Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{5}{2}x-2 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }- \frac{ 30 }{ \color{blue}{70} })& = & (\frac{175}{ \color{blue}{70} }x-\frac{140}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x-30& = & 175x-140 \\\Leftrightarrow & 14x \color{red}{-30} \color{blue}{+30} \color{blue}{-175x} & = & \color{red}{175x} -140 \color{blue}{-175x} \color{blue}{+30} \\\Leftrightarrow & -161x& = & -110 \\\Leftrightarrow & \frac{-161x}{ \color{red}{-161} }& = & \frac{-110}{-161} \\\Leftrightarrow & x = \frac{110}{161} & & \\ & V = \left\{ \frac{110}{161} \right\} & \\\end{align}\)
2. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{5}{4}x+1 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{195}{ \color{blue}{156} }x+\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-36& = & 195x+156 \\\Leftrightarrow & 52x \color{red}{-36} \color{blue}{+36} \color{blue}{-195x} & = & \color{red}{195x} +156 \color{blue}{-195x} \color{blue}{+36} \\\Leftrightarrow & -143x& = & 192 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{192}{-143} \\\Leftrightarrow & x = \frac{-192}{143} & & \\ & V = \left\{ \frac{-192}{143} \right\} & \\\end{align}\)
3. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{13}& = & \frac{-2}{3}x+1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 30 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x+\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-30& = & -52x+78 \\\Leftrightarrow & 39x \color{red}{-30} \color{blue}{+30} \color{blue}{+52x} & = & \color{red}{-52x} +78 \color{blue}{+52x} \color{blue}{+30} \\\Leftrightarrow & 91x& = & 108 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{108}{91} \\\Leftrightarrow & x = \frac{108}{91} & & \\ & V = \left\{ \frac{108}{91} \right\} & \\\end{align}\)
4. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }+ \frac{ 75 }{ \color{blue}{165} })& = & (\frac{55}{ \color{blue}{165} }x+\frac{330}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x+75& = & 55x+330 \\\Leftrightarrow & 33x \color{red}{+75} \color{blue}{-75} \color{blue}{-55x} & = & \color{red}{55x} +330 \color{blue}{-55x} \color{blue}{-75} \\\Leftrightarrow & -22x& = & 255 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{255}{-22} \\\Leftrightarrow & x = \frac{-255}{22} & & \\ & V = \left\{ \frac{-255}{22} \right\} & \\\end{align}\)
5. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{16}& = & \frac{-2}{5}x-2 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 45 }{ \color{blue}{240} })& = & (\frac{-96}{ \color{blue}{240} }x-\frac{480}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-45& = & -96x-480 \\\Leftrightarrow & 40x \color{red}{-45} \color{blue}{+45} \color{blue}{+96x} & = & \color{red}{-96x} -480 \color{blue}{+96x} \color{blue}{+45} \\\Leftrightarrow & 136x& = & -435 \\\Leftrightarrow & \frac{136x}{ \color{red}{136} }& = & \frac{-435}{136} \\\Leftrightarrow & x = \frac{-435}{136} & & \\ & V = \left\{ \frac{-435}{136} \right\} & \\\end{align}\)
6. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}-\frac{3}{7}& = & \frac{5}{2}x+6 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }- \frac{ 6 }{ \color{blue}{14} })& = & (\frac{35}{ \color{blue}{14} }x+\frac{84}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x-6& = & 35x+84 \\\Leftrightarrow & 2x \color{red}{-6} \color{blue}{+6} \color{blue}{-35x} & = & \color{red}{35x} +84 \color{blue}{-35x} \color{blue}{+6} \\\Leftrightarrow & -33x& = & 90 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{90}{-33} \\\Leftrightarrow & x = \frac{-30}{11} & & \\ & V = \left\{ \frac{-30}{11} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-12& = & 21x-294 \\\Leftrightarrow & 14x \color{red}{-12} \color{blue}{+12} \color{blue}{-21x} & = & \color{red}{21x} -294 \color{blue}{-21x} \color{blue}{+12} \\\Leftrightarrow & -7x& = & -282 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-282}{-7} \\\Leftrightarrow & x = \frac{282}{7} & & \\ & V = \left\{ \frac{282}{7} \right\} & \\\end{align}\)
8. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x-\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+9& = & 12x-72 \\\Leftrightarrow & 8x \color{red}{+9} \color{blue}{-9} \color{blue}{-12x} & = & \color{red}{12x} -72 \color{blue}{-12x} \color{blue}{-9} \\\Leftrightarrow & -4x& = & -81 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-81}{-4} \\\Leftrightarrow & x = \frac{81}{4} & & \\ & V = \left\{ \frac{81}{4} \right\} & \\\end{align}\)
9. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{9}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }+ \frac{ 80 }{ \color{blue}{180} })& = & (\frac{-144}{ \color{blue}{180} }x-\frac{1080}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x+80& = & -144x-1080 \\\Leftrightarrow & 45x \color{red}{+80} \color{blue}{-80} \color{blue}{+144x} & = & \color{red}{-144x} -1080 \color{blue}{+144x} \color{blue}{-80} \\\Leftrightarrow & 189x& = & -1160 \\\Leftrightarrow & \frac{189x}{ \color{red}{189} }& = & \frac{-1160}{189} \\\Leftrightarrow & x = \frac{-1160}{189} & & \\ & V = \left\{ \frac{-1160}{189} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{7}& = & \frac{1}{5}x-8 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 150 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{1680}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-150& = & 42x-1680 \\\Leftrightarrow & 35x \color{red}{-150} \color{blue}{+150} \color{blue}{-42x} & = & \color{red}{42x} -1680 \color{blue}{-42x} \color{blue}{+150} \\\Leftrightarrow & -7x& = & -1530 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-1530}{-7} \\\Leftrightarrow & x = \frac{1530}{7} & & \\ & V = \left\{ \frac{1530}{7} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 4, 6 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{-2}{5}x-8 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 50 }{ \color{blue}{60} })& = & (\frac{-24}{ \color{blue}{60} }x-\frac{480}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+50& = & -24x-480 \\\Leftrightarrow & 15x \color{red}{+50} \color{blue}{-50} \color{blue}{+24x} & = & \color{red}{-24x} -480 \color{blue}{+24x} \color{blue}{-50} \\\Leftrightarrow & 39x& = & -530 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-530}{39} \\\Leftrightarrow & x = \frac{-530}{39} & & \\ & V = \left\{ \frac{-530}{39} \right\} & \\\end{align}\)
12. \(\text{120 is het kleinste gemene veelvoud van 5, 8 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{8}& = & \frac{5}{6}x+4 \\\Leftrightarrow & \color{blue}{120.} (\frac{24x}{ \color{blue}{120} }- \frac{ 45 }{ \color{blue}{120} })& = & (\frac{100}{ \color{blue}{120} }x+\frac{480}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 24x-45& = & 100x+480 \\\Leftrightarrow & 24x \color{red}{-45} \color{blue}{+45} \color{blue}{-100x} & = & \color{red}{100x} +480 \color{blue}{-100x} \color{blue}{+45} \\\Leftrightarrow & -76x& = & 525 \\\Leftrightarrow & \frac{-76x}{ \color{red}{-76} }& = & \frac{525}{-76} \\\Leftrightarrow & x = \frac{-525}{76} & & \\ & V = \left\{ \frac{-525}{76} \right\} & \\\end{align}\)