Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{7}& = & \frac{3}{2}x-2 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 30 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x-\frac{84}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+30& = & 63x-84 \\\Leftrightarrow & 14x \color{red}{+30} \color{blue}{-30} \color{blue}{-63x} & = & \color{red}{63x} -84 \color{blue}{-63x} \color{blue}{-30} \\\Leftrightarrow & -49x& = & -114 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-114}{-49} \\\Leftrightarrow & x = \frac{114}{49} & & \\ & V = \left\{ \frac{114}{49} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 42x-1050 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-42x} & = & \color{red}{42x} -1050 \color{blue}{-42x} \color{blue}{-120} \\\Leftrightarrow & -7x& = & -1170 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-1170}{-7} \\\Leftrightarrow & x = \frac{1170}{7} & & \\ & V = \left\{ \frac{1170}{7} \right\} & \\\end{align}\)
3. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{3}{13}& = & \frac{7}{6}x-3 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 126 }{ \color{blue}{546} })& = & (\frac{637}{ \color{blue}{546} }x-\frac{1638}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-126& = & 637x-1638 \\\Leftrightarrow & 78x \color{red}{-126} \color{blue}{+126} \color{blue}{-637x} & = & \color{red}{637x} -1638 \color{blue}{-637x} \color{blue}{+126} \\\Leftrightarrow & -559x& = & -1512 \\\Leftrightarrow & \frac{-559x}{ \color{red}{-559} }& = & \frac{-1512}{-559} \\\Leftrightarrow & x = \frac{1512}{559} & & \\ & V = \left\{ \frac{1512}{559} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{2}{5}x-5 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 60 }{ \color{blue}{140} })& = & (\frac{56}{ \color{blue}{140} }x-\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-60& = & 56x-700 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-56x} & = & \color{red}{56x} -700 \color{blue}{-56x} \color{blue}{+60} \\\Leftrightarrow & -21x& = & -640 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-640}{-21} \\\Leftrightarrow & x = \frac{640}{21} & & \\ & V = \left\{ \frac{640}{21} \right\} & \\\end{align}\)
5. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{5}{3}x+2 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{110}{ \color{blue}{66} }x+\frac{132}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+12& = & 110x+132 \\\Leftrightarrow & 33x \color{red}{+12} \color{blue}{-12} \color{blue}{-110x} & = & \color{red}{110x} +132 \color{blue}{-110x} \color{blue}{-12} \\\Leftrightarrow & -77x& = & 120 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{120}{-77} \\\Leftrightarrow & x = \frac{-120}{77} & & \\ & V = \left\{ \frac{-120}{77} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{7}& = & \frac{1}{3}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 60 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+60& = & 28x+252 \\\Leftrightarrow & 21x \color{red}{+60} \color{blue}{-60} \color{blue}{-28x} & = & \color{red}{28x} +252 \color{blue}{-28x} \color{blue}{-60} \\\Leftrightarrow & -7x& = & 192 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{192}{-7} \\\Leftrightarrow & x = \frac{-192}{7} & & \\ & V = \left\{ \frac{-192}{7} \right\} & \\\end{align}\)
7. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{1}{3}x+3 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 30 }{ \color{blue}{195} })& = & (\frac{65}{ \color{blue}{195} }x+\frac{585}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+30& = & 65x+585 \\\Leftrightarrow & 39x \color{red}{+30} \color{blue}{-30} \color{blue}{-65x} & = & \color{red}{65x} +585 \color{blue}{-65x} \color{blue}{-30} \\\Leftrightarrow & -26x& = & 555 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{555}{-26} \\\Leftrightarrow & x = \frac{-555}{26} & & \\ & V = \left\{ \frac{-555}{26} \right\} & \\\end{align}\)
8. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{16}& = & \frac{4}{5}x+4 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{192}{ \color{blue}{240} }x+\frac{960}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-75& = & 192x+960 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{-192x} & = & \color{red}{192x} +960 \color{blue}{-192x} \color{blue}{+75} \\\Leftrightarrow & -152x& = & 1035 \\\Leftrightarrow & \frac{-152x}{ \color{red}{-152} }& = & \frac{1035}{-152} \\\Leftrightarrow & x = \frac{-1035}{152} & & \\ & V = \left\{ \frac{-1035}{152} \right\} & \\\end{align}\)
9. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+24& = & 21x+168 \\\Leftrightarrow & 14x \color{red}{+24} \color{blue}{-24} \color{blue}{-21x} & = & \color{red}{21x} +168 \color{blue}{-21x} \color{blue}{-24} \\\Leftrightarrow & -7x& = & 144 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{144}{-7} \\\Leftrightarrow & x = \frac{-144}{7} & & \\ & V = \left\{ \frac{-144}{7} \right\} & \\\end{align}\)
10. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{11}& = & \frac{3}{4}x-1 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }+ \frac{ 60 }{ \color{blue}{220} })& = & (\frac{165}{ \color{blue}{220} }x-\frac{220}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x+60& = & 165x-220 \\\Leftrightarrow & 44x \color{red}{+60} \color{blue}{-60} \color{blue}{-165x} & = & \color{red}{165x} -220 \color{blue}{-165x} \color{blue}{-60} \\\Leftrightarrow & -121x& = & -280 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{-280}{-121} \\\Leftrightarrow & x = \frac{280}{121} & & \\ & V = \left\{ \frac{280}{121} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{7}{3}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & 196x-588 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-196x} & = & \color{red}{196x} -588 \color{blue}{-196x} \color{blue}{-24} \\\Leftrightarrow & -175x& = & -612 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{-612}{-175} \\\Leftrightarrow & x = \frac{612}{175} & & \\ & V = \left\{ \frac{612}{175} \right\} & \\\end{align}\)
12. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{5}{6}x+1 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }- \frac{ 84 }{ \color{blue}{546} })& = & (\frac{455}{ \color{blue}{546} }x+\frac{546}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x-84& = & 455x+546 \\\Leftrightarrow & 78x \color{red}{-84} \color{blue}{+84} \color{blue}{-455x} & = & \color{red}{455x} +546 \color{blue}{-455x} \color{blue}{+84} \\\Leftrightarrow & -377x& = & 630 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{630}{-377} \\\Leftrightarrow & x = \frac{-630}{377} & & \\ & V = \left\{ \frac{-630}{377} \right\} & \\\end{align}\)