Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-36}{ \color{blue}{48} }x-\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & -36x-288 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{+36x} & = & \color{red}{-36x} -288 \color{blue}{+36x} \color{blue}{+9} \\\Leftrightarrow & 52x& = & -279 \\\Leftrightarrow & \frac{52x}{ \color{red}{52} }& = & \frac{-279}{52} \\\Leftrightarrow & x = \frac{-279}{52} & & \\ & V = \left\{ \frac{-279}{52} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{6}{5}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 252x+840 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-252x} & = & \color{red}{252x} +840 \color{blue}{-252x} \color{blue}{+120} \\\Leftrightarrow & -217x& = & 960 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{960}{-217} \\\Leftrightarrow & x = \frac{-960}{217} & & \\ & V = \left\{ \frac{-960}{217} \right\} & \\\end{align}\)
3. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{8}& = & \frac{-8}{3}x-5 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }- \frac{ 9 }{ \color{blue}{24} })& = & (\frac{-64}{ \color{blue}{24} }x-\frac{120}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x-9& = & -64x-120 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{+64x} & = & \color{red}{-64x} -120 \color{blue}{+64x} \color{blue}{+9} \\\Leftrightarrow & 70x& = & -111 \\\Leftrightarrow & \frac{70x}{ \color{red}{70} }& = & \frac{-111}{70} \\\Leftrightarrow & x = \frac{-111}{70} & & \\ & V = \left\{ \frac{-111}{70} \right\} & \\\end{align}\)
4. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{780}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+30& = & 65x-780 \\\Leftrightarrow & 26x \color{red}{+30} \color{blue}{-30} \color{blue}{-65x} & = & \color{red}{65x} -780 \color{blue}{-65x} \color{blue}{-30} \\\Leftrightarrow & -39x& = & -810 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-810}{-39} \\\Leftrightarrow & x = \frac{270}{13} & & \\ & V = \left\{ \frac{270}{13} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 20x-180 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-20x} & = & \color{red}{20x} -180 \color{blue}{-20x} \color{blue}{+16} \\\Leftrightarrow & -5x& = & -164 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-164}{-5} \\\Leftrightarrow & x = \frac{164}{5} & & \\ & V = \left\{ \frac{164}{5} \right\} & \\\end{align}\)
6. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{13}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 80 }{ \color{blue}{260} })& = & (\frac{-208}{ \color{blue}{260} }x+\frac{1560}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+80& = & -208x+1560 \\\Leftrightarrow & 65x \color{red}{+80} \color{blue}{-80} \color{blue}{+208x} & = & \color{red}{-208x} +1560 \color{blue}{+208x} \color{blue}{-80} \\\Leftrightarrow & 273x& = & 1480 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{1480}{273} \\\Leftrightarrow & x = \frac{1480}{273} & & \\ & V = \left\{ \frac{1480}{273} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{14}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & 14x-168 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-14x} & = & \color{red}{14x} -168 \color{blue}{-14x} \color{blue}{+24} \\\Leftrightarrow & 7x& = & -144 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-144}{7} \\\Leftrightarrow & x = \frac{-144}{7} & & \\ & V = \left\{ \frac{-144}{7} \right\} & \\\end{align}\)
8. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{-5}{6}x+2 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-325}{ \color{blue}{390} }x+\frac{780}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+120& = & -325x+780 \\\Leftrightarrow & 78x \color{red}{+120} \color{blue}{-120} \color{blue}{+325x} & = & \color{red}{-325x} +780 \color{blue}{+325x} \color{blue}{-120} \\\Leftrightarrow & 403x& = & 660 \\\Leftrightarrow & \frac{403x}{ \color{red}{403} }& = & \frac{660}{403} \\\Leftrightarrow & x = \frac{660}{403} & & \\ & V = \left\{ \frac{660}{403} \right\} & \\\end{align}\)
9. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{6}{5}x+4 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{126}{ \color{blue}{105} }x+\frac{420}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+60& = & 126x+420 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-126x} & = & \color{red}{126x} +420 \color{blue}{-126x} \color{blue}{-60} \\\Leftrightarrow & -91x& = & 360 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{360}{-91} \\\Leftrightarrow & x = \frac{-360}{91} & & \\ & V = \left\{ \frac{-360}{91} \right\} & \\\end{align}\)
10. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{9}& = & \frac{5}{3}x+1 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 8 }{ \color{blue}{18} })& = & (\frac{30}{ \color{blue}{18} }x+\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-8& = & 30x+18 \\\Leftrightarrow & 9x \color{red}{-8} \color{blue}{+8} \color{blue}{-30x} & = & \color{red}{30x} +18 \color{blue}{-30x} \color{blue}{+8} \\\Leftrightarrow & -21x& = & 26 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{26}{-21} \\\Leftrightarrow & x = \frac{-26}{21} & & \\ & V = \left\{ \frac{-26}{21} \right\} & \\\end{align}\)
11. \(\text{12 is het kleinste gemene veelvoud van 3, 6 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{6}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 10 }{ \color{blue}{12} })& = & (\frac{-9}{ \color{blue}{12} }x+\frac{36}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+10& = & -9x+36 \\\Leftrightarrow & 4x \color{red}{+10} \color{blue}{-10} \color{blue}{+9x} & = & \color{red}{-9x} +36 \color{blue}{+9x} \color{blue}{-10} \\\Leftrightarrow & 13x& = & 26 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{26}{13} \\\Leftrightarrow & x = 2 & & \\ & V = \left\{ 2 \right\} & \\\end{align}\)
12. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x+\frac{14}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+4& = & 7x+14 \\\Leftrightarrow & 2x \color{red}{+4} \color{blue}{-4} \color{blue}{-7x} & = & \color{red}{7x} +14 \color{blue}{-7x} \color{blue}{-4} \\\Leftrightarrow & -5x& = & 10 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{10}{-5} \\\Leftrightarrow & x = -2 & & \\ & V = \left\{ -2 \right\} & \\\end{align}\)