Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{-2}{5}x-1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-132}{ \color{blue}{330} }x-\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & -132x-330 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{+132x} & = & \color{red}{-132x} -330 \color{blue}{+132x} \color{blue}{-120} \\\Leftrightarrow & 187x& = & -450 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{-450}{187} \\\Leftrightarrow & x = \frac{-450}{187} & & \\ & V = \left\{ \frac{-450}{187} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{15}& = & \frac{7}{2}x-1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{105}{ \color{blue}{30} }x-\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-4& = & 105x-30 \\\Leftrightarrow & 6x \color{red}{-4} \color{blue}{+4} \color{blue}{-105x} & = & \color{red}{105x} -30 \color{blue}{-105x} \color{blue}{+4} \\\Leftrightarrow & -99x& = & -26 \\\Leftrightarrow & \frac{-99x}{ \color{red}{-99} }& = & \frac{-26}{-99} \\\Leftrightarrow & x = \frac{26}{99} & & \\ & V = \left\{ \frac{26}{99} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & 98x+294 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{-98x} & = & \color{red}{98x} +294 \color{blue}{-98x} \color{blue}{+12} \\\Leftrightarrow & -77x& = & 306 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{306}{-77} \\\Leftrightarrow & x = \frac{-306}{77} & & \\ & V = \left\{ \frac{-306}{77} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{-7}{4}x+5 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }+ \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-245}{ \color{blue}{140} }x+\frac{700}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x+60& = & -245x+700 \\\Leftrightarrow & 28x \color{red}{+60} \color{blue}{-60} \color{blue}{+245x} & = & \color{red}{-245x} +700 \color{blue}{+245x} \color{blue}{-60} \\\Leftrightarrow & 273x& = & 640 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{640}{273} \\\Leftrightarrow & x = \frac{640}{273} & & \\ & V = \left\{ \frac{640}{273} \right\} & \\\end{align}\)
5. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{4}{3}x-8 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 10 }{ \color{blue}{12} })& = & (\frac{16}{ \color{blue}{12} }x-\frac{96}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+10& = & 16x-96 \\\Leftrightarrow & 3x \color{red}{+10} \color{blue}{-10} \color{blue}{-16x} & = & \color{red}{16x} -96 \color{blue}{-16x} \color{blue}{-10} \\\Leftrightarrow & -13x& = & -106 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-106}{-13} \\\Leftrightarrow & x = \frac{106}{13} & & \\ & V = \left\{ \frac{106}{13} \right\} & \\\end{align}\)
6. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{9}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-27}{ \color{blue}{36} }x+\frac{108}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x-16& = & -27x+108 \\\Leftrightarrow & 12x \color{red}{-16} \color{blue}{+16} \color{blue}{+27x} & = & \color{red}{-27x} +108 \color{blue}{+27x} \color{blue}{+16} \\\Leftrightarrow & 39x& = & 124 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{124}{39} \\\Leftrightarrow & x = \frac{124}{39} & & \\ & V = \left\{ \frac{124}{39} \right\} & \\\end{align}\)
7. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{7}{2}x-6 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{231}{ \color{blue}{66} }x-\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+24& = & 231x-396 \\\Leftrightarrow & 22x \color{red}{+24} \color{blue}{-24} \color{blue}{-231x} & = & \color{red}{231x} -396 \color{blue}{-231x} \color{blue}{-24} \\\Leftrightarrow & -209x& = & -420 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-420}{-209} \\\Leftrightarrow & x = \frac{420}{209} & & \\ & V = \left\{ \frac{420}{209} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{15}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+4& = & 6x+210 \\\Leftrightarrow & 15x \color{red}{+4} \color{blue}{-4} \color{blue}{-6x} & = & \color{red}{6x} +210 \color{blue}{-6x} \color{blue}{-4} \\\Leftrightarrow & 9x& = & 206 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{206}{9} \\\Leftrightarrow & x = \frac{206}{9} & & \\ & V = \left\{ \frac{206}{9} \right\} & \\\end{align}\)
9. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{28}{ \color{blue}{140} }x-\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-80& = & 28x-560 \\\Leftrightarrow & 35x \color{red}{-80} \color{blue}{+80} \color{blue}{-28x} & = & \color{red}{28x} -560 \color{blue}{-28x} \color{blue}{+80} \\\Leftrightarrow & 7x& = & -480 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-480}{7} \\\Leftrightarrow & x = \frac{-480}{7} & & \\ & V = \left\{ \frac{-480}{7} \right\} & \\\end{align}\)
10. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{14}& = & \frac{7}{2}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 15 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+15& = & 147x-168 \\\Leftrightarrow & 14x \color{red}{+15} \color{blue}{-15} \color{blue}{-147x} & = & \color{red}{147x} -168 \color{blue}{-147x} \color{blue}{-15} \\\Leftrightarrow & -133x& = & -183 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-183}{-133} \\\Leftrightarrow & x = \frac{183}{133} & & \\ & V = \left\{ \frac{183}{133} \right\} & \\\end{align}\)
11. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{21}{ \color{blue}{105} }x-\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+60& = & 21x-735 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-21x} & = & \color{red}{21x} -735 \color{blue}{-21x} \color{blue}{-60} \\\Leftrightarrow & 14x& = & -795 \\\Leftrightarrow & \frac{14x}{ \color{red}{14} }& = & \frac{-795}{14} \\\Leftrightarrow & x = \frac{-795}{14} & & \\ & V = \left\{ \frac{-795}{14} \right\} & \\\end{align}\)
12. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{13}& = & \frac{-8}{3}x+7 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{-416}{ \color{blue}{156} }x+\frac{1092}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-60& = & -416x+1092 \\\Leftrightarrow & 39x \color{red}{-60} \color{blue}{+60} \color{blue}{+416x} & = & \color{red}{-416x} +1092 \color{blue}{+416x} \color{blue}{+60} \\\Leftrightarrow & 455x& = & 1152 \\\Leftrightarrow & \frac{455x}{ \color{red}{455} }& = & \frac{1152}{455} \\\Leftrightarrow & x = \frac{1152}{455} & & \\ & V = \left\{ \frac{1152}{455} \right\} & \\\end{align}\)