Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{-7}{4}x-4 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }- \frac{ 112 }{ \color{blue}{420} })& = & (\frac{-735}{ \color{blue}{420} }x-\frac{1680}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x-112& = & -735x-1680 \\\Leftrightarrow & 60x \color{red}{-112} \color{blue}{+112} \color{blue}{+735x} & = & \color{red}{-735x} -1680 \color{blue}{+735x} \color{blue}{+112} \\\Leftrightarrow & 795x& = & -1568 \\\Leftrightarrow & \frac{795x}{ \color{red}{795} }& = & \frac{-1568}{795} \\\Leftrightarrow & x = \frac{-1568}{795} & & \\ & V = \left\{ \frac{-1568}{795} \right\} & \\\end{align}\)
2. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }+ \frac{ 60 }{ \color{blue}{165} })& = & (\frac{-66}{ \color{blue}{165} }x+\frac{495}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x+60& = & -66x+495 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{+66x} & = & \color{red}{-66x} +495 \color{blue}{+66x} \color{blue}{-60} \\\Leftrightarrow & 121x& = & 435 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{435}{121} \\\Leftrightarrow & x = \frac{435}{121} & & \\ & V = \left\{ \frac{435}{121} \right\} & \\\end{align}\)
3. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 30 }{ \color{blue}{165} })& = & (\frac{33}{ \color{blue}{165} }x+\frac{1320}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-30& = & 33x+1320 \\\Leftrightarrow & 55x \color{red}{-30} \color{blue}{+30} \color{blue}{-33x} & = & \color{red}{33x} +1320 \color{blue}{-33x} \color{blue}{+30} \\\Leftrightarrow & 22x& = & 1350 \\\Leftrightarrow & \frac{22x}{ \color{red}{22} }& = & \frac{1350}{22} \\\Leftrightarrow & x = \frac{675}{11} & & \\ & V = \left\{ \frac{675}{11} \right\} & \\\end{align}\)
4. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{5}{6}x+6 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 84 }{ \color{blue}{546} })& = & (\frac{455}{ \color{blue}{546} }x+\frac{3276}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-84& = & 455x+3276 \\\Leftrightarrow & 78x \color{red}{-84} \color{blue}{+84} \color{blue}{-455x} & = & \color{red}{455x} +3276 \color{blue}{-455x} \color{blue}{+84} \\\Leftrightarrow & -377x& = & 3360 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{3360}{-377} \\\Leftrightarrow & x = \frac{-3360}{377} & & \\ & V = \left\{ \frac{-3360}{377} \right\} & \\\end{align}\)
5. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 2} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{56}{ \color{blue}{112} }x-\frac{672}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & 56x-672 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{-56x} & = & \color{red}{56x} -672 \color{blue}{-56x} \color{blue}{-35} \\\Leftrightarrow & -40x& = & -707 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{-707}{-40} \\\Leftrightarrow & x = \frac{707}{40} & & \\ & V = \left\{ \frac{707}{40} \right\} & \\\end{align}\)
6. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 50 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{220}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+50& = & 55x-220 \\\Leftrightarrow & 22x \color{red}{+50} \color{blue}{-50} \color{blue}{-55x} & = & \color{red}{55x} -220 \color{blue}{-55x} \color{blue}{-50} \\\Leftrightarrow & -33x& = & -270 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-270}{-33} \\\Leftrightarrow & x = \frac{90}{11} & & \\ & V = \left\{ \frac{90}{11} \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 40 }{ \color{blue}{140} })& = & (\frac{28}{ \color{blue}{140} }x+\frac{980}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-40& = & 28x+980 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{-28x} & = & \color{red}{28x} +980 \color{blue}{-28x} \color{blue}{+40} \\\Leftrightarrow & 7x& = & 1020 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{1020}{7} \\\Leftrightarrow & x = \frac{1020}{7} & & \\ & V = \left\{ \frac{1020}{7} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{-98}{ \color{blue}{70} }x+\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-20& = & -98x+420 \\\Leftrightarrow & 35x \color{red}{-20} \color{blue}{+20} \color{blue}{+98x} & = & \color{red}{-98x} +420 \color{blue}{+98x} \color{blue}{+20} \\\Leftrightarrow & 133x& = & 440 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{440}{133} \\\Leftrightarrow & x = \frac{440}{133} & & \\ & V = \left\{ \frac{440}{133} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 3, 6 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{6}& = & \frac{2}{5}x-5 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{12}{ \color{blue}{30} }x-\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+25& = & 12x-150 \\\Leftrightarrow & 10x \color{red}{+25} \color{blue}{-25} \color{blue}{-12x} & = & \color{red}{12x} -150 \color{blue}{-12x} \color{blue}{-25} \\\Leftrightarrow & -2x& = & -175 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-175}{-2} \\\Leftrightarrow & x = \frac{175}{2} & & \\ & V = \left\{ \frac{175}{2} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{7}& = & \frac{5}{3}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{70}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-30& = & 70x-126 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{-70x} & = & \color{red}{70x} -126 \color{blue}{-70x} \color{blue}{+30} \\\Leftrightarrow & -49x& = & -96 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-96}{-49} \\\Leftrightarrow & x = \frac{96}{49} & & \\ & V = \left\{ \frac{96}{49} \right\} & \\\end{align}\)
11. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{13}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 50 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{130}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-50& = & 65x-130 \\\Leftrightarrow & 26x \color{red}{-50} \color{blue}{+50} \color{blue}{-65x} & = & \color{red}{65x} -130 \color{blue}{-65x} \color{blue}{+50} \\\Leftrightarrow & -39x& = & -80 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-80}{-39} \\\Leftrightarrow & x = \frac{80}{39} & & \\ & V = \left\{ \frac{80}{39} \right\} & \\\end{align}\)
12. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 84 }{ \color{blue}{273} })& = & (\frac{-455}{ \color{blue}{273} }x+\frac{1911}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-84& = & -455x+1911 \\\Leftrightarrow & 39x \color{red}{-84} \color{blue}{+84} \color{blue}{+455x} & = & \color{red}{-455x} +1911 \color{blue}{+455x} \color{blue}{+84} \\\Leftrightarrow & 494x& = & 1995 \\\Leftrightarrow & \frac{494x}{ \color{red}{494} }& = & \frac{1995}{494} \\\Leftrightarrow & x = \frac{105}{26} & & \\ & V = \left\{ \frac{105}{26} \right\} & \\\end{align}\)