Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{3}-\frac{3}{7}=\frac{1}{4}x+8\)
2. \(\frac{x}{5}+\frac{5}{16}=\frac{3}{4}x+2\)
3. \(\frac{x}{6}-\frac{2}{13}=\frac{-4}{5}x-4\)
4. \(\frac{x}{7}+\frac{2}{7}=\frac{7}{2}x-3\)
5. \(\frac{x}{5}-\frac{3}{13}=\frac{-8}{3}x-3\)
6. \(\frac{x}{7}-\frac{5}{9}=\frac{1}{2}x+2\)
7. \(\frac{x}{4}+\frac{3}{16}=\frac{7}{5}x+8\)
8. \(\frac{x}{4}-\frac{2}{9}=\frac{-2}{3}x-1\)
9. \(\frac{x}{7}+\frac{5}{16}=\frac{5}{4}x+5\)
10. \(\frac{x}{2}+\frac{5}{7}=\frac{5}{3}x+4\)
11. \(\frac{x}{6}+\frac{3}{14}=\frac{-7}{5}x+4\)
12. \(\frac{x}{7}+\frac{4}{13}=\frac{3}{2}x+3\)

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{4}x+8 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & 21x+672 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{-21x} & = & \color{red}{21x} +672 \color{blue}{-21x} \color{blue}{+36} \\\Leftrightarrow & 7x& = & 708 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{708}{7} \\\Leftrightarrow & x = \frac{708}{7} & & \\ & V = \left\{ \frac{708}{7} \right\} & \\\end{align}\)
2. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 4} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{3}{4}x+2 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{60}{ \color{blue}{80} }x+\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+25& = & 60x+160 \\\Leftrightarrow & 16x \color{red}{+25} \color{blue}{-25} \color{blue}{-60x} & = & \color{red}{60x} +160 \color{blue}{-60x} \color{blue}{-25} \\\Leftrightarrow & -44x& = & 135 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{135}{-44} \\\Leftrightarrow & x = \frac{-135}{44} & & \\ & V = \left\{ \frac{-135}{44} \right\} & \\\end{align}\)
3. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{-4}{5}x-4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x-\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & -312x-1560 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{+312x} & = & \color{red}{-312x} -1560 \color{blue}{+312x} \color{blue}{+60} \\\Leftrightarrow & 377x& = & -1500 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{-1500}{377} \\\Leftrightarrow & x = \frac{-1500}{377} & & \\ & V = \left\{ \frac{-1500}{377} \right\} & \\\end{align}\)
4. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{7}{2}x-3 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 4 }{ \color{blue}{14} })& = & (\frac{49}{ \color{blue}{14} }x-\frac{42}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+4& = & 49x-42 \\\Leftrightarrow & 2x \color{red}{+4} \color{blue}{-4} \color{blue}{-49x} & = & \color{red}{49x} -42 \color{blue}{-49x} \color{blue}{-4} \\\Leftrightarrow & -47x& = & -46 \\\Leftrightarrow & \frac{-47x}{ \color{red}{-47} }& = & \frac{-46}{-47} \\\Leftrightarrow & x = \frac{46}{47} & & \\ & V = \left\{ \frac{46}{47} \right\} & \\\end{align}\)
5. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{-8}{3}x-3 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }- \frac{ 45 }{ \color{blue}{195} })& = & (\frac{-520}{ \color{blue}{195} }x-\frac{585}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x-45& = & -520x-585 \\\Leftrightarrow & 39x \color{red}{-45} \color{blue}{+45} \color{blue}{+520x} & = & \color{red}{-520x} -585 \color{blue}{+520x} \color{blue}{+45} \\\Leftrightarrow & 559x& = & -540 \\\Leftrightarrow & \frac{559x}{ \color{red}{559} }& = & \frac{-540}{559} \\\Leftrightarrow & x = \frac{-540}{559} & & \\ & V = \left\{ \frac{-540}{559} \right\} & \\\end{align}\)
6. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{9}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 70 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x+\frac{252}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-70& = & 63x+252 \\\Leftrightarrow & 18x \color{red}{-70} \color{blue}{+70} \color{blue}{-63x} & = & \color{red}{63x} +252 \color{blue}{-63x} \color{blue}{+70} \\\Leftrightarrow & -45x& = & 322 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{322}{-45} \\\Leftrightarrow & x = \frac{-322}{45} & & \\ & V = \left\{ \frac{-322}{45} \right\} & \\\end{align}\)
7. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{7}{5}x+8 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{112}{ \color{blue}{80} }x+\frac{640}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+15& = & 112x+640 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{-112x} & = & \color{red}{112x} +640 \color{blue}{-112x} \color{blue}{-15} \\\Leftrightarrow & -92x& = & 625 \\\Leftrightarrow & \frac{-92x}{ \color{red}{-92} }& = & \frac{625}{-92} \\\Leftrightarrow & x = \frac{-625}{92} & & \\ & V = \left\{ \frac{-625}{92} \right\} & \\\end{align}\)
8. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{9}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 8 }{ \color{blue}{36} })& = & (\frac{-24}{ \color{blue}{36} }x-\frac{36}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-8& = & -24x-36 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{+24x} & = & \color{red}{-24x} -36 \color{blue}{+24x} \color{blue}{+8} \\\Leftrightarrow & 33x& = & -28 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-28}{33} \\\Leftrightarrow & x = \frac{-28}{33} & & \\ & V = \left\{ \frac{-28}{33} \right\} & \\\end{align}\)
9. \(\text{112 is het kleinste gemene veelvoud van 7, 16 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{16}& = & \frac{5}{4}x+5 \\\Leftrightarrow & \color{blue}{112.} (\frac{16x}{ \color{blue}{112} }+ \frac{ 35 }{ \color{blue}{112} })& = & (\frac{140}{ \color{blue}{112} }x+\frac{560}{ \color{blue}{112} }) \color{blue}{.112} \\\Leftrightarrow & 16x+35& = & 140x+560 \\\Leftrightarrow & 16x \color{red}{+35} \color{blue}{-35} \color{blue}{-140x} & = & \color{red}{140x} +560 \color{blue}{-140x} \color{blue}{-35} \\\Leftrightarrow & -124x& = & 525 \\\Leftrightarrow & \frac{-124x}{ \color{red}{-124} }& = & \frac{525}{-124} \\\Leftrightarrow & x = \frac{-525}{124} & & \\ & V = \left\{ \frac{-525}{124} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{7}& = & \frac{5}{3}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 30 }{ \color{blue}{42} })& = & (\frac{70}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+30& = & 70x+168 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{-70x} & = & \color{red}{70x} +168 \color{blue}{-70x} \color{blue}{-30} \\\Leftrightarrow & -49x& = & 138 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{138}{-49} \\\Leftrightarrow & x = \frac{-138}{49} & & \\ & V = \left\{ \frac{-138}{49} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{14}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+45& = & -294x+840 \\\Leftrightarrow & 35x \color{red}{+45} \color{blue}{-45} \color{blue}{+294x} & = & \color{red}{-294x} +840 \color{blue}{+294x} \color{blue}{-45} \\\Leftrightarrow & 329x& = & 795 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{795}{329} \\\Leftrightarrow & x = \frac{795}{329} & & \\ & V = \left\{ \frac{795}{329} \right\} & \\\end{align}\)
12. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{3}{2}x+3 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }+ \frac{ 56 }{ \color{blue}{182} })& = & (\frac{273}{ \color{blue}{182} }x+\frac{546}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x+56& = & 273x+546 \\\Leftrightarrow & 26x \color{red}{+56} \color{blue}{-56} \color{blue}{-273x} & = & \color{red}{273x} +546 \color{blue}{-273x} \color{blue}{-56} \\\Leftrightarrow & -247x& = & 490 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{490}{-247} \\\Leftrightarrow & x = \frac{-490}{247} & & \\ & V = \left\{ \frac{-490}{247} \right\} & \\\end{align}\)