Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{1}{4}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{15}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+16& = & 15x-180 \\\Leftrightarrow & 20x \color{red}{+16} \color{blue}{-16} \color{blue}{-15x} & = & \color{red}{15x} -180 \color{blue}{-15x} \color{blue}{-16} \\\Leftrightarrow & 5x& = & -196 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{-196}{5} \\\Leftrightarrow & x = \frac{-196}{5} & & \\ & V = \left\{ \frac{-196}{5} \right\} & \\\end{align}\)
2. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 40 }{ \color{blue}{260} })& = & (\frac{-208}{ \color{blue}{260} }x+\frac{2080}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+40& = & -208x+2080 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{+208x} & = & \color{red}{-208x} +2080 \color{blue}{+208x} \color{blue}{-40} \\\Leftrightarrow & 273x& = & 2040 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{2040}{273} \\\Leftrightarrow & x = \frac{680}{91} & & \\ & V = \left\{ \frac{680}{91} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-2}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -84x-1050 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+84x} & = & \color{red}{-84x} -1050 \color{blue}{+84x} \color{blue}{+60} \\\Leftrightarrow & 119x& = & -990 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-990}{119} \\\Leftrightarrow & x = \frac{-990}{119} & & \\ & V = \left\{ \frac{-990}{119} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{-8}{3}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+18& = & -112x+210 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{+112x} & = & \color{red}{-112x} +210 \color{blue}{+112x} \color{blue}{-18} \\\Leftrightarrow & 133x& = & 192 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{192}{133} \\\Leftrightarrow & x = \frac{192}{133} & & \\ & V = \left\{ \frac{192}{133} \right\} & \\\end{align}\)
5. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x-\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-40& = & -105x-840 \\\Leftrightarrow & 28x \color{red}{-40} \color{blue}{+40} \color{blue}{+105x} & = & \color{red}{-105x} -840 \color{blue}{+105x} \color{blue}{+40} \\\Leftrightarrow & 133x& = & -800 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{-800}{133} \\\Leftrightarrow & x = \frac{-800}{133} & & \\ & V = \left\{ \frac{-800}{133} \right\} & \\\end{align}\)
6. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{1}{3}x+1 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }- \frac{ 60 }{ \color{blue}{165} })& = & (\frac{55}{ \color{blue}{165} }x+\frac{165}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x-60& = & 55x+165 \\\Leftrightarrow & 33x \color{red}{-60} \color{blue}{+60} \color{blue}{-55x} & = & \color{red}{55x} +165 \color{blue}{-55x} \color{blue}{+60} \\\Leftrightarrow & -22x& = & 225 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{225}{-22} \\\Leftrightarrow & x = \frac{-225}{22} & & \\ & V = \left\{ \frac{-225}{22} \right\} & \\\end{align}\)
7. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{1}{4}x+5 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }- \frac{ 100 }{ \color{blue}{220} })& = & (\frac{55}{ \color{blue}{220} }x+\frac{1100}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x-100& = & 55x+1100 \\\Leftrightarrow & 44x \color{red}{-100} \color{blue}{+100} \color{blue}{-55x} & = & \color{red}{55x} +1100 \color{blue}{-55x} \color{blue}{+100} \\\Leftrightarrow & -11x& = & 1200 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{1200}{-11} \\\Leftrightarrow & x = \frac{-1200}{11} & & \\ & V = \left\{ \frac{-1200}{11} \right\} & \\\end{align}\)
8. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{13}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 100 }{ \color{blue}{260} })& = & (\frac{52}{ \color{blue}{260} }x+\frac{260}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+100& = & 52x+260 \\\Leftrightarrow & 65x \color{red}{+100} \color{blue}{-100} \color{blue}{-52x} & = & \color{red}{52x} +260 \color{blue}{-52x} \color{blue}{-100} \\\Leftrightarrow & 13x& = & 160 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{160}{13} \\\Leftrightarrow & x = \frac{160}{13} & & \\ & V = \left\{ \frac{160}{13} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+9& = & 15x+90 \\\Leftrightarrow & 10x \color{red}{+9} \color{blue}{-9} \color{blue}{-15x} & = & \color{red}{15x} +90 \color{blue}{-15x} \color{blue}{-9} \\\Leftrightarrow & -5x& = & 81 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{81}{-5} \\\Leftrightarrow & x = \frac{-81}{5} & & \\ & V = \left\{ \frac{-81}{5} \right\} & \\\end{align}\)
10. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x+\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+18& = & -130x+156 \\\Leftrightarrow & 39x \color{red}{+18} \color{blue}{-18} \color{blue}{+130x} & = & \color{red}{-130x} +156 \color{blue}{+130x} \color{blue}{-18} \\\Leftrightarrow & 169x& = & 138 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{138}{169} \\\Leftrightarrow & x = \frac{138}{169} & & \\ & V = \left\{ \frac{138}{169} \right\} & \\\end{align}\)
11. \(\text{105 is het kleinste gemene veelvoud van 5, 7 en 3} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{2}{3}x-7 \\\Leftrightarrow & \color{blue}{105.} (\frac{21x}{ \color{blue}{105} }- \frac{ 45 }{ \color{blue}{105} })& = & (\frac{70}{ \color{blue}{105} }x-\frac{735}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 21x-45& = & 70x-735 \\\Leftrightarrow & 21x \color{red}{-45} \color{blue}{+45} \color{blue}{-70x} & = & \color{red}{70x} -735 \color{blue}{-70x} \color{blue}{+45} \\\Leftrightarrow & -49x& = & -690 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-690}{-49} \\\Leftrightarrow & x = \frac{690}{49} & & \\ & V = \left\{ \frac{690}{49} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{3}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{234}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & 234x+3120 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{-234x} & = & \color{red}{234x} +3120 \color{blue}{-234x} \color{blue}{-90} \\\Leftrightarrow & -169x& = & 3030 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{3030}{-169} \\\Leftrightarrow & x = \frac{-3030}{169} & & \\ & V = \left\{ \frac{-3030}{169} \right\} & \\\end{align}\)