Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}+\frac{5}{11}& = & \frac{-2}{3}x+2 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }+ \frac{ 105 }{ \color{blue}{231} })& = & (\frac{-154}{ \color{blue}{231} }x+\frac{462}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x+105& = & -154x+462 \\\Leftrightarrow & 33x \color{red}{+105} \color{blue}{-105} \color{blue}{+154x} & = & \color{red}{-154x} +462 \color{blue}{+154x} \color{blue}{-105} \\\Leftrightarrow & 187x& = & 357 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{357}{187} \\\Leftrightarrow & x = \frac{21}{11} & & \\ & V = \left\{ \frac{21}{11} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -294x+1680 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+294x} & = & \color{red}{-294x} +1680 \color{blue}{+294x} \color{blue}{-120} \\\Leftrightarrow & 329x& = & 1560 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{1560}{329} \\\Leftrightarrow & x = \frac{1560}{329} & & \\ & V = \left\{ \frac{1560}{329} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 7, 14 en 3} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{7}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }- \frac{ 9 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x-9& = & 98x-42 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{-98x} & = & \color{red}{98x} -42 \color{blue}{-98x} \color{blue}{+9} \\\Leftrightarrow & -92x& = & -33 \\\Leftrightarrow & \frac{-92x}{ \color{red}{-92} }& = & \frac{-33}{-92} \\\Leftrightarrow & x = \frac{33}{92} & & \\ & V = \left\{ \frac{33}{92} \right\} & \\\end{align}\)
4. \(\text{18 is het kleinste gemene veelvoud van 3, 9 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{18.} (\frac{6x}{ \color{blue}{18} }+ \frac{ 4 }{ \color{blue}{18} })& = & (\frac{9}{ \color{blue}{18} }x-\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 6x+4& = & 9x-126 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-9x} & = & \color{red}{9x} -126 \color{blue}{-9x} \color{blue}{-4} \\\Leftrightarrow & -3x& = & -130 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-130}{-3} \\\Leftrightarrow & x = \frac{130}{3} & & \\ & V = \left\{ \frac{130}{3} \right\} & \\\end{align}\)
5. \(\text{10 is het kleinste gemene veelvoud van 5, 10 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{7}{2}x+7 \\\Leftrightarrow & \color{blue}{10.} (\frac{2x}{ \color{blue}{10} }- \frac{ 3 }{ \color{blue}{10} })& = & (\frac{35}{ \color{blue}{10} }x+\frac{70}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 2x-3& = & 35x+70 \\\Leftrightarrow & 2x \color{red}{-3} \color{blue}{+3} \color{blue}{-35x} & = & \color{red}{35x} +70 \color{blue}{-35x} \color{blue}{+3} \\\Leftrightarrow & -33x& = & 73 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{73}{-33} \\\Leftrightarrow & x = \frac{-73}{33} & & \\ & V = \left\{ \frac{-73}{33} \right\} & \\\end{align}\)
6. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{1}{5}x-6 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }+ \frac{ 140 }{ \color{blue}{385} })& = & (\frac{77}{ \color{blue}{385} }x-\frac{2310}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x+140& = & 77x-2310 \\\Leftrightarrow & 55x \color{red}{+140} \color{blue}{-140} \color{blue}{-77x} & = & \color{red}{77x} -2310 \color{blue}{-77x} \color{blue}{-140} \\\Leftrightarrow & -22x& = & -2450 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{-2450}{-22} \\\Leftrightarrow & x = \frac{1225}{11} & & \\ & V = \left\{ \frac{1225}{11} \right\} & \\\end{align}\)
7. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{-7}{4}x-7 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }+ \frac{ 112 }{ \color{blue}{420} })& = & (\frac{-735}{ \color{blue}{420} }x-\frac{2940}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x+112& = & -735x-2940 \\\Leftrightarrow & 60x \color{red}{+112} \color{blue}{-112} \color{blue}{+735x} & = & \color{red}{-735x} -2940 \color{blue}{+735x} \color{blue}{-112} \\\Leftrightarrow & 795x& = & -3052 \\\Leftrightarrow & \frac{795x}{ \color{red}{795} }& = & \frac{-3052}{795} \\\Leftrightarrow & x = \frac{-3052}{795} & & \\ & V = \left\{ \frac{-3052}{795} \right\} & \\\end{align}\)
8. \(\text{260 is het kleinste gemene veelvoud van 4, 13 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{13}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{260.} (\frac{65x}{ \color{blue}{260} }- \frac{ 100 }{ \color{blue}{260} })& = & (\frac{-208}{ \color{blue}{260} }x-\frac{520}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 65x-100& = & -208x-520 \\\Leftrightarrow & 65x \color{red}{-100} \color{blue}{+100} \color{blue}{+208x} & = & \color{red}{-208x} -520 \color{blue}{+208x} \color{blue}{+100} \\\Leftrightarrow & 273x& = & -420 \\\Leftrightarrow & \frac{273x}{ \color{red}{273} }& = & \frac{-420}{273} \\\Leftrightarrow & x = \frac{-20}{13} & & \\ & V = \left\{ \frac{-20}{13} \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{10}& = & \frac{2}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{12}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x+9& = & 12x+150 \\\Leftrightarrow & 10x \color{red}{+9} \color{blue}{-9} \color{blue}{-12x} & = & \color{red}{12x} +150 \color{blue}{-12x} \color{blue}{-9} \\\Leftrightarrow & -2x& = & 141 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{141}{-2} \\\Leftrightarrow & x = \frac{-141}{2} & & \\ & V = \left\{ \frac{-141}{2} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{9}& = & \frac{1}{2}x+7 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x+\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-40& = & 45x+630 \\\Leftrightarrow & 18x \color{red}{-40} \color{blue}{+40} \color{blue}{-45x} & = & \color{red}{45x} +630 \color{blue}{-45x} \color{blue}{+40} \\\Leftrightarrow & -27x& = & 670 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{670}{-27} \\\Leftrightarrow & x = \frac{-670}{27} & & \\ & V = \left\{ \frac{-670}{27} \right\} & \\\end{align}\)
11. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{8}& = & \frac{-5}{6}x+5 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }+ \frac{ 75 }{ \color{blue}{120} })& = & (\frac{-100}{ \color{blue}{120} }x+\frac{600}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x+75& = & -100x+600 \\\Leftrightarrow & 24x \color{red}{+75} \color{blue}{-75} \color{blue}{+100x} & = & \color{red}{-100x} +600 \color{blue}{+100x} \color{blue}{-75} \\\Leftrightarrow & 124x& = & 525 \\\Leftrightarrow & \frac{124x}{ \color{red}{124} }& = & \frac{525}{124} \\\Leftrightarrow & x = \frac{525}{124} & & \\ & V = \left\{ \frac{525}{124} \right\} & \\\end{align}\)
12. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{560}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 35x-560 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-35x} & = & \color{red}{35x} -560 \color{blue}{-35x} \color{blue}{-20} \\\Leftrightarrow & -21x& = & -580 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-580}{-21} \\\Leftrightarrow & x = \frac{580}{21} & & \\ & V = \left\{ \frac{580}{21} \right\} & \\\end{align}\)