Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x+\frac{640}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+15& = & 40x+640 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-40x} & = & \color{red}{40x} +640 \color{blue}{-40x} \color{blue}{-15} \\\Leftrightarrow & -24x& = & 625 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{625}{-24} \\\Leftrightarrow & x = \frac{-625}{24} & & \\ & V = \left\{ \frac{-625}{24} \right\} & \\\end{align}\)
2. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{11}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 60 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x+\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+60& = & -462x+330 \\\Leftrightarrow & 55x \color{red}{+60} \color{blue}{-60} \color{blue}{+462x} & = & \color{red}{-462x} +330 \color{blue}{+462x} \color{blue}{-60} \\\Leftrightarrow & 517x& = & 270 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{270}{517} \\\Leftrightarrow & x = \frac{270}{517} & & \\ & V = \left\{ \frac{270}{517} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 14 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{7}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-9& = & 98x-168 \\\Leftrightarrow & 21x \color{red}{-9} \color{blue}{+9} \color{blue}{-98x} & = & \color{red}{98x} -168 \color{blue}{-98x} \color{blue}{+9} \\\Leftrightarrow & -77x& = & -159 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-159}{-77} \\\Leftrightarrow & x = \frac{159}{77} & & \\ & V = \left\{ \frac{159}{77} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{7}{6}x+7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{245}{ \color{blue}{210} }x+\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+90& = & 245x+1470 \\\Leftrightarrow & 42x \color{red}{+90} \color{blue}{-90} \color{blue}{-245x} & = & \color{red}{245x} +1470 \color{blue}{-245x} \color{blue}{-90} \\\Leftrightarrow & -203x& = & 1380 \\\Leftrightarrow & \frac{-203x}{ \color{red}{-203} }& = & \frac{1380}{-203} \\\Leftrightarrow & x = \frac{-1380}{203} & & \\ & V = \left\{ \frac{-1380}{203} \right\} & \\\end{align}\)
5. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x+\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & 396x+330 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{-396x} & = & \color{red}{396x} +330 \color{blue}{-396x} \color{blue}{-120} \\\Leftrightarrow & -341x& = & 210 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{210}{-341} \\\Leftrightarrow & x = \frac{-210}{341} & & \\ & V = \left\{ \frac{-210}{341} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{12}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & 12x-240 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-12x} & = & \color{red}{12x} -240 \color{blue}{-12x} \color{blue}{+8} \\\Leftrightarrow & 3x& = & -232 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-232}{3} \\\Leftrightarrow & x = \frac{-232}{3} & & \\ & V = \left\{ \frac{-232}{3} \right\} & \\\end{align}\)
7. \(\text{12 is het kleinste gemene veelvoud van 3, 6 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{6}& = & \frac{-7}{4}x+8 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 10 }{ \color{blue}{12} })& = & (\frac{-21}{ \color{blue}{12} }x+\frac{96}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+10& = & -21x+96 \\\Leftrightarrow & 4x \color{red}{+10} \color{blue}{-10} \color{blue}{+21x} & = & \color{red}{-21x} +96 \color{blue}{+21x} \color{blue}{-10} \\\Leftrightarrow & 25x& = & 86 \\\Leftrightarrow & \frac{25x}{ \color{red}{25} }& = & \frac{86}{25} \\\Leftrightarrow & x = \frac{86}{25} & & \\ & V = \left\{ \frac{86}{25} \right\} & \\\end{align}\)
8. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}-\frac{2}{9}& = & \frac{-7}{4}x+1 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }- \frac{ 56 }{ \color{blue}{252} })& = & (\frac{-441}{ \color{blue}{252} }x+\frac{252}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x-56& = & -441x+252 \\\Leftrightarrow & 36x \color{red}{-56} \color{blue}{+56} \color{blue}{+441x} & = & \color{red}{-441x} +252 \color{blue}{+441x} \color{blue}{+56} \\\Leftrightarrow & 477x& = & 308 \\\Leftrightarrow & \frac{477x}{ \color{red}{477} }& = & \frac{308}{477} \\\Leftrightarrow & x = \frac{308}{477} & & \\ & V = \left\{ \frac{308}{477} \right\} & \\\end{align}\)
9. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{11}& = & \frac{1}{5}x-6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x-\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+150& = & 66x-1980 \\\Leftrightarrow & 55x \color{red}{+150} \color{blue}{-150} \color{blue}{-66x} & = & \color{red}{66x} -1980 \color{blue}{-66x} \color{blue}{-150} \\\Leftrightarrow & -11x& = & -2130 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-2130}{-11} \\\Leftrightarrow & x = \frac{2130}{11} & & \\ & V = \left\{ \frac{2130}{11} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{3}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-24& = & 63x-252 \\\Leftrightarrow & 14x \color{red}{-24} \color{blue}{+24} \color{blue}{-63x} & = & \color{red}{63x} -252 \color{blue}{-63x} \color{blue}{+24} \\\Leftrightarrow & -49x& = & -228 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-228}{-49} \\\Leftrightarrow & x = \frac{228}{49} & & \\ & V = \left\{ \frac{228}{49} \right\} & \\\end{align}\)
11. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{11}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 18 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+18& = & 33x+396 \\\Leftrightarrow & 22x \color{red}{+18} \color{blue}{-18} \color{blue}{-33x} & = & \color{red}{33x} +396 \color{blue}{-33x} \color{blue}{-18} \\\Leftrightarrow & -11x& = & 378 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{378}{-11} \\\Leftrightarrow & x = \frac{-378}{11} & & \\ & V = \left\{ \frac{-378}{11} \right\} & \\\end{align}\)
12. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{-4}{5}x+5 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x+\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & -72x+450 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{+72x} & = & \color{red}{-72x} +450 \color{blue}{+72x} \color{blue}{+40} \\\Leftrightarrow & 87x& = & 490 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{490}{87} \\\Leftrightarrow & x = \frac{490}{87} & & \\ & V = \left\{ \frac{490}{87} \right\} & \\\end{align}\)