Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{9}& = & \frac{7}{2}x+4 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{315}{ \color{blue}{90} }x+\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-50& = & 315x+360 \\\Leftrightarrow & 18x \color{red}{-50} \color{blue}{+50} \color{blue}{-315x} & = & \color{red}{315x} +360 \color{blue}{-315x} \color{blue}{+50} \\\Leftrightarrow & -297x& = & 410 \\\Leftrightarrow & \frac{-297x}{ \color{red}{-297} }& = & \frac{410}{-297} \\\Leftrightarrow & x = \frac{-410}{297} & & \\ & V = \left\{ \frac{-410}{297} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{-5}{3}x+7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 36 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x+\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-36& = & -140x+588 \\\Leftrightarrow & 21x \color{red}{-36} \color{blue}{+36} \color{blue}{+140x} & = & \color{red}{-140x} +588 \color{blue}{+140x} \color{blue}{+36} \\\Leftrightarrow & 161x& = & 624 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{624}{161} \\\Leftrightarrow & x = \frac{624}{161} & & \\ & V = \left\{ \frac{624}{161} \right\} & \\\end{align}\)
3. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & -28x+294 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{+28x} & = & \color{red}{-28x} +294 \color{blue}{+28x} \color{blue}{-12} \\\Leftrightarrow & 49x& = & 282 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{282}{49} \\\Leftrightarrow & x = \frac{282}{49} & & \\ & V = \left\{ \frac{282}{49} \right\} & \\\end{align}\)
4. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 4 }{ \color{blue}{14} })& = & (\frac{7}{ \color{blue}{14} }x-\frac{28}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+4& = & 7x-28 \\\Leftrightarrow & 2x \color{red}{+4} \color{blue}{-4} \color{blue}{-7x} & = & \color{red}{7x} -28 \color{blue}{-7x} \color{blue}{-4} \\\Leftrightarrow & -5x& = & -32 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-32}{-5} \\\Leftrightarrow & x = \frac{32}{5} & & \\ & V = \left\{ \frac{32}{5} \right\} & \\\end{align}\)
5. \(\text{130 is het kleinste gemene veelvoud van 2, 13 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{13}& = & \frac{-2}{5}x-3 \\\Leftrightarrow & \color{blue}{130.} (\frac{65x}{ \color{blue}{130} }- \frac{ 20 }{ \color{blue}{130} })& = & (\frac{-52}{ \color{blue}{130} }x-\frac{390}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 65x-20& = & -52x-390 \\\Leftrightarrow & 65x \color{red}{-20} \color{blue}{+20} \color{blue}{+52x} & = & \color{red}{-52x} -390 \color{blue}{+52x} \color{blue}{+20} \\\Leftrightarrow & 117x& = & -370 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-370}{117} \\\Leftrightarrow & x = \frac{-370}{117} & & \\ & V = \left\{ \frac{-370}{117} \right\} & \\\end{align}\)
6. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{-5}{6}x-5 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{-275}{ \color{blue}{330} }x-\frac{1650}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+150& = & -275x-1650 \\\Leftrightarrow & 66x \color{red}{+150} \color{blue}{-150} \color{blue}{+275x} & = & \color{red}{-275x} -1650 \color{blue}{+275x} \color{blue}{-150} \\\Leftrightarrow & 341x& = & -1800 \\\Leftrightarrow & \frac{341x}{ \color{red}{341} }& = & \frac{-1800}{341} \\\Leftrightarrow & x = \frac{-1800}{341} & & \\ & V = \left\{ \frac{-1800}{341} \right\} & \\\end{align}\)
7. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 40 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-40& = & 14x+490 \\\Leftrightarrow & 35x \color{red}{-40} \color{blue}{+40} \color{blue}{-14x} & = & \color{red}{14x} +490 \color{blue}{-14x} \color{blue}{+40} \\\Leftrightarrow & 21x& = & 530 \\\Leftrightarrow & \frac{21x}{ \color{red}{21} }& = & \frac{530}{21} \\\Leftrightarrow & x = \frac{530}{21} & & \\ & V = \left\{ \frac{530}{21} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-84}{ \color{blue}{60} }x+\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & -84x+60 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{+84x} & = & \color{red}{-84x} +60 \color{blue}{+84x} \color{blue}{+16} \\\Leftrightarrow & 99x& = & 76 \\\Leftrightarrow & \frac{99x}{ \color{red}{99} }& = & \frac{76}{99} \\\Leftrightarrow & x = \frac{76}{99} & & \\ & V = \left\{ \frac{76}{99} \right\} & \\\end{align}\)
9. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{2}{5}x+6 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{56}{ \color{blue}{140} }x+\frac{840}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-80& = & 56x+840 \\\Leftrightarrow & 35x \color{red}{-80} \color{blue}{+80} \color{blue}{-56x} & = & \color{red}{56x} +840 \color{blue}{-56x} \color{blue}{+80} \\\Leftrightarrow & -21x& = & 920 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{920}{-21} \\\Leftrightarrow & x = \frac{-920}{21} & & \\ & V = \left\{ \frac{-920}{21} \right\} & \\\end{align}\)
10. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{-112}{ \color{blue}{80} }x-\frac{640}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-25& = & -112x-640 \\\Leftrightarrow & 40x \color{red}{-25} \color{blue}{+25} \color{blue}{+112x} & = & \color{red}{-112x} -640 \color{blue}{+112x} \color{blue}{+25} \\\Leftrightarrow & 152x& = & -615 \\\Leftrightarrow & \frac{152x}{ \color{red}{152} }& = & \frac{-615}{152} \\\Leftrightarrow & x = \frac{-615}{152} & & \\ & V = \left\{ \frac{-615}{152} \right\} & \\\end{align}\)
11. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{16}& = & \frac{3}{5}x+2 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 15 }{ \color{blue}{80} })& = & (\frac{48}{ \color{blue}{80} }x+\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+15& = & 48x+160 \\\Leftrightarrow & 40x \color{red}{+15} \color{blue}{-15} \color{blue}{-48x} & = & \color{red}{48x} +160 \color{blue}{-48x} \color{blue}{-15} \\\Leftrightarrow & -8x& = & 145 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{145}{-8} \\\Leftrightarrow & x = \frac{-145}{8} & & \\ & V = \left\{ \frac{-145}{8} \right\} & \\\end{align}\)
12. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{7}& = & \frac{3}{5}x+3 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 75 }{ \color{blue}{105} })& = & (\frac{63}{ \color{blue}{105} }x+\frac{315}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-75& = & 63x+315 \\\Leftrightarrow & 35x \color{red}{-75} \color{blue}{+75} \color{blue}{-63x} & = & \color{red}{63x} +315 \color{blue}{-63x} \color{blue}{+75} \\\Leftrightarrow & -28x& = & 390 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{390}{-28} \\\Leftrightarrow & x = \frac{-195}{14} & & \\ & V = \left\{ \frac{-195}{14} \right\} & \\\end{align}\)