Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{308 is het kleinste gemene veelvoud van 7, 11 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{1}{4}x-2 \\\Leftrightarrow & \color{blue}{308.} (\frac{44x}{ \color{blue}{308} }+ \frac{ 112 }{ \color{blue}{308} })& = & (\frac{77}{ \color{blue}{308} }x-\frac{616}{ \color{blue}{308} }) \color{blue}{.308} \\\Leftrightarrow & 44x+112& = & 77x-616 \\\Leftrightarrow & 44x \color{red}{+112} \color{blue}{-112} \color{blue}{-77x} & = & \color{red}{77x} -616 \color{blue}{-77x} \color{blue}{-112} \\\Leftrightarrow & -33x& = & -728 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-728}{-33} \\\Leftrightarrow & x = \frac{728}{33} & & \\ & V = \left\{ \frac{728}{33} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{7}{6}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{35}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+4& = & 35x+150 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-35x} & = & \color{red}{35x} +150 \color{blue}{-35x} \color{blue}{-4} \\\Leftrightarrow & -29x& = & 146 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{146}{-29} \\\Leftrightarrow & x = \frac{-146}{29} & & \\ & V = \left\{ \frac{-146}{29} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{14}& = & \frac{7}{6}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{245}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+75& = & 245x+420 \\\Leftrightarrow & 42x \color{red}{+75} \color{blue}{-75} \color{blue}{-245x} & = & \color{red}{245x} +420 \color{blue}{-245x} \color{blue}{-75} \\\Leftrightarrow & -203x& = & 345 \\\Leftrightarrow & \frac{-203x}{ \color{red}{-203} }& = & \frac{345}{-203} \\\Leftrightarrow & x = \frac{-345}{203} & & \\ & V = \left\{ \frac{-345}{203} \right\} & \\\end{align}\)
4. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{15}& = & \frac{5}{3}x-8 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{100}{ \color{blue}{60} }x-\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-8& = & 100x-480 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-100x} & = & \color{red}{100x} -480 \color{blue}{-100x} \color{blue}{+8} \\\Leftrightarrow & -85x& = & -472 \\\Leftrightarrow & \frac{-85x}{ \color{red}{-85} }& = & \frac{-472}{-85} \\\Leftrightarrow & x = \frac{472}{85} & & \\ & V = \left\{ \frac{472}{85} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{6}{5}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 36x-210 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-36x} & = & \color{red}{36x} -210 \color{blue}{-36x} \color{blue}{-9} \\\Leftrightarrow & -31x& = & -219 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{-219}{-31} \\\Leftrightarrow & x = \frac{219}{31} & & \\ & V = \left\{ \frac{219}{31} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-8}{3}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & -112x+168 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+112x} & = & \color{red}{-112x} +168 \color{blue}{+112x} \color{blue}{+18} \\\Leftrightarrow & 133x& = & 186 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{186}{133} \\\Leftrightarrow & x = \frac{186}{133} & & \\ & V = \left\{ \frac{186}{133} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{7}& = & \frac{-5}{6}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 150 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-150& = & -175x+630 \\\Leftrightarrow & 42x \color{red}{-150} \color{blue}{+150} \color{blue}{+175x} & = & \color{red}{-175x} +630 \color{blue}{+175x} \color{blue}{+150} \\\Leftrightarrow & 217x& = & 780 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{780}{217} \\\Leftrightarrow & x = \frac{780}{217} & & \\ & V = \left\{ \frac{780}{217} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-8}{3}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -112x+252 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+112x} & = & \color{red}{-112x} +252 \color{blue}{+112x} \color{blue}{-24} \\\Leftrightarrow & 133x& = & 228 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{228}{133} \\\Leftrightarrow & x = \frac{12}{7} & & \\ & V = \left\{ \frac{12}{7} \right\} & \\\end{align}\)
9. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{7}{4}x-8 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{455}{ \color{blue}{260} }x-\frac{2080}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-80& = & 455x-2080 \\\Leftrightarrow & 52x \color{red}{-80} \color{blue}{+80} \color{blue}{-455x} & = & \color{red}{455x} -2080 \color{blue}{-455x} \color{blue}{+80} \\\Leftrightarrow & -403x& = & -2000 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{-2000}{-403} \\\Leftrightarrow & x = \frac{2000}{403} & & \\ & V = \left\{ \frac{2000}{403} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{-8}{3}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-160}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+18& = & -160x+360 \\\Leftrightarrow & 15x \color{red}{+18} \color{blue}{-18} \color{blue}{+160x} & = & \color{red}{-160x} +360 \color{blue}{+160x} \color{blue}{-18} \\\Leftrightarrow & 175x& = & 342 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{342}{175} \\\Leftrightarrow & x = \frac{342}{175} & & \\ & V = \left\{ \frac{342}{175} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 5, 12 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{12}& = & \frac{1}{6}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{10}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-25& = & 10x+420 \\\Leftrightarrow & 12x \color{red}{-25} \color{blue}{+25} \color{blue}{-10x} & = & \color{red}{10x} +420 \color{blue}{-10x} \color{blue}{+25} \\\Leftrightarrow & 2x& = & 445 \\\Leftrightarrow & \frac{2x}{ \color{red}{2} }& = & \frac{445}{2} \\\Leftrightarrow & x = \frac{445}{2} & & \\ & V = \left\{ \frac{445}{2} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x-\frac{320}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-25& = & 40x-320 \\\Leftrightarrow & 16x \color{red}{-25} \color{blue}{+25} \color{blue}{-40x} & = & \color{red}{40x} -320 \color{blue}{-40x} \color{blue}{+25} \\\Leftrightarrow & -24x& = & -295 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{-295}{-24} \\\Leftrightarrow & x = \frac{295}{24} & & \\ & V = \left\{ \frac{295}{24} \right\} & \\\end{align}\)