Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{4}+\frac{5}{13}=\frac{-4}{5}x+3\)
2. \(\frac{x}{7}-\frac{3}{7}=\frac{1}{2}x-1\)
3. \(\frac{x}{6}-\frac{4}{15}=\frac{2}{5}x+1\)
4. \(\frac{x}{5}+\frac{4}{11}=\frac{1}{2}x-5\)
5. \(\frac{x}{3}+\frac{3}{16}=\frac{1}{2}x+6\)
6. \(\frac{x}{6}+\frac{3}{13}=\frac{-2}{5}x+4\)
7. \(\frac{x}{4}+\frac{2}{9}=\frac{-2}{5}x+5\)
8. \(\frac{x}{2}-\frac{3}{7}=\frac{-5}{3}x-5\)
9. \(\frac{x}{5}-\frac{3}{7}=\frac{-5}{6}x-7\)
10. \(\frac{x}{5}-\frac{3}{14}=\frac{7}{2}x-5\)
11. \(\frac{x}{5}+\frac{4}{7}=\frac{5}{6}x-4\)
12. \(\frac{x}{2}+\frac{5}{16}=\frac{6}{5}x+7\)

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

Verbetersleutel

1. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{13}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 100 }{ \color{blue}{260} })& = & (\frac{-208}{ \color{blue}{260} }x+\frac{780}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+100& = & -208x+780 \\\Leftrightarrow & 65x \color{red}{+100} \color{blue}{-100} \color{blue}{+208x} & = & \color{red}{-208x} +780 \color{blue}{+208x} \color{blue}{-100} \\\Leftrightarrow & 273x& = & 680 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{680}{273} \\\Leftrightarrow & x = \frac{680}{273} & & \\ & V = \left\{ \frac{680}{273} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 6 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{14}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-6& = & 7x-14 \\\Leftrightarrow & 2x \color{red}{-6} \color{blue}{+6} \color{blue}{-7x} & = & \color{red}{7x} -14 \color{blue}{-7x} \color{blue}{+6} \\\Leftrightarrow & -5x& = & -8 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-8}{-5} \\\Leftrightarrow & x = \frac{8}{5} & & \\ & V = \left\{ \frac{8}{5} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{15}& = & \frac{2}{5}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{12}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-8& = & 12x+30 \\\Leftrightarrow & 5x \color{red}{-8} \color{blue}{+8} \color{blue}{-12x} & = & \color{red}{12x} +30 \color{blue}{-12x} \color{blue}{+8} \\\Leftrightarrow & -7x& = & 38 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{38}{-7} \\\Leftrightarrow & x = \frac{-38}{7} & & \\ & V = \left\{ \frac{-38}{7} \right\} & \\\end{align}\)
4. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{11}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }+ \frac{ 40 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{550}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x+40& = & 55x-550 \\\Leftrightarrow & 22x \color{red}{+40} \color{blue}{-40} \color{blue}{-55x} & = & \color{red}{55x} -550 \color{blue}{-55x} \color{blue}{-40} \\\Leftrightarrow & -33x& = & -590 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-590}{-33} \\\Leftrightarrow & x = \frac{590}{33} & & \\ & V = \left\{ \frac{590}{33} \right\} & \\\end{align}\)
5. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x+\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 24x+288 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-24x} & = & \color{red}{24x} +288 \color{blue}{-24x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & 279 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{279}{-8} \\\Leftrightarrow & x = \frac{-279}{8} & & \\ & V = \left\{ \frac{-279}{8} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x+\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & -156x+1560 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{+156x} & = & \color{red}{-156x} +1560 \color{blue}{+156x} \color{blue}{-90} \\\Leftrightarrow & 221x& = & 1470 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{1470}{221} \\\Leftrightarrow & x = \frac{1470}{221} & & \\ & V = \left\{ \frac{1470}{221} \right\} & \\\end{align}\)
7. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{9}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 40 }{ \color{blue}{180} })& = & (\frac{-72}{ \color{blue}{180} }x+\frac{900}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+40& = & -72x+900 \\\Leftrightarrow & 45x \color{red}{+40} \color{blue}{-40} \color{blue}{+72x} & = & \color{red}{-72x} +900 \color{blue}{+72x} \color{blue}{-40} \\\Leftrightarrow & 117x& = & 860 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{860}{117} \\\Leftrightarrow & x = \frac{860}{117} & & \\ & V = \left\{ \frac{860}{117} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-5}{3}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & -70x-210 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+70x} & = & \color{red}{-70x} -210 \color{blue}{+70x} \color{blue}{+18} \\\Leftrightarrow & 91x& = & -192 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-192}{91} \\\Leftrightarrow & x = \frac{-192}{91} & & \\ & V = \left\{ \frac{-192}{91} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-90& = & -175x-1470 \\\Leftrightarrow & 42x \color{red}{-90} \color{blue}{+90} \color{blue}{+175x} & = & \color{red}{-175x} -1470 \color{blue}{+175x} \color{blue}{+90} \\\Leftrightarrow & 217x& = & -1380 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{-1380}{217} \\\Leftrightarrow & x = \frac{-1380}{217} & & \\ & V = \left\{ \frac{-1380}{217} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 5, 14 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{14}& = & \frac{7}{2}x-5 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{245}{ \color{blue}{70} }x-\frac{350}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-15& = & 245x-350 \\\Leftrightarrow & 14x \color{red}{-15} \color{blue}{+15} \color{blue}{-245x} & = & \color{red}{245x} -350 \color{blue}{-245x} \color{blue}{+15} \\\Leftrightarrow & -231x& = & -335 \\\Leftrightarrow & \frac{-231x}{ \color{red}{-231} }& = & \frac{-335}{-231} \\\Leftrightarrow & x = \frac{335}{231} & & \\ & V = \left\{ \frac{335}{231} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{5}{6}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+120& = & 175x-840 \\\Leftrightarrow & 42x \color{red}{+120} \color{blue}{-120} \color{blue}{-175x} & = & \color{red}{175x} -840 \color{blue}{-175x} \color{blue}{-120} \\\Leftrightarrow & -133x& = & -960 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-960}{-133} \\\Leftrightarrow & x = \frac{960}{133} & & \\ & V = \left\{ \frac{960}{133} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{6}{5}x+7 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{96}{ \color{blue}{80} }x+\frac{560}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+25& = & 96x+560 \\\Leftrightarrow & 40x \color{red}{+25} \color{blue}{-25} \color{blue}{-96x} & = & \color{red}{96x} +560 \color{blue}{-96x} \color{blue}{-25} \\\Leftrightarrow & -56x& = & 535 \\\Leftrightarrow & \frac{-56x}{ \color{red}{-56} }& = & \frac{535}{-56} \\\Leftrightarrow & x = \frac{-535}{56} & & \\ & V = \left\{ \frac{-535}{56} \right\} & \\\end{align}\)