Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{1}{4}x+7 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{35}{ \color{blue}{140} }x+\frac{980}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+60& = & 35x+980 \\\Leftrightarrow & 28x \color{red}{+60} \color{blue}{-60} \color{blue}{-35x} & = & \color{red}{35x} +980 \color{blue}{-35x} \color{blue}{-60} \\\Leftrightarrow & -7x& = & 920 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{920}{-7} \\\Leftrightarrow & x = \frac{-920}{7} & & \\ & V = \left\{ \frac{-920}{7} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{1}{6}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+24& = & 7x-336 \\\Leftrightarrow & 6x \color{red}{+24} \color{blue}{-24} \color{blue}{-7x} & = & \color{red}{7x} -336 \color{blue}{-7x} \color{blue}{-24} \\\Leftrightarrow & -x& = & -360 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-360}{-1} \\\Leftrightarrow & x = 360 & & \\ & V = \left\{ 360 \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{12}& = & \frac{4}{5}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{48}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x+25& = & 48x-60 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{-48x} & = & \color{red}{48x} -60 \color{blue}{-48x} \color{blue}{-25} \\\Leftrightarrow & -38x& = & -85 \\\Leftrightarrow & \frac{-38x}{ \color{red}{-38} }& = & \frac{-85}{-38} \\\Leftrightarrow & x = \frac{85}{38} & & \\ & V = \left\{ \frac{85}{38} \right\} & \\\end{align}\)
4. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }+ \frac{ 8 }{ \color{blue}{36} })& = & (\frac{12}{ \color{blue}{36} }x+\frac{72}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x+8& = & 12x+72 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{-12x} & = & \color{red}{12x} +72 \color{blue}{-12x} \color{blue}{-8} \\\Leftrightarrow & -3x& = & 64 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{64}{-3} \\\Leftrightarrow & x = \frac{-64}{3} & & \\ & V = \left\{ \frac{-64}{3} \right\} & \\\end{align}\)
5. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{13}& = & \frac{5}{4}x+1 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 40 }{ \color{blue}{260} })& = & (\frac{325}{ \color{blue}{260} }x+\frac{260}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-40& = & 325x+260 \\\Leftrightarrow & 52x \color{red}{-40} \color{blue}{+40} \color{blue}{-325x} & = & \color{red}{325x} +260 \color{blue}{-325x} \color{blue}{+40} \\\Leftrightarrow & -273x& = & 300 \\\Leftrightarrow & \frac{-273x}{ \color{red}{-273} }& = & \frac{300}{-273} \\\Leftrightarrow & x = \frac{-100}{91} & & \\ & V = \left\{ \frac{-100}{91} \right\} & \\\end{align}\)
6. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-24& = & 33x+528 \\\Leftrightarrow & 22x \color{red}{-24} \color{blue}{+24} \color{blue}{-33x} & = & \color{red}{33x} +528 \color{blue}{-33x} \color{blue}{+24} \\\Leftrightarrow & -11x& = & 552 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{552}{-11} \\\Leftrightarrow & x = \frac{-552}{11} & & \\ & V = \left\{ \frac{-552}{11} \right\} & \\\end{align}\)
7. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{8}& = & \frac{-2}{3}x+8 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-16}{ \color{blue}{24} }x+\frac{192}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-15& = & -16x+192 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{+16x} & = & \color{red}{-16x} +192 \color{blue}{+16x} \color{blue}{+15} \\\Leftrightarrow & 28x& = & 207 \\\Leftrightarrow & \frac{28x}{ \color{red}{28} }& = & \frac{207}{28} \\\Leftrightarrow & x = \frac{207}{28} & & \\ & V = \left\{ \frac{207}{28} \right\} & \\\end{align}\)
8. \(\text{12 is het kleinste gemene veelvoud van 4, 12 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{12}& = & \frac{7}{3}x+6 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{28}{ \color{blue}{12} }x+\frac{72}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+5& = & 28x+72 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-28x} & = & \color{red}{28x} +72 \color{blue}{-28x} \color{blue}{-5} \\\Leftrightarrow & -25x& = & 67 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{67}{-25} \\\Leftrightarrow & x = \frac{-67}{25} & & \\ & V = \left\{ \frac{-67}{25} \right\} & \\\end{align}\)
9. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{11}& = & \frac{6}{5}x-8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 90 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x-\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-90& = & 396x-2640 \\\Leftrightarrow & 55x \color{red}{-90} \color{blue}{+90} \color{blue}{-396x} & = & \color{red}{396x} -2640 \color{blue}{-396x} \color{blue}{+90} \\\Leftrightarrow & -341x& = & -2550 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{-2550}{-341} \\\Leftrightarrow & x = \frac{2550}{341} & & \\ & V = \left\{ \frac{2550}{341} \right\} & \\\end{align}\)
10. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x-\frac{1120}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & -196x-1120 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{+196x} & = & \color{red}{-196x} -1120 \color{blue}{+196x} \color{blue}{-80} \\\Leftrightarrow & 231x& = & -1200 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{-1200}{231} \\\Leftrightarrow & x = \frac{-400}{77} & & \\ & V = \left\{ \frac{-400}{77} \right\} & \\\end{align}\)
11. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }- \frac{ 28 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x+\frac{546}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x-28& = & 91x+546 \\\Leftrightarrow & 26x \color{red}{-28} \color{blue}{+28} \color{blue}{-91x} & = & \color{red}{91x} +546 \color{blue}{-91x} \color{blue}{+28} \\\Leftrightarrow & -65x& = & 574 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{574}{-65} \\\Leftrightarrow & x = \frac{-574}{65} & & \\ & V = \left\{ \frac{-574}{65} \right\} & \\\end{align}\)
12. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{22}{ \color{blue}{110} }x+\frac{660}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-40& = & 22x+660 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{-22x} & = & \color{red}{22x} +660 \color{blue}{-22x} \color{blue}{+40} \\\Leftrightarrow & 33x& = & 700 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{700}{33} \\\Leftrightarrow & x = \frac{700}{33} & & \\ & V = \left\{ \frac{700}{33} \right\} & \\\end{align}\)