Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }+ \frac{ 40 }{ \color{blue}{260} })& = & (\frac{-364}{ \color{blue}{260} }x-\frac{780}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x+40& = & -364x-780 \\\Leftrightarrow & 65x \color{red}{+40} \color{blue}{-40} \color{blue}{+364x} & = & \color{red}{-364x} -780 \color{blue}{+364x} \color{blue}{-40} \\\Leftrightarrow & 429x& = & -820 \\\Leftrightarrow & \frac{429x}{ \color{red}{429} }& = & \frac{-820}{429} \\\Leftrightarrow & x = \frac{-820}{429} & & \\ & V = \left\{ \frac{-820}{429} \right\} & \\\end{align}\)
2. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 8 }{ \color{blue}{28} })& = & (\frac{7}{ \color{blue}{28} }x+\frac{112}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+8& = & 7x+112 \\\Leftrightarrow & 4x \color{red}{+8} \color{blue}{-8} \color{blue}{-7x} & = & \color{red}{7x} +112 \color{blue}{-7x} \color{blue}{-8} \\\Leftrightarrow & -3x& = & 104 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{104}{-3} \\\Leftrightarrow & x = \frac{-104}{3} & & \\ & V = \left\{ \frac{-104}{3} \right\} & \\\end{align}\)
3. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{9}& = & \frac{3}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{54}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x-50& = & 54x-630 \\\Leftrightarrow & 45x \color{red}{-50} \color{blue}{+50} \color{blue}{-54x} & = & \color{red}{54x} -630 \color{blue}{-54x} \color{blue}{+50} \\\Leftrightarrow & -9x& = & -580 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-580}{-9} \\\Leftrightarrow & x = \frac{580}{9} & & \\ & V = \left\{ \frac{580}{9} \right\} & \\\end{align}\)
4. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{11}& = & \frac{3}{5}x+4 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 30 }{ \color{blue}{110} })& = & (\frac{66}{ \color{blue}{110} }x+\frac{440}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+30& = & 66x+440 \\\Leftrightarrow & 55x \color{red}{+30} \color{blue}{-30} \color{blue}{-66x} & = & \color{red}{66x} +440 \color{blue}{-66x} \color{blue}{-30} \\\Leftrightarrow & -11x& = & 410 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{410}{-11} \\\Leftrightarrow & x = \frac{-410}{11} & & \\ & V = \left\{ \frac{-410}{11} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 2, 12 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{12}& = & \frac{1}{5}x+4 \\\Leftrightarrow & \color{blue}{60.} (\frac{30x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{12}{ \color{blue}{60} }x+\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 30x-25& = & 12x+240 \\\Leftrightarrow & 30x \color{red}{-25} \color{blue}{+25} \color{blue}{-12x} & = & \color{red}{12x} +240 \color{blue}{-12x} \color{blue}{+25} \\\Leftrightarrow & 18x& = & 265 \\\Leftrightarrow & \frac{18x}{ \color{red}{18} }& = & \frac{265}{18} \\\Leftrightarrow & x = \frac{265}{18} & & \\ & V = \left\{ \frac{265}{18} \right\} & \\\end{align}\)
6. \(\text{182 is het kleinste gemene veelvoud van 7, 13 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{13}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{182.} (\frac{26x}{ \color{blue}{182} }+ \frac{ 56 }{ \color{blue}{182} })& = & (\frac{91}{ \color{blue}{182} }x-\frac{1274}{ \color{blue}{182} }) \color{blue}{.182} \\\Leftrightarrow & 26x+56& = & 91x-1274 \\\Leftrightarrow & 26x \color{red}{+56} \color{blue}{-56} \color{blue}{-91x} & = & \color{red}{91x} -1274 \color{blue}{-91x} \color{blue}{-56} \\\Leftrightarrow & -65x& = & -1330 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-1330}{-65} \\\Leftrightarrow & x = \frac{266}{13} & & \\ & V = \left\{ \frac{266}{13} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x+\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 45x+630 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-45x} & = & \color{red}{45x} +630 \color{blue}{-45x} \color{blue}{+40} \\\Leftrightarrow & -27x& = & 670 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{670}{-27} \\\Leftrightarrow & x = \frac{-670}{27} & & \\ & V = \left\{ \frac{-670}{27} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{7}{5}x-4 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{294}{ \color{blue}{210} }x-\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 294x-840 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-294x} & = & \color{red}{294x} -840 \color{blue}{-294x} \color{blue}{+60} \\\Leftrightarrow & -259x& = & -780 \\\Leftrightarrow & \frac{-259x}{ \color{red}{-259} }& = & \frac{-780}{-259} \\\Leftrightarrow & x = \frac{780}{259} & & \\ & V = \left\{ \frac{780}{259} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{1}{4}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{15}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+16& = & 15x+360 \\\Leftrightarrow & 20x \color{red}{+16} \color{blue}{-16} \color{blue}{-15x} & = & \color{red}{15x} +360 \color{blue}{-15x} \color{blue}{-16} \\\Leftrightarrow & 5x& = & 344 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{344}{5} \\\Leftrightarrow & x = \frac{344}{5} & & \\ & V = \left\{ \frac{344}{5} \right\} & \\\end{align}\)
10. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{-2}{3}x-3 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{-88}{ \color{blue}{132} }x-\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & -88x-396 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{+88x} & = & \color{red}{-88x} -396 \color{blue}{+88x} \color{blue}{-24} \\\Leftrightarrow & 121x& = & -420 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{-420}{121} \\\Leftrightarrow & x = \frac{-420}{121} & & \\ & V = \left\{ \frac{-420}{121} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{14}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 30 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x+\frac{420}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+30& = & -140x+420 \\\Leftrightarrow & 21x \color{red}{+30} \color{blue}{-30} \color{blue}{+140x} & = & \color{red}{-140x} +420 \color{blue}{+140x} \color{blue}{-30} \\\Leftrightarrow & 161x& = & 390 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{390}{161} \\\Leftrightarrow & x = \frac{390}{161} & & \\ & V = \left\{ \frac{390}{161} \right\} & \\\end{align}\)
12. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{-8}{3}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-160}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & -160x-240 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+160x} & = & \color{red}{-160x} -240 \color{blue}{+160x} \color{blue}{-8} \\\Leftrightarrow & 175x& = & -248 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{-248}{175} \\\Leftrightarrow & x = \frac{-248}{175} & & \\ & V = \left\{ \frac{-248}{175} \right\} & \\\end{align}\)