & = & \color{red}{234x} +975 \color{blue}{-234x} \color{blue}{-60} \\\Leftrightarrow & -169x& = & 915 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{915}{-169} \\\Leftrightarrow & x = \frac{-915}{169} & & \\ & V = \left\{ \frac{-915}{169} \right\} & \\\end{align}\)
4. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 40 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+40& = & -56x-490 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{+56x} & = & \color{red}{-56x} -490 \color{blue}{+56x} \color{blue}{-40} \\\Leftrightarrow & 91x& = & -530 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-530}{91} \\\Leftrightarrow & x = \frac{-530}{91} & & \\ & V = \left\{ \frac{-530}{91} \right\} & \\\end{align}\)
5. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{11}& = & \frac{7}{3}x-6 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x-\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-30& = & 154x-396 \\\Leftrightarrow & 33x \color{red}{-30} \color{blue}{+30} \color{blue}{-154x} & = & \color{red}{154x} -396 \color{blue}{-154x} \color{blue}{+30} \\\Leftrightarrow & -121x& = & -366 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{-366}{-121} \\\Leftrightarrow & x = \frac{366}{121} & & \\ & V = \left\{ \frac{366}{121} \right\} & \\\end{align}\)
6. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{11}& = & \frac{6}{5}x-8 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 30 }{ \color{blue}{165} })& = & (\frac{198}{ \color{blue}{165} }x-\frac{1320}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+30& = & 198x-1320 \\\Leftrightarrow & 55x \color{red}{+30} \color{blue}{-30} \color{blue}{-198x} & = & \color{red}{198x} -1320 \color{blue}{-198x} \color{blue}{-30} \\\Leftrightarrow & -143x& = & -1350 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-1350}{-143} \\\Leftrightarrow & x = \frac{1350}{143} & & \\ & V = \left\{ \frac{1350}{143} \right\} & \\\end{align}\)
7. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x+\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & 396x+1650 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{-396x} & = & \color{red}{396x} +1650 \color{blue}{-396x} \color{blue}{-120} \\\Leftrightarrow & -341x& = & 1530 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{1530}{-341} \\\Leftrightarrow & x = \frac{-1530}{341} & & \\ & V = \left\{ \frac{-1530}{341} \right\} & \\\end{align}\)
8. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{2}{3}x+2 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{104}{ \color{blue}{156} }x+\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+48& = & 104x+312 \\\Leftrightarrow & 39x \color{red}{+48} \color{blue}{-48} \color{blue}{-104x} & = & \color{red}{104x} +312 \color{blue}{-104x} \color{blue}{-48} \\\Leftrightarrow & -65x& = & 264 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{264}{-65} \\\Leftrightarrow & x = \frac{-264}{65} & & \\ & V = \left\{ \frac{-264}{65} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 15x-150 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-15x} & = & \color{red}{15x} -150 \color{blue}{-15x} \color{blue}{-25} \\\Leftrightarrow & -9x& = & -175 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-175}{-9} \\\Leftrightarrow & x = \frac{175}{9} & & \\ & V = \left\{ \frac{175}{9} \right\} & \\\end{align}\)
10. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{16}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x-\frac{80}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-15& = & 40x-80 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{-40x} & = & \color{red}{40x} -80 \color{blue}{-40x} \color{blue}{+15} \\\Leftrightarrow & -24x& = & -65 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{-65}{-24} \\\Leftrightarrow & x = \frac{65}{24} & & \\ & V = \left\{ \frac{65}{24} \right\} & \\\end{align}\)
11. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{11}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 42 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x-\frac{616}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-42& = & 77x-616 \\\Leftrightarrow & 22x \color{red}{-42} \color{blue}{+42} \color{blue}{-77x} & = & \color{red}{77x} -616 \color{blue}{-77x} \color{blue}{+42} \\\Leftrightarrow & -55x& = & -574 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-574}{-55} \\\Leftrightarrow & x = \frac{574}{55} & & \\ & V = \left\{ \frac{574}{55} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{5}{4}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{105}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-24& = & 105x-252 \\\Leftrightarrow & 28x \color{red}{-24} \color{blue}{+24} \color{blue}{-105x} & = & \color{red}{105x} -252 \color{blue}{-105x} \color{blue}{+24} \\\Leftrightarrow & -77x& = & -228 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-228}{-77} \\\Leftrightarrow & x = \frac{228}{77} & & \\ & V = \left\{ \frac{228}{77} \right\} & \\\end{align}\)