Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{4}{5}x+6 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{64}{ \color{blue}{80} }x+\frac{480}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-15& = & 64x+480 \\\Leftrightarrow & 40x \color{red}{-15} \color{blue}{+15} \color{blue}{-64x} & = & \color{red}{64x} +480 \color{blue}{-64x} \color{blue}{+15} \\\Leftrightarrow & -24x& = & 495 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{495}{-24} \\\Leftrightarrow & x = \frac{-165}{8} & & \\ & V = \left\{ \frac{-165}{8} \right\} & \\\end{align}\)
2. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{7}{2}x+8 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 8 }{ \color{blue}{14} })& = & (\frac{49}{ \color{blue}{14} }x+\frac{112}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+8& = & 49x+112 \\\Leftrightarrow & 2x \color{red}{+8} \color{blue}{-8} \color{blue}{-49x} & = & \color{red}{49x} +112 \color{blue}{-49x} \color{blue}{-8} \\\Leftrightarrow & -47x& = & 104 \\\Leftrightarrow & \frac{-47x}{ \color{red}{-47} }& = & \frac{104}{-47} \\\Leftrightarrow & x = \frac{-104}{47} & & \\ & V = \left\{ \frac{-104}{47} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 7, 6 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{7}{4}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{12x}{ \color{blue}{84} }+ \frac{ 70 }{ \color{blue}{84} })& = & (\frac{147}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 12x+70& = & 147x-252 \\\Leftrightarrow & 12x \color{red}{+70} \color{blue}{-70} \color{blue}{-147x} & = & \color{red}{147x} -252 \color{blue}{-147x} \color{blue}{-70} \\\Leftrightarrow & -135x& = & -322 \\\Leftrightarrow & \frac{-135x}{ \color{red}{-135} }& = & \frac{-322}{-135} \\\Leftrightarrow & x = \frac{322}{135} & & \\ & V = \left\{ \frac{322}{135} \right\} & \\\end{align}\)
4. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{312}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x+18& = & 39x-312 \\\Leftrightarrow & 26x \color{red}{+18} \color{blue}{-18} \color{blue}{-39x} & = & \color{red}{39x} -312 \color{blue}{-39x} \color{blue}{-18} \\\Leftrightarrow & -13x& = & -330 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-330}{-13} \\\Leftrightarrow & x = \frac{330}{13} & & \\ & V = \left\{ \frac{330}{13} \right\} & \\\end{align}\)
5. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}-\frac{5}{9}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }- \frac{ 100 }{ \color{blue}{180} })& = & (\frac{-144}{ \color{blue}{180} }x+\frac{540}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x-100& = & -144x+540 \\\Leftrightarrow & 45x \color{red}{-100} \color{blue}{+100} \color{blue}{+144x} & = & \color{red}{-144x} +540 \color{blue}{+144x} \color{blue}{+100} \\\Leftrightarrow & 189x& = & 640 \\\Leftrightarrow & \frac{189x}{ \color{red}{189} }& = & \frac{640}{189} \\\Leftrightarrow & x = \frac{640}{189} & & \\ & V = \left\{ \frac{640}{189} \right\} & \\\end{align}\)
6. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{4}{5}x+6 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }+ \frac{ 140 }{ \color{blue}{385} })& = & (\frac{308}{ \color{blue}{385} }x+\frac{2310}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x+140& = & 308x+2310 \\\Leftrightarrow & 55x \color{red}{+140} \color{blue}{-140} \color{blue}{-308x} & = & \color{red}{308x} +2310 \color{blue}{-308x} \color{blue}{-140} \\\Leftrightarrow & -253x& = & 2170 \\\Leftrightarrow & \frac{-253x}{ \color{red}{-253} }& = & \frac{2170}{-253} \\\Leftrightarrow & x = \frac{-2170}{253} & & \\ & V = \left\{ \frac{-2170}{253} \right\} & \\\end{align}\)
7. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{13}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x+\frac{1170}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+150& = & -156x+1170 \\\Leftrightarrow & 65x \color{red}{+150} \color{blue}{-150} \color{blue}{+156x} & = & \color{red}{-156x} +1170 \color{blue}{+156x} \color{blue}{-150} \\\Leftrightarrow & 221x& = & 1020 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{1020}{221} \\\Leftrightarrow & x = \frac{60}{13} & & \\ & V = \left\{ \frac{60}{13} \right\} & \\\end{align}\)
8. \(\text{48 is het kleinste gemene veelvoud van 4, 16 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{16}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{48.} (\frac{12x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{112}{ \color{blue}{48} }x-\frac{240}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 12x+9& = & 112x-240 \\\Leftrightarrow & 12x \color{red}{+9} \color{blue}{-9} \color{blue}{-112x} & = & \color{red}{112x} -240 \color{blue}{-112x} \color{blue}{-9} \\\Leftrightarrow & -100x& = & -249 \\\Leftrightarrow & \frac{-100x}{ \color{red}{-100} }& = & \frac{-249}{-100} \\\Leftrightarrow & x = \frac{249}{100} & & \\ & V = \left\{ \frac{249}{100} \right\} & \\\end{align}\)
9. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}+\frac{4}{9}& = & \frac{7}{3}x+4 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }+ \frac{ 8 }{ \color{blue}{18} })& = & (\frac{42}{ \color{blue}{18} }x+\frac{72}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x+8& = & 42x+72 \\\Leftrightarrow & 9x \color{red}{+8} \color{blue}{-8} \color{blue}{-42x} & = & \color{red}{42x} +72 \color{blue}{-42x} \color{blue}{-8} \\\Leftrightarrow & -33x& = & 64 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{64}{-33} \\\Leftrightarrow & x = \frac{-64}{33} & & \\ & V = \left\{ \frac{-64}{33} \right\} & \\\end{align}\)
10. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{-5}{6}x-1 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-75}{ \color{blue}{90} }x-\frac{90}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+20& = & -75x-90 \\\Leftrightarrow & 18x \color{red}{+20} \color{blue}{-20} \color{blue}{+75x} & = & \color{red}{-75x} -90 \color{blue}{+75x} \color{blue}{-20} \\\Leftrightarrow & 93x& = & -110 \\\Leftrightarrow & \frac{93x}{ \color{red}{93} }& = & \frac{-110}{93} \\\Leftrightarrow & x = \frac{-110}{93} & & \\ & V = \left\{ \frac{-110}{93} \right\} & \\\end{align}\)
11. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{-4}{5}x-4 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 40 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x-\frac{560}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+40& = & -112x-560 \\\Leftrightarrow & 35x \color{red}{+40} \color{blue}{-40} \color{blue}{+112x} & = & \color{red}{-112x} -560 \color{blue}{+112x} \color{blue}{-40} \\\Leftrightarrow & 147x& = & -600 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{-600}{147} \\\Leftrightarrow & x = \frac{-200}{49} & & \\ & V = \left\{ \frac{-200}{49} \right\} & \\\end{align}\)
12. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{11}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{66}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+150& = & 66x+2640 \\\Leftrightarrow & 55x \color{red}{+150} \color{blue}{-150} \color{blue}{-66x} & = & \color{red}{66x} +2640 \color{blue}{-66x} \color{blue}{-150} \\\Leftrightarrow & -11x& = & 2490 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{2490}{-11} \\\Leftrightarrow & x = \frac{-2490}{11} & & \\ & V = \left\{ \frac{-2490}{11} \right\} & \\\end{align}\)