Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{7}& = & \frac{-2}{5}x+1 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 75 }{ \color{blue}{105} })& = & (\frac{-42}{ \color{blue}{105} }x+\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-75& = & -42x+105 \\\Leftrightarrow & 35x \color{red}{-75} \color{blue}{+75} \color{blue}{+42x} & = & \color{red}{-42x} +105 \color{blue}{+42x} \color{blue}{+75} \\\Leftrightarrow & 77x& = & 180 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{180}{77} \\\Leftrightarrow & x = \frac{180}{77} & & \\ & V = \left\{ \frac{180}{77} \right\} & \\\end{align}\)
2. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{7}{3}x+1 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{42}{ \color{blue}{18} }x+\frac{18}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-4& = & 42x+18 \\\Leftrightarrow & 9x \color{red}{-4} \color{blue}{+4} \color{blue}{-42x} & = & \color{red}{42x} +18 \color{blue}{-42x} \color{blue}{+4} \\\Leftrightarrow & -33x& = & 22 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{22}{-33} \\\Leftrightarrow & x = \frac{-2}{3} & & \\ & V = \left\{ \frac{-2}{3} \right\} & \\\end{align}\)
3. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{-7}{5}x-1 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x-\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & -462x-330 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{+462x} & = & \color{red}{-462x} -330 \color{blue}{+462x} \color{blue}{+150} \\\Leftrightarrow & 517x& = & -180 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{-180}{517} \\\Leftrightarrow & x = \frac{-180}{517} & & \\ & V = \left\{ \frac{-180}{517} \right\} & \\\end{align}\)
4. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 50 }{ \color{blue}{110} })& = & (\frac{-154}{ \color{blue}{110} }x+\frac{660}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+50& = & -154x+660 \\\Leftrightarrow & 55x \color{red}{+50} \color{blue}{-50} \color{blue}{+154x} & = & \color{red}{-154x} +660 \color{blue}{+154x} \color{blue}{-50} \\\Leftrightarrow & 209x& = & 610 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{610}{209} \\\Leftrightarrow & x = \frac{610}{209} & & \\ & V = \left\{ \frac{610}{209} \right\} & \\\end{align}\)
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. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{-4}{5}x-4 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }- \frac{ 10 }{ \color{blue}{35} })& = & (\frac{-28}{ \color{blue}{35} }x-\frac{140}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x-10& = & -28x-140 \\\Leftrightarrow & 5x \color{red}{-10} \color{blue}{+10} \color{blue}{+28x} & = & \color{red}{-28x} -140 \color{blue}{+28x} \color{blue}{+10} \\\Leftrightarrow & 33x& = & -130 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{-130}{33} \\\Leftrightarrow & x = \frac{-130}{33} & & \\ & V = \left\{ \frac{-130}{33} \right\} & \\\end{align}\)
7. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{-2}{5}x+7 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-12}{ \color{blue}{30} }x+\frac{210}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & -12x+210 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{+12x} & = & \color{red}{-12x} +210 \color{blue}{+12x} \color{blue}{+4} \\\Leftrightarrow & 27x& = & 214 \\\Leftrightarrow & \frac{27x}{ \color{red}{27} }& = & \frac{214}{27} \\\Leftrightarrow & x = \frac{214}{27} & & \\ & V = \left\{ \frac{214}{27} \right\} & \\\end{align}\)
8. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{8}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-16}{ \color{blue}{24} }x-\frac{48}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x+15& = & -16x-48 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{+16x} & = & \color{red}{-16x} -48 \color{blue}{+16x} \color{blue}{-15} \\\Leftrightarrow & 28x& = & -63 \\\Leftrightarrow & \frac{28x}{ \color{red}{28} }& = & \frac{-63}{28} \\\Leftrightarrow & x = \frac{-9}{4} & & \\ & V = \left\{ \frac{-9}{4} \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{7}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x+\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-20& = & 35x+280 \\\Leftrightarrow & 14x \color{red}{-20} \color{blue}{+20} \color{blue}{-35x} & = & \color{red}{35x} +280 \color{blue}{-35x} \color{blue}{+20} \\\Leftrightarrow & -21x& = & 300 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{300}{-21} \\\Leftrightarrow & x = \frac{-100}{7} & & \\ & V = \left\{ \frac{-100}{7} \right\} & \\\end{align}\)
10. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{-5}{3}x-4 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x-\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-24& = & -130x-312 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{+130x} & = & \color{red}{-130x} -312 \color{blue}{+130x} \color{blue}{+24} \\\Leftrightarrow & 169x& = & -288 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-288}{169} \\\Leftrightarrow & x = \frac{-288}{169} & & \\ & V = \left\{ \frac{-288}{169} \right\} & \\\end{align}\)
11. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{-2}{5}x+8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x+\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & -156x+3120 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{+156x} & = & \color{red}{-156x} +3120 \color{blue}{+156x} \color{blue}{+120} \\\Leftrightarrow & 221x& = & 3240 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{3240}{221} \\\Leftrightarrow & x = \frac{3240}{221} & & \\ & V = \left\{ \frac{3240}{221} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{10}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 9 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+9& = & 6x+90 \\\Leftrightarrow & 5x \color{red}{+9} \color{blue}{-9} \color{blue}{-6x} & = & \color{red}{6x} +90 \color{blue}{-6x} \color{blue}{-9} \\\Leftrightarrow & -x& = & 81 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{81}{-1} \\\Leftrightarrow & x = -81 & & \\ & V = \left\{ -81 \right\} & \\\end{align}\)