Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{7}& = & \frac{5}{2}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-30& = & 105x+210 \\\Leftrightarrow & 14x \color{red}{-30} \color{blue}{+30} \color{blue}{-105x} & = & \color{red}{105x} +210 \color{blue}{-105x} \color{blue}{+30} \\\Leftrightarrow & -91x& = & 240 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{240}{-91} \\\Leftrightarrow & x = \frac{-240}{91} & & \\ & V = \left\{ \frac{-240}{91} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{13}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x+\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+150& = & -156x+780 \\\Leftrightarrow & 65x \color{red}{+150} \color{blue}{-150} \color{blue}{+156x} & = & \color{red}{-156x} +780 \color{blue}{+156x} \color{blue}{-150} \\\Leftrightarrow & 221x& = & 630 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{630}{221} \\\Leftrightarrow & x = \frac{630}{221} & & \\ & V = \left\{ \frac{630}{221} \right\} & \\\end{align}\)
3. \(\text{336 is het kleinste gemene veelvoud van 7, 16 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{16}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{336.} (\frac{48x}{ \color{blue}{336} }+ \frac{ 63 }{ \color{blue}{336} })& = & (\frac{-280}{ \color{blue}{336} }x-\frac{2352}{ \color{blue}{336} }) \color{blue}{.336} \\\Leftrightarrow & 48x+63& = & -280x-2352 \\\Leftrightarrow & 48x \color{red}{+63} \color{blue}{-63} \color{blue}{+280x} & = & \color{red}{-280x} -2352 \color{blue}{+280x} \color{blue}{-63} \\\Leftrightarrow & 328x& = & -2415 \\\Leftrightarrow & \frac{328x}{ \color{red}{328} }& = & \frac{-2415}{328} \\\Leftrightarrow & x = \frac{-2415}{328} & & \\ & V = \left\{ \frac{-2415}{328} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & -42x+180 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+42x} & = & \color{red}{-42x} +180 \color{blue}{+42x} \color{blue}{-8} \\\Leftrightarrow & 57x& = & 172 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{172}{57} \\\Leftrightarrow & x = \frac{172}{57} & & \\ & V = \left\{ \frac{172}{57} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-5}{3}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & -140x+168 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{+140x} & = & \color{red}{-140x} +168 \color{blue}{+140x} \color{blue}{+48} \\\Leftrightarrow & 161x& = & 216 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{216}{161} \\\Leftrightarrow & x = \frac{216}{161} & & \\ & V = \left\{ \frac{216}{161} \right\} & \\\end{align}\)
6. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{13}& = & \frac{4}{3}x-3 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 60 }{ \color{blue}{156} })& = & (\frac{208}{ \color{blue}{156} }x-\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+60& = & 208x-468 \\\Leftrightarrow & 39x \color{red}{+60} \color{blue}{-60} \color{blue}{-208x} & = & \color{red}{208x} -468 \color{blue}{-208x} \color{blue}{-60} \\\Leftrightarrow & -169x& = & -528 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{-528}{-169} \\\Leftrightarrow & x = \frac{528}{169} & & \\ & V = \left\{ \frac{528}{169} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{7}& = & \frac{1}{3}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-12& = & 14x-336 \\\Leftrightarrow & 21x \color{red}{-12} \color{blue}{+12} \color{blue}{-14x} & = & \color{red}{14x} -336 \color{blue}{-14x} \color{blue}{+12} \\\Leftrightarrow & 7x& = & -324 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-324}{7} \\\Leftrightarrow & x = \frac{-324}{7} & & \\ & V = \left\{ \frac{-324}{7} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{3}{2}x+1 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{63}{ \color{blue}{42} }x+\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 63x+42 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-63x} & = & \color{red}{63x} +42 \color{blue}{-63x} \color{blue}{+12} \\\Leftrightarrow & -49x& = & 54 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{54}{-49} \\\Leftrightarrow & x = \frac{-54}{49} & & \\ & V = \left\{ \frac{-54}{49} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{13}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{468}{ \color{blue}{390} }x-\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+120& = & 468x-1560 \\\Leftrightarrow & 65x \color{red}{+120} \color{blue}{-120} \color{blue}{-468x} & = & \color{red}{468x} -1560 \color{blue}{-468x} \color{blue}{-120} \\\Leftrightarrow & -403x& = & -1680 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{-1680}{-403} \\\Leftrightarrow & x = \frac{1680}{403} & & \\ & V = \left\{ \frac{1680}{403} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{1}{5}x-3 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x-\frac{270}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 18x-270 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-18x} & = & \color{red}{18x} -270 \color{blue}{-18x} \color{blue}{-40} \\\Leftrightarrow & -3x& = & -310 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-310}{-3} \\\Leftrightarrow & x = \frac{310}{3} & & \\ & V = \left\{ \frac{310}{3} \right\} & \\\end{align}\)
11. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{3}{2}x-5 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 28 }{ \color{blue}{154} })& = & (\frac{231}{ \color{blue}{154} }x-\frac{770}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+28& = & 231x-770 \\\Leftrightarrow & 22x \color{red}{+28} \color{blue}{-28} \color{blue}{-231x} & = & \color{red}{231x} -770 \color{blue}{-231x} \color{blue}{-28} \\\Leftrightarrow & -209x& = & -798 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-798}{-209} \\\Leftrightarrow & x = \frac{42}{11} & & \\ & V = \left\{ \frac{42}{11} \right\} & \\\end{align}\)
12. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{26}{ \color{blue}{130} }x-\frac{130}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-20& = & 26x-130 \\\Leftrightarrow & 65x \color{red}{-20} \color{blue}{+20} \color{blue}{-26x} & = & \color{red}{26x} -130 \color{blue}{-26x} \color{blue}{+20} \\\Leftrightarrow & 39x& = & -110 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-110}{39} \\\Leftrightarrow & x = \frac{-110}{39} & & \\ & V = \left\{ \frac{-110}{39} \right\} & \\\end{align}\)