Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{7}{2}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{245}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+30& = & 245x-350 \\\Leftrightarrow & 14x \color{red}{+30} \color{blue}{-30} \color{blue}{-245x} & = & \color{red}{245x} -350 \color{blue}{-245x} \color{blue}{-30} \\\Leftrightarrow & -231x& = & -380 \\\Leftrightarrow & \frac{-231x}{ \color{red}{-231} }& = & \frac{-380}{-231} \\\Leftrightarrow & x = \frac{380}{231} & & \\ & V = \left\{ \frac{380}{231} \right\} & \\\end{align}\)
2. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{8}& = & \frac{4}{3}x+6 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{32}{ \color{blue}{24} }x+\frac{144}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-15& = & 32x+144 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{-32x} & = & \color{red}{32x} +144 \color{blue}{-32x} \color{blue}{+15} \\\Leftrightarrow & -20x& = & 159 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{159}{-20} \\\Leftrightarrow & x = \frac{-159}{20} & & \\ & V = \left\{ \frac{-159}{20} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x+\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & -140x+672 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+140x} & = & \color{red}{-140x} +672 \color{blue}{+140x} \color{blue}{+24} \\\Leftrightarrow & 161x& = & 696 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{696}{161} \\\Leftrightarrow & x = \frac{696}{161} & & \\ & V = \left\{ \frac{696}{161} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -294x-420 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+294x} & = & \color{red}{-294x} -420 \color{blue}{+294x} \color{blue}{-60} \\\Leftrightarrow & 329x& = & -480 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-480}{329} \\\Leftrightarrow & x = \frac{-480}{329} & & \\ & V = \left\{ \frac{-480}{329} \right\} & \\\end{align}\)
5. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{-5}{6}x+6 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 28 }{ \color{blue}{126} })& = & (\frac{-105}{ \color{blue}{126} }x+\frac{756}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+28& = & -105x+756 \\\Leftrightarrow & 18x \color{red}{+28} \color{blue}{-28} \color{blue}{+105x} & = & \color{red}{-105x} +756 \color{blue}{+105x} \color{blue}{-28} \\\Leftrightarrow & 123x& = & 728 \\\Leftrightarrow & \frac{123x}{ \color{red}{123} }& = & \frac{728}{123} \\\Leftrightarrow & x = \frac{728}{123} & & \\ & V = \left\{ \frac{728}{123} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 7, 6 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{3}{4}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }+ \frac{ 70 }{ \color{blue}{84} })& = & (\frac{63}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x+70& = & 63x+336 \\\Leftrightarrow & 12x \color{red}{+70} \color{blue}{-70} \color{blue}{-63x} & = & \color{red}{63x} +336 \color{blue}{-63x} \color{blue}{-70} \\\Leftrightarrow & -51x& = & 266 \\\Leftrightarrow & \frac{-51x}{ \color{red}{-51} }& = & \frac{266}{-51} \\\Leftrightarrow & x = \frac{-266}{51} & & \\ & V = \left\{ \frac{-266}{51} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 7, 14 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{14}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 15 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+15& = & 14x-168 \\\Leftrightarrow & 6x \color{red}{+15} \color{blue}{-15} \color{blue}{-14x} & = & \color{red}{14x} -168 \color{blue}{-14x} \color{blue}{-15} \\\Leftrightarrow & -8x& = & -183 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-183}{-8} \\\Leftrightarrow & x = \frac{183}{8} & & \\ & V = \left\{ \frac{183}{8} \right\} & \\\end{align}\)
8. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{13}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }+ \frac{ 50 }{ \color{blue}{130} })& = & (\frac{156}{ \color{blue}{130} }x+\frac{780}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x+50& = & 156x+780 \\\Leftrightarrow & 65x \color{red}{+50} \color{blue}{-50} \color{blue}{-156x} & = & \color{red}{156x} +780 \color{blue}{-156x} \color{blue}{-50} \\\Leftrightarrow & -91x& = & 730 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{730}{-91} \\\Leftrightarrow & x = \frac{-730}{91} & & \\ & V = \left\{ \frac{-730}{91} \right\} & \\\end{align}\)
9. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 60 }{ \color{blue}{195} })& = & (\frac{65}{ \color{blue}{195} }x+\frac{1365}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+60& = & 65x+1365 \\\Leftrightarrow & 39x \color{red}{+60} \color{blue}{-60} \color{blue}{-65x} & = & \color{red}{65x} +1365 \color{blue}{-65x} \color{blue}{-60} \\\Leftrightarrow & -26x& = & 1305 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{1305}{-26} \\\Leftrightarrow & x = \frac{-1305}{26} & & \\ & V = \left\{ \frac{-1305}{26} \right\} & \\\end{align}\)
10. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{6}& = & \frac{4}{3}x+8 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{8}{ \color{blue}{6} }x+\frac{48}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x-5& = & 8x+48 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{-8x} & = & \color{red}{8x} +48 \color{blue}{-8x} \color{blue}{+5} \\\Leftrightarrow & -5x& = & 53 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{53}{-5} \\\Leftrightarrow & x = \frac{-53}{5} & & \\ & V = \left\{ \frac{-53}{5} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{5}{4}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{105}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-48& = & 105x+420 \\\Leftrightarrow & 28x \color{red}{-48} \color{blue}{+48} \color{blue}{-105x} & = & \color{red}{105x} +420 \color{blue}{-105x} \color{blue}{+48} \\\Leftrightarrow & -77x& = & 468 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{468}{-77} \\\Leftrightarrow & x = \frac{-468}{77} & & \\ & V = \left\{ \frac{-468}{77} \right\} & \\\end{align}\)
12. \(\text{6 is het kleinste gemene veelvoud van 3, 6 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{6}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{6.} (\frac{2x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{3}{ \color{blue}{6} }x-\frac{36}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 2x-5& = & 3x-36 \\\Leftrightarrow & 2x \color{red}{-5} \color{blue}{+5} \color{blue}{-3x} & = & \color{red}{3x} -36 \color{blue}{-3x} \color{blue}{+5} \\\Leftrightarrow & -x& = & -31 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-31}{-1} \\\Leftrightarrow & x = 31 & & \\ & V = \left\{ 31 \right\} & \\\end{align}\)