Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 3, 15 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{1}{2}x+6 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-4& = & 15x+180 \\\Leftrightarrow & 10x \color{red}{-4} \color{blue}{+4} \color{blue}{-15x} & = & \color{red}{15x} +180 \color{blue}{-15x} \color{blue}{+4} \\\Leftrightarrow & -5x& = & 184 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{184}{-5} \\\Leftrightarrow & x = \frac{-184}{5} & & \\ & V = \left\{ \frac{-184}{5} \right\} & \\\end{align}\)
2. \(\text{70 is het kleinste gemene veelvoud van 5, 7 en 2} \\ \begin{align} & \frac{x}{5}+\frac{2}{7}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{70.} (\frac{14x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{35}{ \color{blue}{70} }x-\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 14x+20& = & 35x-490 \\\Leftrightarrow & 14x \color{red}{+20} \color{blue}{-20} \color{blue}{-35x} & = & \color{red}{35x} -490 \color{blue}{-35x} \color{blue}{-20} \\\Leftrightarrow & -21x& = & -510 \\\Leftrightarrow & \frac{-21x}{ \color{red}{-21} }& = & \frac{-510}{-21} \\\Leftrightarrow & x = \frac{170}{7} & & \\ & V = \left\{ \frac{170}{7} \right\} & \\\end{align}\)
3. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{8}& = & \frac{4}{3}x-5 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{32}{ \color{blue}{24} }x-\frac{120}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x-15& = & 32x-120 \\\Leftrightarrow & 6x \color{red}{-15} \color{blue}{+15} \color{blue}{-32x} & = & \color{red}{32x} -120 \color{blue}{-32x} \color{blue}{+15} \\\Leftrightarrow & -26x& = & -105 \\\Leftrightarrow & \frac{-26x}{ \color{red}{-26} }& = & \frac{-105}{-26} \\\Leftrightarrow & x = \frac{105}{26} & & \\ & V = \left\{ \frac{105}{26} \right\} & \\\end{align}\)
4. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{4}{3}x-8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x-\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & 56x-336 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-56x} & = & \color{red}{56x} -336 \color{blue}{-56x} \color{blue}{+24} \\\Leftrightarrow & -35x& = & -312 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{-312}{-35} \\\Leftrightarrow & x = \frac{312}{35} & & \\ & V = \left\{ \frac{312}{35} \right\} & \\\end{align}\)
5. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{4}{5}x-3 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{192}{ \color{blue}{240} }x-\frac{720}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & 192x-720 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{-192x} & = & \color{red}{192x} -720 \color{blue}{-192x} \color{blue}{-75} \\\Leftrightarrow & -152x& = & -795 \\\Leftrightarrow & \frac{-152x}{ \color{red}{-152} }& = & \frac{-795}{-152} \\\Leftrightarrow & x = \frac{795}{152} & & \\ & V = \left\{ \frac{795}{152} \right\} & \\\end{align}\)
6. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{-8}{3}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-160}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & -160x+180 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{+160x} & = & \color{red}{-160x} +180 \color{blue}{+160x} \color{blue}{-16} \\\Leftrightarrow & 175x& = & 164 \\\Leftrightarrow & \frac{175x}{ \color{red}{175} }& = & \frac{164}{175} \\\Leftrightarrow & x = \frac{164}{175} & & \\ & V = \left\{ \frac{164}{175} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{13}& = & \frac{5}{3}x-3 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 12 }{ \color{blue}{78} })& = & (\frac{130}{ \color{blue}{78} }x-\frac{234}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+12& = & 130x-234 \\\Leftrightarrow & 39x \color{red}{+12} \color{blue}{-12} \color{blue}{-130x} & = & \color{red}{130x} -234 \color{blue}{-130x} \color{blue}{-12} \\\Leftrightarrow & -91x& = & -246 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-246}{-91} \\\Leftrightarrow & x = \frac{246}{91} & & \\ & V = \left\{ \frac{246}{91} \right\} & \\\end{align}\)
8. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{-2}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & -84x+420 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{+84x} & = & \color{red}{-84x} +420 \color{blue}{+84x} \color{blue}{-120} \\\Leftrightarrow & 119x& = & 300 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{300}{119} \\\Leftrightarrow & x = \frac{300}{119} & & \\ & V = \left\{ \frac{300}{119} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{7}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+150& = & 252x-1050 \\\Leftrightarrow & 35x \color{red}{+150} \color{blue}{-150} \color{blue}{-252x} & = & \color{red}{252x} -1050 \color{blue}{-252x} \color{blue}{-150} \\\Leftrightarrow & -217x& = & -1200 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{-1200}{-217} \\\Leftrightarrow & x = \frac{1200}{217} & & \\ & V = \left\{ \frac{1200}{217} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{1}{6}x+8 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{55}{ \color{blue}{330} }x+\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+150& = & 55x+2640 \\\Leftrightarrow & 66x \color{red}{+150} \color{blue}{-150} \color{blue}{-55x} & = & \color{red}{55x} +2640 \color{blue}{-55x} \color{blue}{-150} \\\Leftrightarrow & 11x& = & 2490 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{2490}{11} \\\Leftrightarrow & x = \frac{2490}{11} & & \\ & V = \left\{ \frac{2490}{11} \right\} & \\\end{align}\)
11. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{-4}{5}x-3 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }+ \frac{ 140 }{ \color{blue}{385} })& = & (\frac{-308}{ \color{blue}{385} }x-\frac{1155}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x+140& = & -308x-1155 \\\Leftrightarrow & 55x \color{red}{+140} \color{blue}{-140} \color{blue}{+308x} & = & \color{red}{-308x} -1155 \color{blue}{+308x} \color{blue}{-140} \\\Leftrightarrow & 363x& = & -1295 \\\Leftrightarrow & \frac{363x}{ \color{red}{363} }& = & \frac{-1295}{363} \\\Leftrightarrow & x = \frac{-1295}{363} & & \\ & V = \left\{ \frac{-1295}{363} \right\} & \\\end{align}\)
12. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{3}{16}& = & \frac{-7}{4}x-8 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 9 }{ \color{blue}{48} })& = & (\frac{-84}{ \color{blue}{48} }x-\frac{384}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-9& = & -84x-384 \\\Leftrightarrow & 16x \color{red}{-9} \color{blue}{+9} \color{blue}{+84x} & = & \color{red}{-84x} -384 \color{blue}{+84x} \color{blue}{+9} \\\Leftrightarrow & 100x& = & -375 \\\Leftrightarrow & \frac{100x}{ \color{red}{100} }& = & \frac{-375}{100} \\\Leftrightarrow & x = \frac{-15}{4} & & \\ & V = \left\{ \frac{-15}{4} \right\} & \\\end{align}\)