Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{96}{ \color{blue}{80} }x+\frac{80}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+15& = & 96x+80 \\\Leftrightarrow & 20x \color{red}{+15} \color{blue}{-15} \color{blue}{-96x} & = & \color{red}{96x} +80 \color{blue}{-96x} \color{blue}{-15} \\\Leftrightarrow & -76x& = & 65 \\\Leftrightarrow & \frac{-76x}{ \color{red}{-76} }& = & \frac{65}{-76} \\\Leftrightarrow & x = \frac{-65}{76} & & \\ & V = \left\{ \frac{-65}{76} \right\} & \\\end{align}\)
2. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{13}& = & \frac{-5}{3}x+4 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 60 }{ \color{blue}{156} })& = & (\frac{-260}{ \color{blue}{156} }x+\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+60& = & -260x+624 \\\Leftrightarrow & 39x \color{red}{+60} \color{blue}{-60} \color{blue}{+260x} & = & \color{red}{-260x} +624 \color{blue}{+260x} \color{blue}{-60} \\\Leftrightarrow & 299x& = & 564 \\\Leftrightarrow & \frac{299x}{ \color{red}{299} }& = & \frac{564}{299} \\\Leftrightarrow & x = \frac{564}{299} & & \\ & V = \left\{ \frac{564}{299} \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 5, 6 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{6}& = & \frac{-3}{4}x+8 \\\Leftrightarrow & \color{blue}{60.} (\frac{12x}{ \color{blue}{60} }- \frac{ 50 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x+\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 12x-50& = & -45x+480 \\\Leftrightarrow & 12x \color{red}{-50} \color{blue}{+50} \color{blue}{+45x} & = & \color{red}{-45x} +480 \color{blue}{+45x} \color{blue}{+50} \\\Leftrightarrow & 57x& = & 530 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{530}{57} \\\Leftrightarrow & x = \frac{530}{57} & & \\ & V = \left\{ \frac{530}{57} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{4}{3}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{64}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-9& = & 64x+336 \\\Leftrightarrow & 24x \color{red}{-9} \color{blue}{+9} \color{blue}{-64x} & = & \color{red}{64x} +336 \color{blue}{-64x} \color{blue}{+9} \\\Leftrightarrow & -40x& = & 345 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{345}{-40} \\\Leftrightarrow & x = \frac{-69}{8} & & \\ & V = \left\{ \frac{-69}{8} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{-8}{3}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & -112x+336 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{+112x} & = & \color{red}{-112x} +336 \color{blue}{+112x} \color{blue}{+12} \\\Leftrightarrow & 133x& = & 348 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{348}{133} \\\Leftrightarrow & x = \frac{348}{133} & & \\ & V = \left\{ \frac{348}{133} \right\} & \\\end{align}\)
6. \(\text{80 is het kleinste gemene veelvoud van 4, 16 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{16}& = & \frac{-4}{5}x+7 \\\Leftrightarrow & \color{blue}{80.} (\frac{20x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-64}{ \color{blue}{80} }x+\frac{560}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 20x+25& = & -64x+560 \\\Leftrightarrow & 20x \color{red}{+25} \color{blue}{-25} \color{blue}{+64x} & = & \color{red}{-64x} +560 \color{blue}{+64x} \color{blue}{-25} \\\Leftrightarrow & 84x& = & 535 \\\Leftrightarrow & \frac{84x}{ \color{red}{84} }& = & \frac{535}{84} \\\Leftrightarrow & x = \frac{535}{84} & & \\ & V = \left\{ \frac{535}{84} \right\} & \\\end{align}\)
7. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{-7}{4}x-8 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }- \frac{ 8 }{ \color{blue}{28} })& = & (\frac{-49}{ \color{blue}{28} }x-\frac{224}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x-8& = & -49x-224 \\\Leftrightarrow & 4x \color{red}{-8} \color{blue}{+8} \color{blue}{+49x} & = & \color{red}{-49x} -224 \color{blue}{+49x} \color{blue}{+8} \\\Leftrightarrow & 53x& = & -216 \\\Leftrightarrow & \frac{53x}{ \color{red}{53} }& = & \frac{-216}{53} \\\Leftrightarrow & x = \frac{-216}{53} & & \\ & V = \left\{ \frac{-216}{53} \right\} & \\\end{align}\)
8. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{9}& = & \frac{3}{2}x+3 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 70 }{ \color{blue}{126} })& = & (\frac{189}{ \color{blue}{126} }x+\frac{378}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-70& = & 189x+378 \\\Leftrightarrow & 18x \color{red}{-70} \color{blue}{+70} \color{blue}{-189x} & = & \color{red}{189x} +378 \color{blue}{-189x} \color{blue}{+70} \\\Leftrightarrow & -171x& = & 448 \\\Leftrightarrow & \frac{-171x}{ \color{red}{-171} }& = & \frac{448}{-171} \\\Leftrightarrow & x = \frac{-448}{171} & & \\ & V = \left\{ \frac{-448}{171} \right\} & \\\end{align}\)
9. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{9}& = & \frac{5}{2}x-6 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 28 }{ \color{blue}{126} })& = & (\frac{315}{ \color{blue}{126} }x-\frac{756}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+28& = & 315x-756 \\\Leftrightarrow & 18x \color{red}{+28} \color{blue}{-28} \color{blue}{-315x} & = & \color{red}{315x} -756 \color{blue}{-315x} \color{blue}{-28} \\\Leftrightarrow & -297x& = & -784 \\\Leftrightarrow & \frac{-297x}{ \color{red}{-297} }& = & \frac{-784}{-297} \\\Leftrightarrow & x = \frac{784}{297} & & \\ & V = \left\{ \frac{784}{297} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{11}& = & \frac{6}{5}x-6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 120 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x-\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-120& = & 396x-1980 \\\Leftrightarrow & 55x \color{red}{-120} \color{blue}{+120} \color{blue}{-396x} & = & \color{red}{396x} -1980 \color{blue}{-396x} \color{blue}{+120} \\\Leftrightarrow & -341x& = & -1860 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{-1860}{-341} \\\Leftrightarrow & x = \frac{60}{11} & & \\ & V = \left\{ \frac{60}{11} \right\} & \\\end{align}\)
11. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{4}{3}x+2 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }+ \frac{ 14 }{ \color{blue}{105} })& = & (\frac{140}{ \color{blue}{105} }x+\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x+14& = & 140x+210 \\\Leftrightarrow & 15x \color{red}{+14} \color{blue}{-14} \color{blue}{-140x} & = & \color{red}{140x} +210 \color{blue}{-140x} \color{blue}{-14} \\\Leftrightarrow & -125x& = & 196 \\\Leftrightarrow & \frac{-125x}{ \color{red}{-125} }& = & \frac{196}{-125} \\\Leftrightarrow & x = \frac{-196}{125} & & \\ & V = \left\{ \frac{-196}{125} \right\} & \\\end{align}\)
12. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{-5}{6}x+7 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-275}{ \color{blue}{330} }x+\frac{2310}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+120& = & -275x+2310 \\\Leftrightarrow & 66x \color{red}{+120} \color{blue}{-120} \color{blue}{+275x} & = & \color{red}{-275x} +2310 \color{blue}{+275x} \color{blue}{-120} \\\Leftrightarrow & 341x& = & 2190 \\\Leftrightarrow & \frac{341x}{ \color{red}{341} }& = & \frac{2190}{341} \\\Leftrightarrow & x = \frac{2190}{341} & & \\ & V = \left\{ \frac{2190}{341} \right\} & \\\end{align}\)