Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{13}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 90 }{ \color{blue}{390} })& = & (\frac{78}{ \color{blue}{390} }x-\frac{2730}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-90& = & 78x-2730 \\\Leftrightarrow & 65x \color{red}{-90} \color{blue}{+90} \color{blue}{-78x} & = & \color{red}{78x} -2730 \color{blue}{-78x} \color{blue}{+90} \\\Leftrightarrow & -13x& = & -2640 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-2640}{-13} \\\Leftrightarrow & x = \frac{2640}{13} & & \\ & V = \left\{ \frac{2640}{13} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{7}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+24& = & -28x-168 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+28x} & = & \color{red}{-28x} -168 \color{blue}{+28x} \color{blue}{-24} \\\Leftrightarrow & 49x& = & -192 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-192}{49} \\\Leftrightarrow & x = \frac{-192}{49} & & \\ & V = \left\{ \frac{-192}{49} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{7}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & 98x-42 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{-98x} & = & \color{red}{98x} -42 \color{blue}{-98x} \color{blue}{-12} \\\Leftrightarrow & -77x& = & -54 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-54}{-77} \\\Leftrightarrow & x = \frac{54}{77} & & \\ & V = \left\{ \frac{54}{77} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{-3}{4}x+3 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-63}{ \color{blue}{84} }x+\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+48& = & -63x+252 \\\Leftrightarrow & 28x \color{red}{+48} \color{blue}{-48} \color{blue}{+63x} & = & \color{red}{-63x} +252 \color{blue}{+63x} \color{blue}{-48} \\\Leftrightarrow & 91x& = & 204 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{204}{91} \\\Leftrightarrow & x = \frac{204}{91} & & \\ & V = \left\{ \frac{204}{91} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-7}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -294x-210 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+294x} & = & \color{red}{-294x} -210 \color{blue}{+294x} \color{blue}{+60} \\\Leftrightarrow & 329x& = & -150 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{-150}{329} \\\Leftrightarrow & x = \frac{-150}{329} & & \\ & V = \left\{ \frac{-150}{329} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{4}{7}& = & \frac{1}{6}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-120& = & 35x-1260 \\\Leftrightarrow & 42x \color{red}{-120} \color{blue}{+120} \color{blue}{-35x} & = & \color{red}{35x} -1260 \color{blue}{-35x} \color{blue}{+120} \\\Leftrightarrow & 7x& = & -1140 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-1140}{7} \\\Leftrightarrow & x = \frac{-1140}{7} & & \\ & V = \left\{ \frac{-1140}{7} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{1}{6}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{15}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-20& = & 15x-90 \\\Leftrightarrow & 18x \color{red}{-20} \color{blue}{+20} \color{blue}{-15x} & = & \color{red}{15x} -90 \color{blue}{-15x} \color{blue}{+20} \\\Leftrightarrow & 3x& = & -70 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-70}{3} \\\Leftrightarrow & x = \frac{-70}{3} & & \\ & V = \left\{ \frac{-70}{3} \right\} & \\\end{align}\)
8. \(\text{10 is het kleinste gemene veelvoud van 5, 10 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{10.} (\frac{2x}{ \color{blue}{10} }- \frac{ 3 }{ \color{blue}{10} })& = & (\frac{5}{ \color{blue}{10} }x-\frac{10}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 2x-3& = & 5x-10 \\\Leftrightarrow & 2x \color{red}{-3} \color{blue}{+3} \color{blue}{-5x} & = & \color{red}{5x} -10 \color{blue}{-5x} \color{blue}{+3} \\\Leftrightarrow & -3x& = & -7 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-7}{-3} \\\Leftrightarrow & x = \frac{7}{3} & & \\ & V = \left\{ \frac{7}{3} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-100}{ \color{blue}{60} }x+\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & -100x+480 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{+100x} & = & \color{red}{-100x} +480 \color{blue}{+100x} \color{blue}{-16} \\\Leftrightarrow & 115x& = & 464 \\\Leftrightarrow & \frac{115x}{ \color{red}{115} }& = & \frac{464}{115} \\\Leftrightarrow & x = \frac{464}{115} & & \\ & V = \left\{ \frac{464}{115} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{7}{5}x+6 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{126}{ \color{blue}{90} }x+\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & 126x+540 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{-126x} & = & \color{red}{126x} +540 \color{blue}{-126x} \color{blue}{-20} \\\Leftrightarrow & -111x& = & 520 \\\Leftrightarrow & \frac{-111x}{ \color{red}{-111} }& = & \frac{520}{-111} \\\Leftrightarrow & x = \frac{-520}{111} & & \\ & V = \left\{ \frac{-520}{111} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 7, 7 en 6} \\ \begin{align} & \frac{x}{7}+\frac{3}{7}& = & \frac{7}{6}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{49}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+18& = & 49x-252 \\\Leftrightarrow & 6x \color{red}{+18} \color{blue}{-18} \color{blue}{-49x} & = & \color{red}{49x} -252 \color{blue}{-49x} \color{blue}{-18} \\\Leftrightarrow & -43x& = & -270 \\\Leftrightarrow & \frac{-43x}{ \color{red}{-43} }& = & \frac{-270}{-43} \\\Leftrightarrow & x = \frac{270}{43} & & \\ & V = \left\{ \frac{270}{43} \right\} & \\\end{align}\)
12. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{6}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }+ \frac{ 5 }{ \color{blue}{6} })& = & (\frac{2}{ \color{blue}{6} }x+\frac{12}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x+5& = & 2x+12 \\\Leftrightarrow & 3x \color{red}{+5} \color{blue}{-5} \color{blue}{-2x} & = & \color{red}{2x} +12 \color{blue}{-2x} \color{blue}{-5} \\\Leftrightarrow & x& = & 7 \\ & V = \left\{ 7 \right\} & \\\end{align}\)