Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}-\frac{4}{15}=\frac{7}{5}x+6\)
2. \(\frac{x}{5}+\frac{2}{13}=\frac{1}{6}x+8\)
3. \(\frac{x}{2}+\frac{5}{16}=\frac{1}{5}x-3\)
4. \(\frac{x}{4}+\frac{3}{11}=\frac{4}{3}x-8\)
5. \(\frac{x}{2}-\frac{2}{11}=\frac{5}{3}x+1\)
6. \(\frac{x}{3}-\frac{4}{7}=\frac{1}{2}x-6\)
7. \(\frac{x}{2}-\frac{3}{7}=\frac{4}{3}x-4\)
8. \(\frac{x}{7}-\frac{4}{11}=\frac{3}{2}x+4\)
9. \(\frac{x}{6}-\frac{5}{14}=\frac{4}{5}x-6\)
10. \(\frac{x}{6}+\frac{5}{16}=\frac{-2}{5}x+1\)
11. \(\frac{x}{4}-\frac{4}{15}=\frac{1}{3}x+5\)
12. \(\frac{x}{4}+\frac{5}{12}=\frac{-7}{5}x-4\)

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{7}{5}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{42}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & 42x+180 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-42x} & = & \color{red}{42x} +180 \color{blue}{-42x} \color{blue}{+8} \\\Leftrightarrow & -27x& = & 188 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{188}{-27} \\\Leftrightarrow & x = \frac{-188}{27} & & \\ & V = \left\{ \frac{-188}{27} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{1}{6}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+60& = & 65x+3120 \\\Leftrightarrow & 78x \color{red}{+60} \color{blue}{-60} \color{blue}{-65x} & = & \color{red}{65x} +3120 \color{blue}{-65x} \color{blue}{-60} \\\Leftrightarrow & 13x& = & 3060 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{3060}{13} \\\Leftrightarrow & x = \frac{3060}{13} & & \\ & V = \left\{ \frac{3060}{13} \right\} & \\\end{align}\)
3. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{16}{ \color{blue}{80} }x-\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+25& = & 16x-240 \\\Leftrightarrow & 40x \color{red}{+25} \color{blue}{-25} \color{blue}{-16x} & = & \color{red}{16x} -240 \color{blue}{-16x} \color{blue}{-25} \\\Leftrightarrow & 24x& = & -265 \\\Leftrightarrow & \frac{24x}{ \color{red}{24} }& = & \frac{-265}{24} \\\Leftrightarrow & x = \frac{-265}{24} & & \\ & V = \left\{ \frac{-265}{24} \right\} & \\\end{align}\)
4. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{11}& = & \frac{4}{3}x-8 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 36 }{ \color{blue}{132} })& = & (\frac{176}{ \color{blue}{132} }x-\frac{1056}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+36& = & 176x-1056 \\\Leftrightarrow & 33x \color{red}{+36} \color{blue}{-36} \color{blue}{-176x} & = & \color{red}{176x} -1056 \color{blue}{-176x} \color{blue}{-36} \\\Leftrightarrow & -143x& = & -1092 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-1092}{-143} \\\Leftrightarrow & x = \frac{84}{11} & & \\ & V = \left\{ \frac{84}{11} \right\} & \\\end{align}\)
5. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{5}{3}x+1 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{110}{ \color{blue}{66} }x+\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & 110x+66 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{-110x} & = & \color{red}{110x} +66 \color{blue}{-110x} \color{blue}{+12} \\\Leftrightarrow & -77x& = & 78 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{78}{-77} \\\Leftrightarrow & x = \frac{-78}{77} & & \\ & V = \left\{ \frac{-78}{77} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 21x-252 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} -252 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & -228 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-228}{-7} \\\Leftrightarrow & x = \frac{228}{7} & & \\ & V = \left\{ \frac{228}{7} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{4}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & 56x-168 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-56x} & = & \color{red}{56x} -168 \color{blue}{-56x} \color{blue}{+18} \\\Leftrightarrow & -35x& = & -150 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-150}{-35} \\\Leftrightarrow & x = \frac{30}{7} & & \\ & V = \left\{ \frac{30}{7} \right\} & \\\end{align}\)
8. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{11}& = & \frac{3}{2}x+4 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 56 }{ \color{blue}{154} })& = & (\frac{231}{ \color{blue}{154} }x+\frac{616}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-56& = & 231x+616 \\\Leftrightarrow & 22x \color{red}{-56} \color{blue}{+56} \color{blue}{-231x} & = & \color{red}{231x} +616 \color{blue}{-231x} \color{blue}{+56} \\\Leftrightarrow & -209x& = & 672 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{672}{-209} \\\Leftrightarrow & x = \frac{-672}{209} & & \\ & V = \left\{ \frac{-672}{209} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{14}& = & \frac{4}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-75& = & 168x-1260 \\\Leftrightarrow & 35x \color{red}{-75} \color{blue}{+75} \color{blue}{-168x} & = & \color{red}{168x} -1260 \color{blue}{-168x} \color{blue}{+75} \\\Leftrightarrow & -133x& = & -1185 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-1185}{-133} \\\Leftrightarrow & x = \frac{1185}{133} & & \\ & V = \left\{ \frac{1185}{133} \right\} & \\\end{align}\)
10. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{-2}{5}x+1 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{-96}{ \color{blue}{240} }x+\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & -96x+240 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{+96x} & = & \color{red}{-96x} +240 \color{blue}{+96x} \color{blue}{-75} \\\Leftrightarrow & 136x& = & 165 \\\Leftrightarrow & \frac{136x}{ \color{red}{136} }& = & \frac{165}{136} \\\Leftrightarrow & x = \frac{165}{136} & & \\ & V = \left\{ \frac{165}{136} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x+\frac{300}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 20x+300 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-20x} & = & \color{red}{20x} +300 \color{blue}{-20x} \color{blue}{+16} \\\Leftrightarrow & -5x& = & 316 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{316}{-5} \\\Leftrightarrow & x = \frac{-316}{5} & & \\ & V = \left\{ \frac{-316}{5} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 4, 12 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{12}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-84}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+25& = & -84x-240 \\\Leftrightarrow & 15x \color{red}{+25} \color{blue}{-25} \color{blue}{+84x} & = & \color{red}{-84x} -240 \color{blue}{+84x} \color{blue}{-25} \\\Leftrightarrow & 99x& = & -265 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{-265}{99} \\\Leftrightarrow & x = \frac{-265}{99} & & \\ & V = \left\{ \frac{-265}{99} \right\} & \\\end{align}\)