Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 30 }{ \color{blue}{130} })& = & (\frac{156}{ \color{blue}{130} }x+\frac{130}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-30& = & 156x+130 \\\Leftrightarrow & 65x \color{red}{-30} \color{blue}{+30} \color{blue}{-156x} & = & \color{red}{156x} +130 \color{blue}{-156x} \color{blue}{+30} \\\Leftrightarrow & -91x& = & 160 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{160}{-91} \\\Leftrightarrow & x = \frac{-160}{91} & & \\ & V = \left\{ \frac{-160}{91} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 42x-1050 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-42x} & = & \color{red}{42x} -1050 \color{blue}{-42x} \color{blue}{+120} \\\Leftrightarrow & -7x& = & -930 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-930}{-7} \\\Leftrightarrow & x = \frac{930}{7} & & \\ & V = \left\{ \frac{930}{7} \right\} & \\\end{align}\)
3. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{4}{3}x-7 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{24}{ \color{blue}{18} }x-\frac{126}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-4& = & 24x-126 \\\Leftrightarrow & 9x \color{red}{-4} \color{blue}{+4} \color{blue}{-24x} & = & \color{red}{24x} -126 \color{blue}{-24x} \color{blue}{+4} \\\Leftrightarrow & -15x& = & -122 \\\Leftrightarrow & \frac{-15x}{ \color{red}{-15} }& = & \frac{-122}{-15} \\\Leftrightarrow & x = \frac{122}{15} & & \\ & V = \left\{ \frac{122}{15} \right\} & \\\end{align}\)
4. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }- \frac{ 70 }{ \color{blue}{385} })& = & (\frac{-154}{ \color{blue}{385} }x+\frac{1540}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x-70& = & -154x+1540 \\\Leftrightarrow & 55x \color{red}{-70} \color{blue}{+70} \color{blue}{+154x} & = & \color{red}{-154x} +1540 \color{blue}{+154x} \color{blue}{+70} \\\Leftrightarrow & 209x& = & 1610 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{1610}{209} \\\Leftrightarrow & x = \frac{1610}{209} & & \\ & V = \left\{ \frac{1610}{209} \right\} & \\\end{align}\)
5. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{-260}{ \color{blue}{156} }x+\frac{1092}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & -260x+1092 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{+260x} & = & \color{red}{-260x} +1092 \color{blue}{+260x} \color{blue}{+36} \\\Leftrightarrow & 299x& = & 1128 \\\Leftrightarrow & \frac{299x}{ \color{red}{299} }& = & \frac{1128}{299} \\\Leftrightarrow & x = \frac{1128}{299} & & \\ & V = \left\{ \frac{1128}{299} \right\} & \\\end{align}\)
6. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 16 }{ \color{blue}{36} })& = & (\frac{-24}{ \color{blue}{36} }x-\frac{144}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-16& = & -24x-144 \\\Leftrightarrow & 9x \color{red}{-16} \color{blue}{+16} \color{blue}{+24x} & = & \color{red}{-24x} -144 \color{blue}{+24x} \color{blue}{+16} \\\Leftrightarrow & 33x& = & -128 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-128}{33} \\\Leftrightarrow & x = \frac{-128}{33} & & \\ & V = \left\{ \frac{-128}{33} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{14}& = & \frac{-7}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 45 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-45& = & -294x+420 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{+294x} & = & \color{red}{-294x} +420 \color{blue}{+294x} \color{blue}{+45} \\\Leftrightarrow & 329x& = & 465 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{465}{329} \\\Leftrightarrow & x = \frac{465}{329} & & \\ & V = \left\{ \frac{465}{329} \right\} & \\\end{align}\)
8. \(\text{10 is het kleinste gemene veelvoud van 5, 10 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{10.} (\frac{2x}{ \color{blue}{10} }- \frac{ 3 }{ \color{blue}{10} })& = & (\frac{5}{ \color{blue}{10} }x-\frac{40}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 2x-3& = & 5x-40 \\\Leftrightarrow & 2x \color{red}{-3} \color{blue}{+3} \color{blue}{-5x} & = & \color{red}{5x} -40 \color{blue}{-5x} \color{blue}{+3} \\\Leftrightarrow & -3x& = & -37 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-37}{-3} \\\Leftrightarrow & x = \frac{37}{3} & & \\ & V = \left\{ \frac{37}{3} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-48& = & 28x+588 \\\Leftrightarrow & 21x \color{red}{-48} \color{blue}{+48} \color{blue}{-28x} & = & \color{red}{28x} +588 \color{blue}{-28x} \color{blue}{+48} \\\Leftrightarrow & -7x& = & 636 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{636}{-7} \\\Leftrightarrow & x = \frac{-636}{7} & & \\ & V = \left\{ \frac{-636}{7} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-30& = & 35x-280 \\\Leftrightarrow & 14x \color{red}{-30} \color{blue}{+30} \color{blue}{-35x} & = & \color{red}{35x} -280 \color{blue}{-35x} \color{blue}{+30} \\\Leftrightarrow & -21x& = & -250 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-250}{-21} \\\Leftrightarrow & x = \frac{250}{21} & & \\ & V = \left\{ \frac{250}{21} \right\} & \\\end{align}\)
11. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{7}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 100 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x+\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-100& = & -112x+140 \\\Leftrightarrow & 35x \color{red}{-100} \color{blue}{+100} \color{blue}{+112x} & = & \color{red}{-112x} +140 \color{blue}{+112x} \color{blue}{+100} \\\Leftrightarrow & 147x& = & 240 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{240}{147} \\\Leftrightarrow & x = \frac{80}{49} & & \\ & V = \left\{ \frac{80}{49} \right\} & \\\end{align}\)
12. \(\text{12 is het kleinste gemene veelvoud van 2, 12 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{12}& = & \frac{4}{3}x+4 \\\Leftrightarrow & \color{blue}{12.} (\frac{6x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{16}{ \color{blue}{12} }x+\frac{48}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 6x+5& = & 16x+48 \\\Leftrightarrow & 6x \color{red}{+5} \color{blue}{-5} \color{blue}{-16x} & = & \color{red}{16x} +48 \color{blue}{-16x} \color{blue}{-5} \\\Leftrightarrow & -10x& = & 43 \\\Leftrightarrow & \frac{-10x}{ \color{red}{-10} }& = & \frac{43}{-10} \\\Leftrightarrow & x = \frac{-43}{10} & & \\ & V = \left\{ \frac{-43}{10} \right\} & \\\end{align}\)