Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{6}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-120& = & 35x+630 \\\Leftrightarrow & 42x \color{red}{-120} \color{blue}{+120} \color{blue}{-35x} & = & \color{red}{35x} +630 \color{blue}{-35x} \color{blue}{+120} \\\Leftrightarrow & 7x& = & 750 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{750}{7} \\\Leftrightarrow & x = \frac{750}{7} & & \\ & V = \left\{ \frac{750}{7} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x-\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & 26x-390 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{-26x} & = & \color{red}{26x} -390 \color{blue}{-26x} \color{blue}{-12} \\\Leftrightarrow & 13x& = & -402 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-402}{13} \\\Leftrightarrow & x = \frac{-402}{13} & & \\ & V = \left\{ \frac{-402}{13} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-2}{5}x+1 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-56}{ \color{blue}{140} }x+\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-80& = & -56x+140 \\\Leftrightarrow & 35x \color{red}{-80} \color{blue}{+80} \color{blue}{+56x} & = & \color{red}{-56x} +140 \color{blue}{+56x} \color{blue}{+80} \\\Leftrightarrow & 91x& = & 220 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{220}{91} \\\Leftrightarrow & x = \frac{220}{91} & & \\ & V = \left\{ \frac{220}{91} \right\} & \\\end{align}\)
4. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{13}& = & \frac{2}{5}x+8 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 100 }{ \color{blue}{260} })& = & (\frac{104}{ \color{blue}{260} }x+\frac{2080}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+100& = & 104x+2080 \\\Leftrightarrow & 65x \color{red}{+100} \color{blue}{-100} \color{blue}{-104x} & = & \color{red}{104x} +2080 \color{blue}{-104x} \color{blue}{-100} \\\Leftrightarrow & -39x& = & 1980 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{1980}{-39} \\\Leftrightarrow & x = \frac{-660}{13} & & \\ & V = \left\{ \frac{-660}{13} \right\} & \\\end{align}\)
5. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{-182}{ \color{blue}{130} }x+\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+30& = & -182x+650 \\\Leftrightarrow & 65x \color{red}{+30} \color{blue}{-30} \color{blue}{+182x} & = & \color{red}{-182x} +650 \color{blue}{+182x} \color{blue}{-30} \\\Leftrightarrow & 247x& = & 620 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{620}{247} \\\Leftrightarrow & x = \frac{620}{247} & & \\ & V = \left\{ \frac{620}{247} \right\} & \\\end{align}\)
6. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{13}& = & \frac{-5}{3}x-6 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{-260}{ \color{blue}{156} }x-\frac{936}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-24& = & -260x-936 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{+260x} & = & \color{red}{-260x} -936 \color{blue}{+260x} \color{blue}{+24} \\\Leftrightarrow & 299x& = & -912 \\\Leftrightarrow & \frac{299x}{ \color{red}{299} }& = & \frac{-912}{299} \\\Leftrightarrow & x = \frac{-912}{299} & & \\ & V = \left\{ \frac{-912}{299} \right\} & \\\end{align}\)
7. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 20 }{ \color{blue}{35} })& = & (\frac{7}{ \color{blue}{35} }x+\frac{35}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+20& = & 7x+35 \\\Leftrightarrow & 5x \color{red}{+20} \color{blue}{-20} \color{blue}{-7x} & = & \color{red}{7x} +35 \color{blue}{-7x} \color{blue}{-20} \\\Leftrightarrow & -2x& = & 15 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{15}{-2} \\\Leftrightarrow & x = \frac{-15}{2} & & \\ & V = \left\{ \frac{-15}{2} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{-8}{3}x+1 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-160}{ \color{blue}{60} }x+\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & -160x+60 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{+160x} & = & \color{red}{-160x} +60 \color{blue}{+160x} \color{blue}{+16} \\\Leftrightarrow & 175x& = & 76 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{76}{175} \\\Leftrightarrow & x = \frac{76}{175} & & \\ & V = \left\{ \frac{76}{175} \right\} & \\\end{align}\)
9. \(\text{280 is het kleinste gemene veelvoud van 7, 8 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{8}& = & \frac{1}{5}x+4 \\\Leftrightarrow & \color{blue}{280.} (\frac{40x}{ \color{blue}{280} }- \frac{ 105 }{ \color{blue}{280} })& = & (\frac{56}{ \color{blue}{280} }x+\frac{1120}{ \color{blue}{280} }) \color{blue}{.280} \\\Leftrightarrow & 40x-105& = & 56x+1120 \\\Leftrightarrow & 40x \color{red}{-105} \color{blue}{+105} \color{blue}{-56x} & = & \color{red}{56x} +1120 \color{blue}{-56x} \color{blue}{+105} \\\Leftrightarrow & -16x& = & 1225 \\\Leftrightarrow & \frac{-16x}{ \color{red}{-16} }& = & \frac{1225}{-16} \\\Leftrightarrow & x = \frac{-1225}{16} & & \\ & V = \left\{ \frac{-1225}{16} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{7}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 294x+1050 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-294x} & = & \color{red}{294x} +1050 \color{blue}{-294x} \color{blue}{+60} \\\Leftrightarrow & -259x& = & 1110 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{1110}{-259} \\\Leftrightarrow & x = \frac{-30}{7} & & \\ & V = \left\{ \frac{-30}{7} \right\} & \\\end{align}\)
11. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{52}{ \color{blue}{156} }x+\frac{780}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+48& = & 52x+780 \\\Leftrightarrow & 39x \color{red}{+48} \color{blue}{-48} \color{blue}{-52x} & = & \color{red}{52x} +780 \color{blue}{-52x} \color{blue}{-48} \\\Leftrightarrow & -13x& = & 732 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{732}{-13} \\\Leftrightarrow & x = \frac{-732}{13} & & \\ & V = \left\{ \frac{-732}{13} \right\} & \\\end{align}\)
12. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x-\frac{264}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & -88x-264 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{+88x} & = & \color{red}{-88x} -264 \color{blue}{+88x} \color{blue}{-48} \\\Leftrightarrow & 121x& = & -312 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{-312}{121} \\\Leftrightarrow & x = \frac{-312}{121} & & \\ & V = \left\{ \frac{-312}{121} \right\} & \\\end{align}\)