Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{4}{9}=\frac{2}{3}x-1\)
2. \(\frac{x}{5}+\frac{4}{13}=\frac{-3}{4}x-2\)
3. \(\frac{x}{5}-\frac{5}{16}=\frac{1}{6}x-3\)
4. \(\frac{x}{3}+\frac{2}{13}=\frac{1}{2}x-2\)
5. \(\frac{x}{5}+\frac{5}{11}=\frac{7}{6}x-6\)
6. \(\frac{x}{7}+\frac{4}{15}=\frac{5}{2}x-6\)
7. \(\frac{x}{3}-\frac{4}{11}=\frac{-3}{4}x-6\)
8. \(\frac{x}{5}+\frac{3}{16}=\frac{-5}{6}x+7\)
9. \(\frac{x}{4}-\frac{3}{7}=\frac{-7}{5}x-3\)
10. \(\frac{x}{6}-\frac{5}{12}=\frac{-2}{5}x-6\)
11. \(\frac{x}{6}+\frac{4}{9}=\frac{1}{5}x-6\)
12. \(\frac{x}{3}-\frac{3}{7}=\frac{1}{4}x-6\)

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

Verbetersleutel

1. \(\text{63 is het kleinste gemene veelvoud van 7, 9 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{2}{3}x-1 \\\Leftrightarrow & \color{blue}{63.} (\frac{9x}{ \color{blue}{63} }+ \frac{ 28 }{ \color{blue}{63} })& = & (\frac{42}{ \color{blue}{63} }x-\frac{63}{ \color{blue}{63} }) \color{blue}{.63} \\\Leftrightarrow & 9x+28& = & 42x-63 \\\Leftrightarrow & 9x \color{red}{+28} \color{blue}{-28} \color{blue}{-42x} & = & \color{red}{42x} -63 \color{blue}{-42x} \color{blue}{-28} \\\Leftrightarrow & -33x& = & -91 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-91}{-33} \\\Leftrightarrow & x = \frac{91}{33} & & \\ & V = \left\{ \frac{91}{33} \right\} & \\\end{align}\)
2. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{-3}{4}x-2 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }+ \frac{ 80 }{ \color{blue}{260} })& = & (\frac{-195}{ \color{blue}{260} }x-\frac{520}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x+80& = & -195x-520 \\\Leftrightarrow & 52x \color{red}{+80} \color{blue}{-80} \color{blue}{+195x} & = & \color{red}{-195x} -520 \color{blue}{+195x} \color{blue}{-80} \\\Leftrightarrow & 247x& = & -600 \\\Leftrightarrow & \frac{247x}{ \color{red}{247} }& = & \frac{-600}{247} \\\Leftrightarrow & x = \frac{-600}{247} & & \\ & V = \left\{ \frac{-600}{247} \right\} & \\\end{align}\)
3. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{1}{6}x-3 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{40}{ \color{blue}{240} }x-\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x-75& = & 40x-720 \\\Leftrightarrow & 48x \color{red}{-75} \color{blue}{+75} \color{blue}{-40x} & = & \color{red}{40x} -720 \color{blue}{-40x} \color{blue}{+75} \\\Leftrightarrow & 8x& = & -645 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{-645}{8} \\\Leftrightarrow & x = \frac{-645}{8} & & \\ & V = \left\{ \frac{-645}{8} \right\} & \\\end{align}\)
4. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{156}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+12& = & 39x-156 \\\Leftrightarrow & 26x \color{red}{+12} \color{blue}{-12} \color{blue}{-39x} & = & \color{red}{39x} -156 \color{blue}{-39x} \color{blue}{-12} \\\Leftrightarrow & -13x& = & -168 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-168}{-13} \\\Leftrightarrow & x = \frac{168}{13} & & \\ & V = \left\{ \frac{168}{13} \right\} & \\\end{align}\)
5. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{7}{6}x-6 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{385}{ \color{blue}{330} }x-\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+150& = & 385x-1980 \\\Leftrightarrow & 66x \color{red}{+150} \color{blue}{-150} \color{blue}{-385x} & = & \color{red}{385x} -1980 \color{blue}{-385x} \color{blue}{-150} \\\Leftrightarrow & -319x& = & -2130 \\\Leftrightarrow & \frac{-319x}{ \color{red}{-319} }& = & \frac{-2130}{-319} \\\Leftrightarrow & x = \frac{2130}{319} & & \\ & V = \left\{ \frac{2130}{319} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{5}{2}x-6 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 56 }{ \color{blue}{210} })& = & (\frac{525}{ \color{blue}{210} }x-\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+56& = & 525x-1260 \\\Leftrightarrow & 30x \color{red}{+56} \color{blue}{-56} \color{blue}{-525x} & = & \color{red}{525x} -1260 \color{blue}{-525x} \color{blue}{-56} \\\Leftrightarrow & -495x& = & -1316 \\\Leftrightarrow & \frac{-495x}{ \color{red}{-495} }& = & \frac{-1316}{-495} \\\Leftrightarrow & x = \frac{1316}{495} & & \\ & V = \left\{ \frac{1316}{495} \right\} & \\\end{align}\)
7. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{-3}{4}x-6 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-99}{ \color{blue}{132} }x-\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-48& = & -99x-792 \\\Leftrightarrow & 44x \color{red}{-48} \color{blue}{+48} \color{blue}{+99x} & = & \color{red}{-99x} -792 \color{blue}{+99x} \color{blue}{+48} \\\Leftrightarrow & 143x& = & -744 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-744}{143} \\\Leftrightarrow & x = \frac{-744}{143} & & \\ & V = \left\{ \frac{-744}{143} \right\} & \\\end{align}\)
8. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{16}& = & \frac{-5}{6}x+7 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-200}{ \color{blue}{240} }x+\frac{1680}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+45& = & -200x+1680 \\\Leftrightarrow & 48x \color{red}{+45} \color{blue}{-45} \color{blue}{+200x} & = & \color{red}{-200x} +1680 \color{blue}{+200x} \color{blue}{-45} \\\Leftrightarrow & 248x& = & 1635 \\\Leftrightarrow & \frac{248x}{ \color{red}{248} }& = & \frac{1635}{248} \\\Leftrightarrow & x = \frac{1635}{248} & & \\ & V = \left\{ \frac{1635}{248} \right\} & \\\end{align}\)
9. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{-7}{5}x-3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x-\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-60& = & -196x-420 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+196x} & = & \color{red}{-196x} -420 \color{blue}{+196x} \color{blue}{+60} \\\Leftrightarrow & 231x& = & -360 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{-360}{231} \\\Leftrightarrow & x = \frac{-120}{77} & & \\ & V = \left\{ \frac{-120}{77} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{12}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-24}{ \color{blue}{60} }x-\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x-25& = & -24x-360 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{+24x} & = & \color{red}{-24x} -360 \color{blue}{+24x} \color{blue}{+25} \\\Leftrightarrow & 34x& = & -335 \\\Leftrightarrow & \frac{34x}{ \color{red}{34} }& = & \frac{-335}{34} \\\Leftrightarrow & x = \frac{-335}{34} & & \\ & V = \left\{ \frac{-335}{34} \right\} & \\\end{align}\)
11. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{9}& = & \frac{1}{5}x-6 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x-\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+40& = & 18x-540 \\\Leftrightarrow & 15x \color{red}{+40} \color{blue}{-40} \color{blue}{-18x} & = & \color{red}{18x} -540 \color{blue}{-18x} \color{blue}{-40} \\\Leftrightarrow & -3x& = & -580 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-580}{-3} \\\Leftrightarrow & x = \frac{580}{3} & & \\ & V = \left\{ \frac{580}{3} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{7}& = & \frac{1}{4}x-6 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x-\frac{504}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-36& = & 21x-504 \\\Leftrightarrow & 28x \color{red}{-36} \color{blue}{+36} \color{blue}{-21x} & = & \color{red}{21x} -504 \color{blue}{-21x} \color{blue}{+36} \\\Leftrightarrow & 7x& = & -468 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-468}{7} \\\Leftrightarrow & x = \frac{-468}{7} & & \\ & V = \left\{ \frac{-468}{7} \right\} & \\\end{align}\)