Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{13}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 24 }{ \color{blue}{156} })& = & (\frac{-104}{ \color{blue}{156} }x+\frac{468}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+24& = & -104x+468 \\\Leftrightarrow & 39x \color{red}{+24} \color{blue}{-24} \color{blue}{+104x} & = & \color{red}{-104x} +468 \color{blue}{+104x} \color{blue}{-24} \\\Leftrightarrow & 143x& = & 444 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{444}{143} \\\Leftrightarrow & x = \frac{444}{143} & & \\ & V = \left\{ \frac{444}{143} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & -70x+336 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{+70x} & = & \color{red}{-70x} +336 \color{blue}{+70x} \color{blue}{-12} \\\Leftrightarrow & 91x& = & 324 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{324}{91} \\\Leftrightarrow & x = \frac{324}{91} & & \\ & V = \left\{ \frac{324}{91} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{2}{5}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{84}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & 84x-420 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{-84x} & = & \color{red}{84x} -420 \color{blue}{-84x} \color{blue}{-60} \\\Leftrightarrow & -49x& = & -480 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{-480}{-49} \\\Leftrightarrow & x = \frac{480}{49} & & \\ & V = \left\{ \frac{480}{49} \right\} & \\\end{align}\)
4. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{130}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+30& = & 65x-130 \\\Leftrightarrow & 26x \color{red}{+30} \color{blue}{-30} \color{blue}{-65x} & = & \color{red}{65x} -130 \color{blue}{-65x} \color{blue}{-30} \\\Leftrightarrow & -39x& = & -160 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-160}{-39} \\\Leftrightarrow & x = \frac{160}{39} & & \\ & V = \left\{ \frac{160}{39} \right\} & \\\end{align}\)
5. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{11}& = & \frac{3}{2}x+6 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{99}{ \color{blue}{66} }x+\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+12& = & 99x+396 \\\Leftrightarrow & 22x \color{red}{+12} \color{blue}{-12} \color{blue}{-99x} & = & \color{red}{99x} +396 \color{blue}{-99x} \color{blue}{-12} \\\Leftrightarrow & -77x& = & 384 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{384}{-77} \\\Leftrightarrow & x = \frac{-384}{77} & & \\ & V = \left\{ \frac{-384}{77} \right\} & \\\end{align}\)
6. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{4}{3}x-2 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{64}{ \color{blue}{48} }x-\frac{96}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & 64x-96 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{-64x} & = & \color{red}{64x} -96 \color{blue}{-64x} \color{blue}{-15} \\\Leftrightarrow & -40x& = & -111 \\\Leftrightarrow & \frac{-40x}{ \color{red}{-40} }& = & \frac{-111}{-40} \\\Leftrightarrow & x = \frac{111}{40} & & \\ & V = \left\{ \frac{111}{40} \right\} & \\\end{align}\)
7. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{-4}{5}x+8 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 100 }{ \color{blue}{220} })& = & (\frac{-176}{ \color{blue}{220} }x+\frac{1760}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+100& = & -176x+1760 \\\Leftrightarrow & 55x \color{red}{+100} \color{blue}{-100} \color{blue}{+176x} & = & \color{red}{-176x} +1760 \color{blue}{+176x} \color{blue}{-100} \\\Leftrightarrow & 231x& = & 1660 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{1660}{231} \\\Leftrightarrow & x = \frac{1660}{231} & & \\ & V = \left\{ \frac{1660}{231} \right\} & \\\end{align}\)
8. \(\text{12 is het kleinste gemene veelvoud van 3, 12 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{1}{4}x+5 \\\Leftrightarrow & \color{blue}{12.} (\frac{4x}{ \color{blue}{12} }+ \frac{ 5 }{ \color{blue}{12} })& = & (\frac{3}{ \color{blue}{12} }x+\frac{60}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 4x+5& = & 3x+60 \\\Leftrightarrow & 4x \color{red}{+5} \color{blue}{-5} \color{blue}{-3x} & = & \color{red}{3x} +60 \color{blue}{-3x} \color{blue}{-5} \\\Leftrightarrow & x& = & 55 \\ & V = \left\{ 55 \right\} & \\\end{align}\)
9. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{8}& = & \frac{-2}{3}x+5 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }+ \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-16}{ \color{blue}{24} }x+\frac{120}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x+15& = & -16x+120 \\\Leftrightarrow & 12x \color{red}{+15} \color{blue}{-15} \color{blue}{+16x} & = & \color{red}{-16x} +120 \color{blue}{+16x} \color{blue}{-15} \\\Leftrightarrow & 28x& = & 105 \\\Leftrightarrow & \frac{28x}{ \color{red}{28} }& = & \frac{105}{28} \\\Leftrightarrow & x = \frac{15}{4} & & \\ & V = \left\{ \frac{15}{4} \right\} & \\\end{align}\)
10. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{7}& = & \frac{6}{5}x+6 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 30 }{ \color{blue}{70} })& = & (\frac{84}{ \color{blue}{70} }x+\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+30& = & 84x+420 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{-84x} & = & \color{red}{84x} +420 \color{blue}{-84x} \color{blue}{-30} \\\Leftrightarrow & -49x& = & 390 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{390}{-49} \\\Leftrightarrow & x = \frac{-390}{49} & & \\ & V = \left\{ \frac{-390}{49} \right\} & \\\end{align}\)
11. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{13}& = & \frac{3}{2}x-3 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 30 }{ \color{blue}{78} })& = & (\frac{117}{ \color{blue}{78} }x-\frac{234}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-30& = & 117x-234 \\\Leftrightarrow & 26x \color{red}{-30} \color{blue}{+30} \color{blue}{-117x} & = & \color{red}{117x} -234 \color{blue}{-117x} \color{blue}{+30} \\\Leftrightarrow & -91x& = & -204 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-204}{-91} \\\Leftrightarrow & x = \frac{204}{91} & & \\ & V = \left\{ \frac{204}{91} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & -24x-90 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{+24x} & = & \color{red}{-24x} -90 \color{blue}{+24x} \color{blue}{+9} \\\Leftrightarrow & 34x& = & -81 \\\Leftrightarrow & \frac{34x}{ \color{red}{34} }& = & \frac{-81}{34} \\\Leftrightarrow & x = \frac{-81}{34} & & \\ & V = \left\{ \frac{-81}{34} \right\} & \\\end{align}\)