Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{3}-\frac{2}{15}=\frac{1}{2}x-3\)
2. \(\frac{x}{6}-\frac{4}{7}=\frac{6}{5}x-1\)
3. \(\frac{x}{7}+\frac{3}{14}=\frac{7}{2}x-6\)
4. \(\frac{x}{7}-\frac{3}{7}=\frac{1}{2}x-7\)
5. \(\frac{x}{2}+\frac{5}{12}=\frac{-7}{5}x-2\)
6. \(\frac{x}{2}+\frac{3}{10}=\frac{-2}{5}x+2\)
7. \(\frac{x}{4}+\frac{3}{14}=\frac{-8}{3}x+5\)
8. \(\frac{x}{4}+\frac{3}{16}=\frac{1}{5}x-2\)
9. \(\frac{x}{3}-\frac{4}{9}=\frac{1}{2}x-6\)
10. \(\frac{x}{7}-\frac{2}{9}=\frac{-7}{4}x-3\)
11. \(\frac{x}{5}-\frac{3}{7}=\frac{-5}{6}x-3\)
12. \(\frac{x}{6}-\frac{2}{11}=\frac{-7}{5}x+3\)

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-4& = & 15x-90 \\\Leftrightarrow & 10x \color{red}{-4} \color{blue}{+4} \color{blue}{-15x} & = & \color{red}{15x} -90 \color{blue}{-15x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & -86 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-86}{-5} \\\Leftrightarrow & x = \frac{86}{5} & & \\ & V = \left\{ \frac{86}{5} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 252x-210 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-252x} & = & \color{red}{252x} -210 \color{blue}{-252x} \color{blue}{+120} \\\Leftrightarrow & -217x& = & -90 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-90}{-217} \\\Leftrightarrow & x = \frac{90}{217} & & \\ & V = \left\{ \frac{90}{217} \right\} & \\\end{align}\)
3. \(\text{14 is het kleinste gemene veelvoud van 7, 14 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{14}& = & \frac{7}{2}x-6 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 3 }{ \color{blue}{14} })& = & (\frac{49}{ \color{blue}{14} }x-\frac{84}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+3& = & 49x-84 \\\Leftrightarrow & 2x \color{red}{+3} \color{blue}{-3} \color{blue}{-49x} & = & \color{red}{49x} -84 \color{blue}{-49x} \color{blue}{-3} \\\Leftrightarrow & -47x& = & -87 \\\Leftrightarrow & \frac{-47x}{ \color{red}{-47} }& = & \frac{-87}{-47} \\\Leftrightarrow & x = \frac{87}{47} & & \\ & V = \left\{ \frac{87}{47} \right\} & \\\end{align}\)
4. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 6 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{98}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-6& = & 7x-98 \\\Leftrightarrow & 2x \color{red}{-6} \color{blue}{+6} \color{blue}{-7x} & = & \color{red}{7x} -98 \color{blue}{-7x} \color{blue}{+6} \\\Leftrightarrow & -5x& = & -92 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-92}{-5} \\\Leftrightarrow & x = \frac{92}{5} & & \\ & V = \left\{ \frac{92}{5} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 2, 12 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{12}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{60.} (\frac{30x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-84}{ \color{blue}{60} }x-\frac{120}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 30x+25& = & -84x-120 \\\Leftrightarrow & 30x \color{red}{+25} \color{blue}{-25} \color{blue}{+84x} & = & \color{red}{-84x} -120 \color{blue}{+84x} \color{blue}{-25} \\\Leftrightarrow & 114x& = & -145 \\\Leftrightarrow & \frac{114x}{ \color{red}{114} }& = & \frac{-145}{114} \\\Leftrightarrow & x = \frac{-145}{114} & & \\ & V = \left\{ \frac{-145}{114} \right\} & \\\end{align}\)
6. \(\text{10 is het kleinste gemene veelvoud van 2, 10 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{10}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{10.} (\frac{5x}{ \color{blue}{10} }+ \frac{ 3 }{ \color{blue}{10} })& = & (\frac{-4}{ \color{blue}{10} }x+\frac{20}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 5x+3& = & -4x+20 \\\Leftrightarrow & 5x \color{red}{+3} \color{blue}{-3} \color{blue}{+4x} & = & \color{red}{-4x} +20 \color{blue}{+4x} \color{blue}{-3} \\\Leftrightarrow & 9x& = & 17 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{17}{9} \\\Leftrightarrow & x = \frac{17}{9} & & \\ & V = \left\{ \frac{17}{9} \right\} & \\\end{align}\)
7. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{14}& = & \frac{-8}{3}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 18 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+18& = & -224x+420 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{+224x} & = & \color{red}{-224x} +420 \color{blue}{+224x} \color{blue}{-18} \\\Leftrightarrow & 245x& = & 402 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{402}{245} \\\Leftrightarrow & x = \frac{402}{245} & & \\ & V = \left\{ \frac{402}{245} \right\} & \\\end{align}\)
8. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{16}{ \color{blue}{80} }x-\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+15& = & 16x-160 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{-16x} & = & \color{red}{16x} -160 \color{blue}{-16x} \color{blue}{-15} \\\Leftrightarrow & 4x& = & -175 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{-175}{4} \\\Leftrightarrow & x = \frac{-175}{4} & & \\ & V = \left\{ \frac{-175}{4} \right\} & \\\end{align}\)
9. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{108}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x-8& = & 9x-108 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-9x} & = & \color{red}{9x} -108 \color{blue}{-9x} \color{blue}{+8} \\\Leftrightarrow & -3x& = & -100 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-100}{-3} \\\Leftrightarrow & x = \frac{100}{3} & & \\ & V = \left\{ \frac{100}{3} \right\} & \\\end{align}\)
10. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{-7}{4}x-3 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }- \frac{ 56 }{ \color{blue}{252} })& = & (\frac{-441}{ \color{blue}{252} }x-\frac{756}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x-56& = & -441x-756 \\\Leftrightarrow & 36x \color{red}{-56} \color{blue}{+56} \color{blue}{+441x} & = & \color{red}{-441x} -756 \color{blue}{+441x} \color{blue}{+56} \\\Leftrightarrow & 477x& = & -700 \\\Leftrightarrow & \frac{477x}{ \color{red}{477} }& = & \frac{-700}{477} \\\Leftrightarrow & x = \frac{-700}{477} & & \\ & V = \left\{ \frac{-700}{477} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{-5}{6}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-90& = & -175x-630 \\\Leftrightarrow & 42x \color{red}{-90} \color{blue}{+90} \color{blue}{+175x} & = & \color{red}{-175x} -630 \color{blue}{+175x} \color{blue}{+90} \\\Leftrightarrow & 217x& = & -540 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{-540}{217} \\\Leftrightarrow & x = \frac{-540}{217} & & \\ & V = \left\{ \frac{-540}{217} \right\} & \\\end{align}\)
12. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{11}& = & \frac{-7}{5}x+3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x+\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-60& = & -462x+990 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{+462x} & = & \color{red}{-462x} +990 \color{blue}{+462x} \color{blue}{+60} \\\Leftrightarrow & 517x& = & 1050 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{1050}{517} \\\Leftrightarrow & x = \frac{1050}{517} & & \\ & V = \left\{ \frac{1050}{517} \right\} & \\\end{align}\)