Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{2}{3}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{28}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & 28x+210 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-28x} & = & \color{red}{28x} +210 \color{blue}{-28x} \color{blue}{+24} \\\Leftrightarrow & -7x& = & 234 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{234}{-7} \\\Leftrightarrow & x = \frac{-234}{7} & & \\ & V = \left\{ \frac{-234}{7} \right\} & \\\end{align}\)
2. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{8}& = & \frac{-5}{3}x+1 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-40}{ \color{blue}{24} }x+\frac{24}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x-15& = & -40x+24 \\\Leftrightarrow & 6x \color{red}{-15} \color{blue}{+15} \color{blue}{+40x} & = & \color{red}{-40x} +24 \color{blue}{+40x} \color{blue}{+15} \\\Leftrightarrow & 46x& = & 39 \\\Leftrightarrow & \frac{46x}{ \color{red}{46} }& = & \frac{39}{46} \\\Leftrightarrow & x = \frac{39}{46} & & \\ & V = \left\{ \frac{39}{46} \right\} & \\\end{align}\)
3. \(\text{36 is het kleinste gemene veelvoud van 3, 9 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{9}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{36.} (\frac{12x}{ \color{blue}{36} }+ \frac{ 16 }{ \color{blue}{36} })& = & (\frac{9}{ \color{blue}{36} }x+\frac{144}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 12x+16& = & 9x+144 \\\Leftrightarrow & 12x \color{red}{+16} \color{blue}{-16} \color{blue}{-9x} & = & \color{red}{9x} +144 \color{blue}{-9x} \color{blue}{-16} \\\Leftrightarrow & 3x& = & 128 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{128}{3} \\\Leftrightarrow & x = \frac{128}{3} & & \\ & V = \left\{ \frac{128}{3} \right\} & \\\end{align}\)
4. \(\text{140 is het kleinste gemene veelvoud van 5, 7 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{1}{4}x-1 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 60 }{ \color{blue}{140} })& = & (\frac{35}{ \color{blue}{140} }x-\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-60& = & 35x-140 \\\Leftrightarrow & 28x \color{red}{-60} \color{blue}{+60} \color{blue}{-35x} & = & \color{red}{35x} -140 \color{blue}{-35x} \color{blue}{+60} \\\Leftrightarrow & -7x& = & -80 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-80}{-7} \\\Leftrightarrow & x = \frac{80}{7} & & \\ & V = \left\{ \frac{80}{7} \right\} & \\\end{align}\)
5. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 45 }{ \color{blue}{105} })& = & (\frac{-42}{ \color{blue}{105} }x+\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-45& = & -42x+210 \\\Leftrightarrow & 35x \color{red}{-45} \color{blue}{+45} \color{blue}{+42x} & = & \color{red}{-42x} +210 \color{blue}{+42x} \color{blue}{+45} \\\Leftrightarrow & 77x& = & 255 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{255}{77} \\\Leftrightarrow & x = \frac{255}{77} & & \\ & V = \left\{ \frac{255}{77} \right\} & \\\end{align}\)
6. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{1}{6}x-6 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{15}{ \color{blue}{90} }x-\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+20& = & 15x-540 \\\Leftrightarrow & 18x \color{red}{+20} \color{blue}{-20} \color{blue}{-15x} & = & \color{red}{15x} -540 \color{blue}{-15x} \color{blue}{-20} \\\Leftrightarrow & 3x& = & -560 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-560}{3} \\\Leftrightarrow & x = \frac{-560}{3} & & \\ & V = \left\{ \frac{-560}{3} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{5}{13}& = & \frac{7}{2}x-7 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 30 }{ \color{blue}{78} })& = & (\frac{273}{ \color{blue}{78} }x-\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+30& = & 273x-546 \\\Leftrightarrow & 26x \color{red}{+30} \color{blue}{-30} \color{blue}{-273x} & = & \color{red}{273x} -546 \color{blue}{-273x} \color{blue}{-30} \\\Leftrightarrow & -247x& = & -576 \\\Leftrightarrow & \frac{-247x}{ \color{red}{-247} }& = & \frac{-576}{-247} \\\Leftrightarrow & x = \frac{576}{247} & & \\ & V = \left\{ \frac{576}{247} \right\} & \\\end{align}\)
8. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 20 }{ \color{blue}{35} })& = & (\frac{7}{ \color{blue}{35} }x-\frac{70}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+20& = & 7x-70 \\\Leftrightarrow & 5x \color{red}{+20} \color{blue}{-20} \color{blue}{-7x} & = & \color{red}{7x} -70 \color{blue}{-7x} \color{blue}{-20} \\\Leftrightarrow & -2x& = & -90 \\\Leftrightarrow & \frac{-2x}{ \color{red}{-2} }& = & \frac{-90}{-2} \\\Leftrightarrow & x = 45 & & \\ & V = \left\{ 45 \right\} & \\\end{align}\)
9. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{7}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{42}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 42x+240 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-42x} & = & \color{red}{42x} +240 \color{blue}{-42x} \color{blue}{-9} \\\Leftrightarrow & -37x& = & 231 \\\Leftrightarrow & \frac{-37x}{ \color{red}{-37} }& = & \frac{231}{-37} \\\Leftrightarrow & x = \frac{-231}{37} & & \\ & V = \left\{ \frac{-231}{37} \right\} & \\\end{align}\)
10. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{-2}{5}x+6 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-56}{ \color{blue}{140} }x+\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & -56x+840 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{+56x} & = & \color{red}{-56x} +840 \color{blue}{+56x} \color{blue}{-40} \\\Leftrightarrow & 91x& = & 800 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{800}{91} \\\Leftrightarrow & x = \frac{800}{91} & & \\ & V = \left\{ \frac{800}{91} \right\} & \\\end{align}\)
11. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-56}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & -56x+588 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{+56x} & = & \color{red}{-56x} +588 \color{blue}{+56x} \color{blue}{-36} \\\Leftrightarrow & 77x& = & 552 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{552}{77} \\\Leftrightarrow & x = \frac{552}{77} & & \\ & V = \left\{ \frac{552}{77} \right\} & \\\end{align}\)
12. \(\text{140 is het kleinste gemene veelvoud van 5, 14 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{14}& = & \frac{1}{4}x+2 \\\Leftrightarrow & \color{blue}{140.} (\frac{28x}{ \color{blue}{140} }- \frac{ 30 }{ \color{blue}{140} })& = & (\frac{35}{ \color{blue}{140} }x+\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 28x-30& = & 35x+280 \\\Leftrightarrow & 28x \color{red}{-30} \color{blue}{+30} \color{blue}{-35x} & = & \color{red}{35x} +280 \color{blue}{-35x} \color{blue}{+30} \\\Leftrightarrow & -7x& = & 310 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{310}{-7} \\\Leftrightarrow & x = \frac{-310}{7} & & \\ & V = \left\{ \frac{-310}{7} \right\} & \\\end{align}\)