Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{180 is het kleinste gemene veelvoud van 5, 9 en 4} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{-7}{4}x-8 \\\Leftrightarrow & \color{blue}{180.} (\frac{36x}{ \color{blue}{180} }+ \frac{ 40 }{ \color{blue}{180} })& = & (\frac{-315}{ \color{blue}{180} }x-\frac{1440}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 36x+40& = & -315x-1440 \\\Leftrightarrow & 36x \color{red}{+40} \color{blue}{-40} \color{blue}{+315x} & = & \color{red}{-315x} -1440 \color{blue}{+315x} \color{blue}{-40} \\\Leftrightarrow & 351x& = & -1480 \\\Leftrightarrow & \frac{351x}{ \color{red}{351} }& = & \frac{-1480}{351} \\\Leftrightarrow & x = \frac{-1480}{351} & & \\ & V = \left\{ \frac{-1480}{351} \right\} & \\\end{align}\)
2. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{4}{7}& = & \frac{6}{5}x-2 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 20 }{ \color{blue}{35} })& = & (\frac{42}{ \color{blue}{35} }x-\frac{70}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-20& = & 42x-70 \\\Leftrightarrow & 5x \color{red}{-20} \color{blue}{+20} \color{blue}{-42x} & = & \color{red}{42x} -70 \color{blue}{-42x} \color{blue}{+20} \\\Leftrightarrow & -37x& = & -50 \\\Leftrightarrow & \frac{-37x}{ \color{red}{-37} }& = & \frac{-50}{-37} \\\Leftrightarrow & x = \frac{50}{37} & & \\ & V = \left\{ \frac{50}{37} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{4}{3}x+6 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{112}{ \color{blue}{84} }x+\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & 112x+504 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{-112x} & = & \color{red}{112x} +504 \color{blue}{-112x} \color{blue}{-36} \\\Leftrightarrow & -91x& = & 468 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{468}{-91} \\\Leftrightarrow & x = \frac{-36}{7} & & \\ & V = \left\{ \frac{-36}{7} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{-5}{6}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-60& = & -175x+210 \\\Leftrightarrow & 42x \color{red}{-60} \color{blue}{+60} \color{blue}{+175x} & = & \color{red}{-175x} +210 \color{blue}{+175x} \color{blue}{+60} \\\Leftrightarrow & 217x& = & 270 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{270}{217} \\\Leftrightarrow & x = \frac{270}{217} & & \\ & V = \left\{ \frac{270}{217} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & 252x-1050 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{-252x} & = & \color{red}{252x} -1050 \color{blue}{-252x} \color{blue}{+90} \\\Leftrightarrow & -217x& = & -960 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-960}{-217} \\\Leftrightarrow & x = \frac{960}{217} & & \\ & V = \left\{ \frac{960}{217} \right\} & \\\end{align}\)
6. \(\text{195 is het kleinste gemene veelvoud van 5, 13 en 3} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{2}{3}x-2 \\\Leftrightarrow & \color{blue}{195.} (\frac{39x}{ \color{blue}{195} }+ \frac{ 30 }{ \color{blue}{195} })& = & (\frac{130}{ \color{blue}{195} }x-\frac{390}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 39x+30& = & 130x-390 \\\Leftrightarrow & 39x \color{red}{+30} \color{blue}{-30} \color{blue}{-130x} & = & \color{red}{130x} -390 \color{blue}{-130x} \color{blue}{-30} \\\Leftrightarrow & -91x& = & -420 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-420}{-91} \\\Leftrightarrow & x = \frac{60}{13} & & \\ & V = \left\{ \frac{60}{13} \right\} & \\\end{align}\)
7. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{15}& = & \frac{-8}{3}x+2 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 14 }{ \color{blue}{105} })& = & (\frac{-280}{ \color{blue}{105} }x+\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-14& = & -280x+210 \\\Leftrightarrow & 15x \color{red}{-14} \color{blue}{+14} \color{blue}{+280x} & = & \color{red}{-280x} +210 \color{blue}{+280x} \color{blue}{+14} \\\Leftrightarrow & 295x& = & 224 \\\Leftrightarrow & \frac{295x}{ \color{red}{295} }& = & \frac{224}{295} \\\Leftrightarrow & x = \frac{224}{295} & & \\ & V = \left\{ \frac{224}{295} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{14}& = & \frac{3}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-45& = & 126x-210 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{-126x} & = & \color{red}{126x} -210 \color{blue}{-126x} \color{blue}{+45} \\\Leftrightarrow & -91x& = & -165 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-165}{-91} \\\Leftrightarrow & x = \frac{165}{91} & & \\ & V = \left\{ \frac{165}{91} \right\} & \\\end{align}\)
9. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{1}{5}x+4 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x+\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 18x+360 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-18x} & = & \color{red}{18x} +360 \color{blue}{-18x} \color{blue}{-40} \\\Leftrightarrow & -3x& = & 320 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{320}{-3} \\\Leftrightarrow & x = \frac{-320}{3} & & \\ & V = \left\{ \frac{-320}{3} \right\} & \\\end{align}\)
10. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{4}{3}x-5 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }- \frac{ 42 }{ \color{blue}{231} })& = & (\frac{308}{ \color{blue}{231} }x-\frac{1155}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x-42& = & 308x-1155 \\\Leftrightarrow & 33x \color{red}{-42} \color{blue}{+42} \color{blue}{-308x} & = & \color{red}{308x} -1155 \color{blue}{-308x} \color{blue}{+42} \\\Leftrightarrow & -275x& = & -1113 \\\Leftrightarrow & \frac{-275x}{ \color{red}{-275} }& = & \frac{-1113}{-275} \\\Leftrightarrow & x = \frac{1113}{275} & & \\ & V = \left\{ \frac{1113}{275} \right\} & \\\end{align}\)
11. \(\text{30 is het kleinste gemene veelvoud van 5, 6 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{6}& = & \frac{5}{2}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 25 }{ \color{blue}{30} })& = & (\frac{75}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+25& = & 75x+30 \\\Leftrightarrow & 6x \color{red}{+25} \color{blue}{-25} \color{blue}{-75x} & = & \color{red}{75x} +30 \color{blue}{-75x} \color{blue}{-25} \\\Leftrightarrow & -69x& = & 5 \\\Leftrightarrow & \frac{-69x}{ \color{red}{-69} }& = & \frac{5}{-69} \\\Leftrightarrow & x = \frac{-5}{69} & & \\ & V = \left\{ \frac{-5}{69} \right\} & \\\end{align}\)
12. \(\text{15 is het kleinste gemene veelvoud van 3, 15 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{15}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{15.} (\frac{5x}{ \color{blue}{15} }+ \frac{ 2 }{ \color{blue}{15} })& = & (\frac{-21}{ \color{blue}{15} }x-\frac{45}{ \color{blue}{15} }) \color{blue}{.15} \\\Leftrightarrow & 5x+2& = & -21x-45 \\\Leftrightarrow & 5x \color{red}{+2} \color{blue}{-2} \color{blue}{+21x} & = & \color{red}{-21x} -45 \color{blue}{+21x} \color{blue}{-2} \\\Leftrightarrow & 26x& = & -47 \\\Leftrightarrow & \frac{26x}{ \color{red}{26} }& = & \frac{-47}{26} \\\Leftrightarrow & x = \frac{-47}{26} & & \\ & V = \left\{ \frac{-47}{26} \right\} & \\\end{align}\)