Vgln. eerste graad (reeks 4)

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

1. \(\frac{x}{3}-\frac{3}{10}=\frac{-3}{4}x-1\)
2. \(\frac{x}{3}-\frac{5}{12}=\frac{-4}{5}x-6\)
3. \(\frac{x}{6}+\frac{3}{13}=\frac{-2}{5}x-6\)
4. \(\frac{x}{2}-\frac{3}{10}=\frac{1}{3}x-6\)
5. \(\frac{x}{6}+\frac{5}{14}=\frac{2}{5}x-7\)
6. \(\frac{x}{4}-\frac{3}{14}=\frac{1}{3}x-3\)
7. \(\frac{x}{2}-\frac{5}{7}=\frac{-2}{3}x-1\)
8. \(\frac{x}{6}-\frac{5}{13}=\frac{-4}{5}x+1\)
9. \(\frac{x}{4}+\frac{2}{15}=\frac{-2}{5}x-7\)
10. \(\frac{x}{4}+\frac{3}{10}=\frac{-2}{3}x+7\)
11. \(\frac{x}{6}-\frac{5}{12}=\frac{4}{5}x+6\)
12. \(\frac{x}{6}-\frac{3}{8}=\frac{4}{5}x+1\)

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 3, 10 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{-3}{4}x-1 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-45}{ \color{blue}{60} }x-\frac{60}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-18& = & -45x-60 \\\Leftrightarrow & 20x \color{red}{-18} \color{blue}{+18} \color{blue}{+45x} & = & \color{red}{-45x} -60 \color{blue}{+45x} \color{blue}{+18} \\\Leftrightarrow & 65x& = & -42 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{-42}{65} \\\Leftrightarrow & x = \frac{-42}{65} & & \\ & V = \left\{ \frac{-42}{65} \right\} & \\\end{align}\)
2. \(\text{60 is het kleinste gemene veelvoud van 3, 12 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{12}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-48}{ \color{blue}{60} }x-\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-25& = & -48x-360 \\\Leftrightarrow & 20x \color{red}{-25} \color{blue}{+25} \color{blue}{+48x} & = & \color{red}{-48x} -360 \color{blue}{+48x} \color{blue}{+25} \\\Leftrightarrow & 68x& = & -335 \\\Leftrightarrow & \frac{68x}{ \color{red}{68} }& = & \frac{-335}{68} \\\Leftrightarrow & x = \frac{-335}{68} & & \\ & V = \left\{ \frac{-335}{68} \right\} & \\\end{align}\)
3. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-156}{ \color{blue}{390} }x-\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & -156x-2340 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{+156x} & = & \color{red}{-156x} -2340 \color{blue}{+156x} \color{blue}{-90} \\\Leftrightarrow & 221x& = & -2430 \\\Leftrightarrow & \frac{221x}{ \color{red}{221} }& = & \frac{-2430}{221} \\\Leftrightarrow & x = \frac{-2430}{221} & & \\ & V = \left\{ \frac{-2430}{221} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{10}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-9& = & 10x-180 \\\Leftrightarrow & 15x \color{red}{-9} \color{blue}{+9} \color{blue}{-10x} & = & \color{red}{10x} -180 \color{blue}{-10x} \color{blue}{+9} \\\Leftrightarrow & 5x& = & -171 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{-171}{5} \\\Leftrightarrow & x = \frac{-171}{5} & & \\ & V = \left\{ \frac{-171}{5} \right\} & \\\end{align}\)
5. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{14}& = & \frac{2}{5}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 75 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+75& = & 84x-1470 \\\Leftrightarrow & 35x \color{red}{+75} \color{blue}{-75} \color{blue}{-84x} & = & \color{red}{84x} -1470 \color{blue}{-84x} \color{blue}{-75} \\\Leftrightarrow & -49x& = & -1545 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-1545}{-49} \\\Leftrightarrow & x = \frac{1545}{49} & & \\ & V = \left\{ \frac{1545}{49} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{14}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }- \frac{ 18 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x-18& = & 28x-252 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{-28x} & = & \color{red}{28x} -252 \color{blue}{-28x} \color{blue}{+18} \\\Leftrightarrow & -7x& = & -234 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-234}{-7} \\\Leftrightarrow & x = \frac{234}{7} & & \\ & V = \left\{ \frac{234}{7} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{7}& = & \frac{-2}{3}x-1 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{-28}{ \color{blue}{42} }x-\frac{42}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-30& = & -28x-42 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{+28x} & = & \color{red}{-28x} -42 \color{blue}{+28x} \color{blue}{+30} \\\Leftrightarrow & 49x& = & -12 \\\Leftrightarrow & \frac{49x}{ \color{red}{49} }& = & \frac{-12}{49} \\\Leftrightarrow & x = \frac{-12}{49} & & \\ & V = \left\{ \frac{-12}{49} \right\} & \\\end{align}\)
8. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{13}& = & \frac{-4}{5}x+1 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 150 }{ \color{blue}{390} })& = & (\frac{-312}{ \color{blue}{390} }x+\frac{390}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-150& = & -312x+390 \\\Leftrightarrow & 65x \color{red}{-150} \color{blue}{+150} \color{blue}{+312x} & = & \color{red}{-312x} +390 \color{blue}{+312x} \color{blue}{+150} \\\Leftrightarrow & 377x& = & 540 \\\Leftrightarrow & \frac{377x}{ \color{red}{377} }& = & \frac{540}{377} \\\Leftrightarrow & x = \frac{540}{377} & & \\ & V = \left\{ \frac{540}{377} \right\} & \\\end{align}\)
9. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{-2}{5}x-7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{-24}{ \color{blue}{60} }x-\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & -24x-420 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+24x} & = & \color{red}{-24x} -420 \color{blue}{+24x} \color{blue}{-8} \\\Leftrightarrow & 39x& = & -428 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-428}{39} \\\Leftrightarrow & x = \frac{-428}{39} & & \\ & V = \left\{ \frac{-428}{39} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 4, 10 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{10}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 18 }{ \color{blue}{60} })& = & (\frac{-40}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+18& = & -40x+420 \\\Leftrightarrow & 15x \color{red}{+18} \color{blue}{-18} \color{blue}{+40x} & = & \color{red}{-40x} +420 \color{blue}{+40x} \color{blue}{-18} \\\Leftrightarrow & 55x& = & 402 \\\Leftrightarrow & \frac{55x}{ \color{red}{55} }& = & \frac{402}{55} \\\Leftrightarrow & x = \frac{402}{55} & & \\ & V = \left\{ \frac{402}{55} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 6, 12 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{12}& = & \frac{4}{5}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{10x}{ \color{blue}{60} }- \frac{ 25 }{ \color{blue}{60} })& = & (\frac{48}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 10x-25& = & 48x+360 \\\Leftrightarrow & 10x \color{red}{-25} \color{blue}{+25} \color{blue}{-48x} & = & \color{red}{48x} +360 \color{blue}{-48x} \color{blue}{+25} \\\Leftrightarrow & -38x& = & 385 \\\Leftrightarrow & \frac{-38x}{ \color{red}{-38} }& = & \frac{385}{-38} \\\Leftrightarrow & x = \frac{-385}{38} & & \\ & V = \left\{ \frac{-385}{38} \right\} & \\\end{align}\)
12. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{8}& = & \frac{4}{5}x+1 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }- \frac{ 45 }{ \color{blue}{120} })& = & (\frac{96}{ \color{blue}{120} }x+\frac{120}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x-45& = & 96x+120 \\\Leftrightarrow & 20x \color{red}{-45} \color{blue}{+45} \color{blue}{-96x} & = & \color{red}{96x} +120 \color{blue}{-96x} \color{blue}{+45} \\\Leftrightarrow & -76x& = & 165 \\\Leftrightarrow & \frac{-76x}{ \color{red}{-76} }& = & \frac{165}{-76} \\\Leftrightarrow & x = \frac{-165}{76} & & \\ & V = \left\{ \frac{-165}{76} \right\} & \\\end{align}\)