Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{13}& = & \frac{-3}{4}x-4 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 48 }{ \color{blue}{156} })& = & (\frac{-117}{ \color{blue}{156} }x-\frac{624}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+48& = & -117x-624 \\\Leftrightarrow & 52x \color{red}{+48} \color{blue}{-48} \color{blue}{+117x} & = & \color{red}{-117x} -624 \color{blue}{+117x} \color{blue}{-48} \\\Leftrightarrow & 169x& = & -672 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-672}{169} \\\Leftrightarrow & x = \frac{-672}{169} & & \\ & V = \left\{ \frac{-672}{169} \right\} & \\\end{align}\)
2. \(\text{30 is het kleinste gemene veelvoud van 6, 6 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{6}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 25 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-25& = & -42x+120 \\\Leftrightarrow & 5x \color{red}{-25} \color{blue}{+25} \color{blue}{+42x} & = & \color{red}{-42x} +120 \color{blue}{+42x} \color{blue}{+25} \\\Leftrightarrow & 47x& = & 145 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{145}{47} \\\Leftrightarrow & x = \frac{145}{47} & & \\ & V = \left\{ \frac{145}{47} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 21x-168 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-21x} & = & \color{red}{21x} -168 \color{blue}{-21x} \color{blue}{+12} \\\Leftrightarrow & -7x& = & -156 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-156}{-7} \\\Leftrightarrow & x = \frac{156}{7} & & \\ & V = \left\{ \frac{156}{7} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{7}{6}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{245}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+60& = & 245x-840 \\\Leftrightarrow & 42x \color{red}{+60} \color{blue}{-60} \color{blue}{-245x} & = & \color{red}{245x} -840 \color{blue}{-245x} \color{blue}{-60} \\\Leftrightarrow & -203x& = & -900 \\\Leftrightarrow & \frac{-203x}{ \color{red}{-203} }& = & \frac{-900}{-203} \\\Leftrightarrow & x = \frac{900}{203} & & \\ & V = \left\{ \frac{900}{203} \right\} & \\\end{align}\)
5. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{208}{ \color{blue}{156} }x+\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & 208x+156 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{-208x} & = & \color{red}{208x} +156 \color{blue}{-208x} \color{blue}{+36} \\\Leftrightarrow & -169x& = & 192 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{192}{-169} \\\Leftrightarrow & x = \frac{-192}{169} & & \\ & V = \left\{ \frac{-192}{169} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{3}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{234}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & 234x+3120 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-234x} & = & \color{red}{234x} +3120 \color{blue}{-234x} \color{blue}{+60} \\\Leftrightarrow & -169x& = & 3180 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{3180}{-169} \\\Leftrightarrow & x = \frac{-3180}{169} & & \\ & V = \left\{ \frac{-3180}{169} \right\} & \\\end{align}\)
7. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & -312x+2340 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{+312x} & = & \color{red}{-312x} +2340 \color{blue}{+312x} \color{blue}{-120} \\\Leftrightarrow & 377x& = & 2220 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{2220}{377} \\\Leftrightarrow & x = \frac{2220}{377} & & \\ & V = \left\{ \frac{2220}{377} \right\} & \\\end{align}\)
8. \(\text{20 is het kleinste gemene veelvoud van 4, 10 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{1}{5}x+6 \\\Leftrightarrow & \color{blue}{20.} (\frac{5x}{ \color{blue}{20} }+ \frac{ 6 }{ \color{blue}{20} })& = & (\frac{4}{ \color{blue}{20} }x+\frac{120}{ \color{blue}{20} }) \color{blue}{.20} \\\Leftrightarrow & 5x+6& = & 4x+120 \\\Leftrightarrow & 5x \color{red}{+6} \color{blue}{-6} \color{blue}{-4x} & = & \color{red}{4x} +120 \color{blue}{-4x} \color{blue}{-6} \\\Leftrightarrow & x& = & 114 \\ & V = \left\{ 114 \right\} & \\\end{align}\)
9. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 40 }{ \color{blue}{260} })& = & (\frac{312}{ \color{blue}{260} }x-\frac{520}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+40& = & 312x-520 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{-312x} & = & \color{red}{312x} -520 \color{blue}{-312x} \color{blue}{-40} \\\Leftrightarrow & -247x& = & -560 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-560}{-247} \\\Leftrightarrow & x = \frac{560}{247} & & \\ & V = \left\{ \frac{560}{247} \right\} & \\\end{align}\)
10. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x+\frac{198}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+12& = & -44x+198 \\\Leftrightarrow & 33x \color{red}{+12} \color{blue}{-12} \color{blue}{+44x} & = & \color{red}{-44x} +198 \color{blue}{+44x} \color{blue}{-12} \\\Leftrightarrow & 77x& = & 186 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{186}{77} \\\Leftrightarrow & x = \frac{186}{77} & & \\ & V = \left\{ \frac{186}{77} \right\} & \\\end{align}\)
11. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{-2}{3}x+5 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 12 }{ \color{blue}{21} })& = & (\frac{-14}{ \color{blue}{21} }x+\frac{105}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-12& = & -14x+105 \\\Leftrightarrow & 3x \color{red}{-12} \color{blue}{+12} \color{blue}{+14x} & = & \color{red}{-14x} +105 \color{blue}{+14x} \color{blue}{+12} \\\Leftrightarrow & 17x& = & 117 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{117}{17} \\\Leftrightarrow & x = \frac{117}{17} & & \\ & V = \left\{ \frac{117}{17} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{1}{5}x+5 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x+\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-30& = & 14x+350 \\\Leftrightarrow & 35x \color{red}{-30} \color{blue}{+30} \color{blue}{-14x} & = & \color{red}{14x} +350 \color{blue}{-14x} \color{blue}{+30} \\\Leftrightarrow & 21x& = & 380 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{380}{21} \\\Leftrightarrow & x = \frac{380}{21} & & \\ & V = \left\{ \frac{380}{21} \right\} & \\\end{align}\)