Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 6} \\ \begin{align} & \frac{x}{7}-\frac{5}{6}& = & \frac{1}{6}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 35 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-35& = & 7x-252 \\\Leftrightarrow & 6x \color{red}{-35} \color{blue}{+35} \color{blue}{-7x} & = & \color{red}{7x} -252 \color{blue}{-7x} \color{blue}{+35} \\\Leftrightarrow & -x& = & -217 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-217}{-1} \\\Leftrightarrow & x = 217 & & \\ & V = \left\{ 217 \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{14}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-9& = & 21x+126 \\\Leftrightarrow & 14x \color{red}{-9} \color{blue}{+9} \color{blue}{-21x} & = & \color{red}{21x} +126 \color{blue}{-21x} \color{blue}{+9} \\\Leftrightarrow & -7x& = & 135 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{135}{-7} \\\Leftrightarrow & x = \frac{-135}{7} & & \\ & V = \left\{ \frac{-135}{7} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{108}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & 108x-90 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{-108x} & = & \color{red}{108x} -90 \color{blue}{-108x} \color{blue}{-20} \\\Leftrightarrow & -93x& = & -110 \\\Leftrightarrow & \frac{-93x}{ \color{red}{-93} }& = & \frac{-110}{-93} \\\Leftrightarrow & x = \frac{110}{93} & & \\ & V = \left\{ \frac{110}{93} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{7}& = & \frac{7}{6}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{245}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+90& = & 245x+1260 \\\Leftrightarrow & 42x \color{red}{+90} \color{blue}{-90} \color{blue}{-245x} & = & \color{red}{245x} +1260 \color{blue}{-245x} \color{blue}{-90} \\\Leftrightarrow & -203x& = & 1170 \\\Leftrightarrow & \frac{-203x}{ \color{red}{-203} }& = & \frac{1170}{-203} \\\Leftrightarrow & x = \frac{-1170}{203} & & \\ & V = \left\{ \frac{-1170}{203} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -168x-1260 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+168x} & = & \color{red}{-168x} -1260 \color{blue}{+168x} \color{blue}{+60} \\\Leftrightarrow & 203x& = & -1200 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-1200}{203} \\\Leftrightarrow & x = \frac{-1200}{203} & & \\ & V = \left\{ \frac{-1200}{203} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{1}{6}x-4 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x-\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+60& = & 65x-1560 \\\Leftrightarrow & 78x \color{red}{+60} \color{blue}{-60} \color{blue}{-65x} & = & \color{red}{65x} -1560 \color{blue}{-65x} \color{blue}{-60} \\\Leftrightarrow & 13x& = & -1620 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-1620}{13} \\\Leftrightarrow & x = \frac{-1620}{13} & & \\ & V = \left\{ \frac{-1620}{13} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+4& = & 15x-90 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-15x} & = & \color{red}{15x} -90 \color{blue}{-15x} \color{blue}{-4} \\\Leftrightarrow & -9x& = & -94 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-94}{-9} \\\Leftrightarrow & x = \frac{94}{9} & & \\ & V = \left\{ \frac{94}{9} \right\} & \\\end{align}\)
8. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 60 }{ \color{blue}{260} })& = & (\frac{65}{ \color{blue}{260} }x+\frac{1040}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-60& = & 65x+1040 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{-65x} & = & \color{red}{65x} +1040 \color{blue}{-65x} \color{blue}{+60} \\\Leftrightarrow & -13x& = & 1100 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{1100}{-13} \\\Leftrightarrow & x = \frac{-1100}{13} & & \\ & V = \left\{ \frac{-1100}{13} \right\} & \\\end{align}\)
9. \(\text{546 is het kleinste gemene veelvoud van 7, 13 en 6} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{5}{6}x-2 \\\Leftrightarrow & \color{blue}{546.} (\frac{78x}{ \color{blue}{546} }+ \frac{ 84 }{ \color{blue}{546} })& = & (\frac{455}{ \color{blue}{546} }x-\frac{1092}{ \color{blue}{546} }) \color{blue}{.546} \\\Leftrightarrow & 78x+84& = & 455x-1092 \\\Leftrightarrow & 78x \color{red}{+84} \color{blue}{-84} \color{blue}{-455x} & = & \color{red}{455x} -1092 \color{blue}{-455x} \color{blue}{-84} \\\Leftrightarrow & -377x& = & -1176 \\\Leftrightarrow & \frac{-377x}{ \color{red}{-377} }& = & \frac{-1176}{-377} \\\Leftrightarrow & x = \frac{1176}{377} & & \\ & V = \left\{ \frac{1176}{377} \right\} & \\\end{align}\)
10. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{13}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 50 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{260}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+50& = & 65x-260 \\\Leftrightarrow & 26x \color{red}{+50} \color{blue}{-50} \color{blue}{-65x} & = & \color{red}{65x} -260 \color{blue}{-65x} \color{blue}{-50} \\\Leftrightarrow & -39x& = & -310 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-310}{-39} \\\Leftrightarrow & x = \frac{310}{39} & & \\ & V = \left\{ \frac{310}{39} \right\} & \\\end{align}\)
11. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{5}{2}x+4 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{45}{ \color{blue}{18} }x+\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+8& = & 45x+72 \\\Leftrightarrow & 6x \color{red}{+8} \color{blue}{-8} \color{blue}{-45x} & = & \color{red}{45x} +72 \color{blue}{-45x} \color{blue}{-8} \\\Leftrightarrow & -39x& = & 64 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{64}{-39} \\\Leftrightarrow & x = \frac{-64}{39} & & \\ & V = \left\{ \frac{-64}{39} \right\} & \\\end{align}\)
12. \(\text{220 is het kleinste gemene veelvoud van 5, 11 en 4} \\ \begin{align} & \frac{x}{5}-\frac{2}{11}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{220.} (\frac{44x}{ \color{blue}{220} }- \frac{ 40 }{ \color{blue}{220} })& = & (\frac{55}{ \color{blue}{220} }x+\frac{880}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 44x-40& = & 55x+880 \\\Leftrightarrow & 44x \color{red}{-40} \color{blue}{+40} \color{blue}{-55x} & = & \color{red}{55x} +880 \color{blue}{-55x} \color{blue}{+40} \\\Leftrightarrow & -11x& = & 920 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{920}{-11} \\\Leftrightarrow & x = \frac{-920}{11} & & \\ & V = \left\{ \frac{-920}{11} \right\} & \\\end{align}\)