Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-64}{ \color{blue}{80} }x+\frac{480}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-25& = & -64x+480 \\\Leftrightarrow & 40x \color{red}{-25} \color{blue}{+25} \color{blue}{+64x} & = & \color{red}{-64x} +480 \color{blue}{+64x} \color{blue}{+25} \\\Leftrightarrow & 104x& = & 505 \\\Leftrightarrow & \frac{104x}{ \color{red}{104} }& = & \frac{505}{104} \\\Leftrightarrow & x = \frac{505}{104} & & \\ & V = \left\{ \frac{505}{104} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{5}{6}x+3 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x+\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-120& = & 175x+630 \\\Leftrightarrow & 42x \color{red}{-120} \color{blue}{+120} \color{blue}{-175x} & = & \color{red}{175x} +630 \color{blue}{-175x} \color{blue}{+120} \\\Leftrightarrow & -133x& = & 750 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{750}{-133} \\\Leftrightarrow & x = \frac{-750}{133} & & \\ & V = \left\{ \frac{-750}{133} \right\} & \\\end{align}\)
3. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{11}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 18 }{ \color{blue}{66} })& = & (\frac{154}{ \color{blue}{66} }x-\frac{330}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+18& = & 154x-330 \\\Leftrightarrow & 33x \color{red}{+18} \color{blue}{-18} \color{blue}{-154x} & = & \color{red}{154x} -330 \color{blue}{-154x} \color{blue}{-18} \\\Leftrightarrow & -121x& = & -348 \\\Leftrightarrow & \frac{-121x}{ \color{red}{-121} }& = & \frac{-348}{-121} \\\Leftrightarrow & x = \frac{348}{121} & & \\ & V = \left\{ \frac{348}{121} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x-\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 168x-280 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-168x} & = & \color{red}{168x} -280 \color{blue}{-168x} \color{blue}{-80} \\\Leftrightarrow & -133x& = & -360 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-360}{-133} \\\Leftrightarrow & x = \frac{360}{133} & & \\ & V = \left\{ \frac{360}{133} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & 20x+360 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-20x} & = & \color{red}{20x} +360 \color{blue}{-20x} \color{blue}{-8} \\\Leftrightarrow & -5x& = & 352 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{352}{-5} \\\Leftrightarrow & x = \frac{-352}{5} & & \\ & V = \left\{ \frac{-352}{5} \right\} & \\\end{align}\)
6. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{4}{3}x+4 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 10 }{ \color{blue}{12} })& = & (\frac{16}{ \color{blue}{12} }x+\frac{48}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+10& = & 16x+48 \\\Leftrightarrow & 3x \color{red}{+10} \color{blue}{-10} \color{blue}{-16x} & = & \color{red}{16x} +48 \color{blue}{-16x} \color{blue}{-10} \\\Leftrightarrow & -13x& = & 38 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{38}{-13} \\\Leftrightarrow & x = \frac{-38}{13} & & \\ & V = \left\{ \frac{-38}{13} \right\} & \\\end{align}\)
7. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{5}{4}x+3 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 12 }{ \color{blue}{28} })& = & (\frac{35}{ \color{blue}{28} }x+\frac{84}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-12& = & 35x+84 \\\Leftrightarrow & 4x \color{red}{-12} \color{blue}{+12} \color{blue}{-35x} & = & \color{red}{35x} +84 \color{blue}{-35x} \color{blue}{+12} \\\Leftrightarrow & -31x& = & 96 \\\Leftrightarrow & \frac{-31x}{ \color{red}{-31} }& = & \frac{96}{-31} \\\Leftrightarrow & x = \frac{-96}{31} & & \\ & V = \left\{ \frac{-96}{31} \right\} & \\\end{align}\)
8. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }- \frac{ 40 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x-40& = & 65x-260 \\\Leftrightarrow & 26x \color{red}{-40} \color{blue}{+40} \color{blue}{-65x} & = & \color{red}{65x} -260 \color{blue}{-65x} \color{blue}{+40} \\\Leftrightarrow & -39x& = & -220 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-220}{-39} \\\Leftrightarrow & x = \frac{220}{39} & & \\ & V = \left\{ \frac{220}{39} \right\} & \\\end{align}\)
9. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{13}& = & \frac{5}{2}x+6 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 30 }{ \color{blue}{78} })& = & (\frac{195}{ \color{blue}{78} }x+\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-30& = & 195x+468 \\\Leftrightarrow & 26x \color{red}{-30} \color{blue}{+30} \color{blue}{-195x} & = & \color{red}{195x} +468 \color{blue}{-195x} \color{blue}{+30} \\\Leftrightarrow & -169x& = & 498 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{498}{-169} \\\Leftrightarrow & x = \frac{-498}{169} & & \\ & V = \left\{ \frac{-498}{169} \right\} & \\\end{align}\)
10. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 60 }{ \color{blue}{105} })& = & (\frac{-147}{ \color{blue}{105} }x+\frac{525}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-60& = & -147x+525 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+147x} & = & \color{red}{-147x} +525 \color{blue}{+147x} \color{blue}{+60} \\\Leftrightarrow & 182x& = & 585 \\\Leftrightarrow & \frac{182x}{ \color{red}{182} }& = & \frac{585}{182} \\\Leftrightarrow & x = \frac{45}{14} & & \\ & V = \left\{ \frac{45}{14} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -168x+1260 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+168x} & = & \color{red}{-168x} +1260 \color{blue}{+168x} \color{blue}{-120} \\\Leftrightarrow & 203x& = & 1140 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{1140}{203} \\\Leftrightarrow & x = \frac{1140}{203} & & \\ & V = \left\{ \frac{1140}{203} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{7}& = & \frac{-4}{5}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 50 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-50& = & -56x+490 \\\Leftrightarrow & 35x \color{red}{-50} \color{blue}{+50} \color{blue}{+56x} & = & \color{red}{-56x} +490 \color{blue}{+56x} \color{blue}{+50} \\\Leftrightarrow & 91x& = & 540 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{540}{91} \\\Leftrightarrow & x = \frac{540}{91} & & \\ & V = \left\{ \frac{540}{91} \right\} & \\\end{align}\)