Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{7}{6}x+8 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{280}{ \color{blue}{240} }x+\frac{1920}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+45& = & 280x+1920 \\\Leftrightarrow & 48x \color{red}{+45} \color{blue}{-45} \color{blue}{-280x} & = & \color{red}{280x} +1920 \color{blue}{-280x} \color{blue}{-45} \\\Leftrightarrow & -232x& = & 1875 \\\Leftrightarrow & \frac{-232x}{ \color{red}{-232} }& = & \frac{1875}{-232} \\\Leftrightarrow & x = \frac{-1875}{232} & & \\ & V = \left\{ \frac{-1875}{232} \right\} & \\\end{align}\)
2. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{-4}{5}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & -72x-90 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{+72x} & = & \color{red}{-72x} -90 \color{blue}{+72x} \color{blue}{-40} \\\Leftrightarrow & 87x& = & -130 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{-130}{87} \\\Leftrightarrow & x = \frac{-130}{87} & & \\ & V = \left\{ \frac{-130}{87} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{6}{5}x+5 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x+\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+40& = & 108x+450 \\\Leftrightarrow & 45x \color{red}{+40} \color{blue}{-40} \color{blue}{-108x} & = & \color{red}{108x} +450 \color{blue}{-108x} \color{blue}{-40} \\\Leftrightarrow & -63x& = & 410 \\\Leftrightarrow & \frac{-63x}{ \color{red}{-63} }& = & \frac{410}{-63} \\\Leftrightarrow & x = \frac{-410}{63} & & \\ & V = \left\{ \frac{-410}{63} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{6}{5}x-7 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{36}{ \color{blue}{30} }x-\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & 36x-210 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-36x} & = & \color{red}{36x} -210 \color{blue}{-36x} \color{blue}{-8} \\\Leftrightarrow & -21x& = & -218 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-218}{-21} \\\Leftrightarrow & x = \frac{218}{21} & & \\ & V = \left\{ \frac{218}{21} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{2}{7}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-24& = & -140x+168 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+140x} & = & \color{red}{-140x} +168 \color{blue}{+140x} \color{blue}{+24} \\\Leftrightarrow & 161x& = & 192 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{192}{161} \\\Leftrightarrow & x = \frac{192}{161} & & \\ & V = \left\{ \frac{192}{161} \right\} & \\\end{align}\)
6. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{1}{3}x+3 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{52}{ \color{blue}{156} }x+\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+48& = & 52x+468 \\\Leftrightarrow & 39x \color{red}{+48} \color{blue}{-48} \color{blue}{-52x} & = & \color{red}{52x} +468 \color{blue}{-52x} \color{blue}{-48} \\\Leftrightarrow & -13x& = & 420 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{420}{-13} \\\Leftrightarrow & x = \frac{-420}{13} & & \\ & V = \left\{ \frac{-420}{13} \right\} & \\\end{align}\)
7. \(\text{60 is het kleinste gemene veelvoud van 3, 10 en 4} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{1}{4}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{15}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+18& = & 15x-240 \\\Leftrightarrow & 20x \color{red}{+18} \color{blue}{-18} \color{blue}{-15x} & = & \color{red}{15x} -240 \color{blue}{-15x} \color{blue}{-18} \\\Leftrightarrow & 5x& = & -258 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{-258}{5} \\\Leftrightarrow & x = \frac{-258}{5} & & \\ & V = \left\{ \frac{-258}{5} \right\} & \\\end{align}\)
8. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{-5}{3}x-6 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 28 }{ \color{blue}{105} })& = & (\frac{-175}{ \color{blue}{105} }x-\frac{630}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-28& = & -175x-630 \\\Leftrightarrow & 15x \color{red}{-28} \color{blue}{+28} \color{blue}{+175x} & = & \color{red}{-175x} -630 \color{blue}{+175x} \color{blue}{+28} \\\Leftrightarrow & 190x& = & -602 \\\Leftrightarrow & \frac{190x}{ \color{red}{190} }& = & \frac{-602}{190} \\\Leftrightarrow & x = \frac{-301}{95} & & \\ & V = \left\{ \frac{-301}{95} \right\} & \\\end{align}\)
9. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{4}{9}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 56 }{ \color{blue}{126} })& = & (\frac{63}{ \color{blue}{126} }x-\frac{1008}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-56& = & 63x-1008 \\\Leftrightarrow & 18x \color{red}{-56} \color{blue}{+56} \color{blue}{-63x} & = & \color{red}{63x} -1008 \color{blue}{-63x} \color{blue}{+56} \\\Leftrightarrow & -45x& = & -952 \\\Leftrightarrow & \frac{-45x}{ \color{red}{-45} }& = & \frac{-952}{-45} \\\Leftrightarrow & x = \frac{952}{45} & & \\ & V = \left\{ \frac{952}{45} \right\} & \\\end{align}\)
10. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{7}{5}x-3 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }+ \frac{ 140 }{ \color{blue}{455} })& = & (\frac{637}{ \color{blue}{455} }x-\frac{1365}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x+140& = & 637x-1365 \\\Leftrightarrow & 65x \color{red}{+140} \color{blue}{-140} \color{blue}{-637x} & = & \color{red}{637x} -1365 \color{blue}{-637x} \color{blue}{-140} \\\Leftrightarrow & -572x& = & -1505 \\\Leftrightarrow & \frac{-572x}{ \color{red}{-572} }& = & \frac{-1505}{-572} \\\Leftrightarrow & x = \frac{1505}{572} & & \\ & V = \left\{ \frac{1505}{572} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & -168x-1260 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{+168x} & = & \color{red}{-168x} -1260 \color{blue}{+168x} \color{blue}{+90} \\\Leftrightarrow & 203x& = & -1170 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1170}{203} \\\Leftrightarrow & x = \frac{-1170}{203} & & \\ & V = \left\{ \frac{-1170}{203} \right\} & \\\end{align}\)
12. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{8}& = & \frac{4}{3}x-4 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{32}{ \color{blue}{24} }x-\frac{96}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x+15& = & 32x-96 \\\Leftrightarrow & 6x \color{red}{+15} \color{blue}{-15} \color{blue}{-32x} & = & \color{red}{32x} -96 \color{blue}{-32x} \color{blue}{-15} \\\Leftrightarrow & -26x& = & -111 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{-111}{-26} \\\Leftrightarrow & x = \frac{111}{26} & & \\ & V = \left\{ \frac{111}{26} \right\} & \\\end{align}\)