Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x+\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 168x+840 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-168x} & = & \color{red}{168x} +840 \color{blue}{-168x} \color{blue}{-80} \\\Leftrightarrow & -133x& = & 760 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{760}{-133} \\\Leftrightarrow & x = \frac{-40}{7} & & \\ & V = \left\{ \frac{-40}{7} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x+\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+18& = & -130x+156 \\\Leftrightarrow & 39x \color{red}{+18} \color{blue}{-18} \color{blue}{+130x} & = & \color{red}{-130x} +156 \color{blue}{+130x} \color{blue}{-18} \\\Leftrightarrow & 169x& = & 138 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{138}{169} \\\Leftrightarrow & x = \frac{138}{169} & & \\ & V = \left\{ \frac{138}{169} \right\} & \\\end{align}\)
3. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{3}{2}x+8 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 12 }{ \color{blue}{78} })& = & (\frac{117}{ \color{blue}{78} }x+\frac{624}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-12& = & 117x+624 \\\Leftrightarrow & 26x \color{red}{-12} \color{blue}{+12} \color{blue}{-117x} & = & \color{red}{117x} +624 \color{blue}{-117x} \color{blue}{+12} \\\Leftrightarrow & -91x& = & 636 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{636}{-91} \\\Leftrightarrow & x = \frac{-636}{91} & & \\ & V = \left\{ \frac{-636}{91} \right\} & \\\end{align}\)
4. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{13}& = & \frac{-7}{4}x+5 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 100 }{ \color{blue}{260} })& = & (\frac{-455}{ \color{blue}{260} }x+\frac{1300}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-100& = & -455x+1300 \\\Leftrightarrow & 52x \color{red}{-100} \color{blue}{+100} \color{blue}{+455x} & = & \color{red}{-455x} +1300 \color{blue}{+455x} \color{blue}{+100} \\\Leftrightarrow & 507x& = & 1400 \\\Leftrightarrow & \frac{507x}{ \color{red}{507} }& = & \frac{1400}{507} \\\Leftrightarrow & x = \frac{1400}{507} & & \\ & V = \left\{ \frac{1400}{507} \right\} & \\\end{align}\)
5. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{9}& = & \frac{3}{2}x+5 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 50 }{ \color{blue}{90} })& = & (\frac{135}{ \color{blue}{90} }x+\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+50& = & 135x+450 \\\Leftrightarrow & 18x \color{red}{+50} \color{blue}{-50} \color{blue}{-135x} & = & \color{red}{135x} +450 \color{blue}{-135x} \color{blue}{-50} \\\Leftrightarrow & -117x& = & 400 \\\Leftrightarrow & \frac{-117x}{ \color{red}{-117} }& = & \frac{400}{-117} \\\Leftrightarrow & x = \frac{-400}{117} & & \\ & V = \left\{ \frac{-400}{117} \right\} & \\\end{align}\)
6. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 80 }{ \color{blue}{180} })& = & (\frac{36}{ \color{blue}{180} }x-\frac{900}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+80& = & 36x-900 \\\Leftrightarrow & 45x \color{red}{+80} \color{blue}{-80} \color{blue}{-36x} & = & \color{red}{36x} -900 \color{blue}{-36x} \color{blue}{-80} \\\Leftrightarrow & 9x& = & -980 \\\Leftrightarrow & \frac{9x}{ \color{red}{9} }& = & \frac{-980}{9} \\\Leftrightarrow & x = \frac{-980}{9} & & \\ & V = \left\{ \frac{-980}{9} \right\} & \\\end{align}\)
7. \(\text{140 is het kleinste gemene veelvoud van 4, 14 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{14}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 50 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x-\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-50& = & -112x-840 \\\Leftrightarrow & 35x \color{red}{-50} \color{blue}{+50} \color{blue}{+112x} & = & \color{red}{-112x} -840 \color{blue}{+112x} \color{blue}{+50} \\\Leftrightarrow & 147x& = & -790 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{-790}{147} \\\Leftrightarrow & x = \frac{-790}{147} & & \\ & V = \left\{ \frac{-790}{147} \right\} & \\\end{align}\)
8. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{11}& = & \frac{2}{3}x-4 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 18 }{ \color{blue}{66} })& = & (\frac{44}{ \color{blue}{66} }x-\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+18& = & 44x-264 \\\Leftrightarrow & 33x \color{red}{+18} \color{blue}{-18} \color{blue}{-44x} & = & \color{red}{44x} -264 \color{blue}{-44x} \color{blue}{-18} \\\Leftrightarrow & -11x& = & -282 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-282}{-11} \\\Leftrightarrow & x = \frac{282}{11} & & \\ & V = \left\{ \frac{282}{11} \right\} & \\\end{align}\)
9. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{-8}{3}x+1 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 48 }{ \color{blue}{156} })& = & (\frac{-416}{ \color{blue}{156} }x+\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-48& = & -416x+156 \\\Leftrightarrow & 39x \color{red}{-48} \color{blue}{+48} \color{blue}{+416x} & = & \color{red}{-416x} +156 \color{blue}{+416x} \color{blue}{+48} \\\Leftrightarrow & 455x& = & 204 \\\Leftrightarrow & \frac{455x}{ \color{red}{455} }& = & \frac{204}{455} \\\Leftrightarrow & x = \frac{204}{455} & & \\ & V = \left\{ \frac{204}{455} \right\} & \\\end{align}\)
10. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 20 }{ \color{blue}{35} })& = & (\frac{7}{ \color{blue}{35} }x-\frac{105}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-20& = & 7x-105 \\\Leftrightarrow & 5x \color{red}{-20} \color{blue}{+20} \color{blue}{-7x} & = & \color{red}{7x} -105 \color{blue}{-7x} \color{blue}{+20} \\\Leftrightarrow & -2x& = & -85 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-85}{-2} \\\Leftrightarrow & x = \frac{85}{2} & & \\ & V = \left\{ \frac{85}{2} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{10}& = & \frac{4}{3}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{80}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-18& = & 80x-420 \\\Leftrightarrow & 15x \color{red}{-18} \color{blue}{+18} \color{blue}{-80x} & = & \color{red}{80x} -420 \color{blue}{-80x} \color{blue}{+18} \\\Leftrightarrow & -65x& = & -402 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-402}{-65} \\\Leftrightarrow & x = \frac{402}{65} & & \\ & V = \left\{ \frac{402}{65} \right\} & \\\end{align}\)
12. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{12}& = & \frac{2}{3}x-8 \\\Leftrightarrow & \color{blue}{12.} (\frac{6x}{ \color{blue}{12} }- \frac{ 5 }{ \color{blue}{12} })& = & (\frac{8}{ \color{blue}{12} }x-\frac{96}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 6x-5& = & 8x-96 \\\Leftrightarrow & 6x \color{red}{-5} \color{blue}{+5} \color{blue}{-8x} & = & \color{red}{8x} -96 \color{blue}{-8x} \color{blue}{+5} \\\Leftrightarrow & -2x& = & -91 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-91}{-2} \\\Leftrightarrow & x = \frac{91}{2} & & \\ & V = \left\{ \frac{91}{2} \right\} & \\\end{align}\)