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

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{9}& = & \frac{1}{6}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{15}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+20& = & 15x-630 \\\Leftrightarrow & 18x \color{red}{+20} \color{blue}{-20} \color{blue}{-15x} & = & \color{red}{15x} -630 \color{blue}{-15x} \color{blue}{-20} \\\Leftrightarrow & 3x& = & -650 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-650}{3} \\\Leftrightarrow & x = \frac{-650}{3} & & \\ & V = \left\{ \frac{-650}{3} \right\} & \\\end{align}\)
2. \(\text{10 is het kleinste gemene veelvoud van 2, 10 en 5} \\ \begin{align} & \frac{x}{2}+\frac{3}{10}& = & \frac{1}{5}x-5 \\\Leftrightarrow & \color{blue}{10.} (\frac{5x}{ \color{blue}{10} }+ \frac{ 3 }{ \color{blue}{10} })& = & (\frac{2}{ \color{blue}{10} }x-\frac{50}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 5x+3& = & 2x-50 \\\Leftrightarrow & 5x \color{red}{+3} \color{blue}{-3} \color{blue}{-2x} & = & \color{red}{2x} -50 \color{blue}{-2x} \color{blue}{-3} \\\Leftrightarrow & 3x& = & -53 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{-53}{3} \\\Leftrightarrow & x = \frac{-53}{3} & & \\ & V = \left\{ \frac{-53}{3} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{15}& = & \frac{2}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{12}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-8& = & 12x+240 \\\Leftrightarrow & 15x \color{red}{-8} \color{blue}{+8} \color{blue}{-12x} & = & \color{red}{12x} +240 \color{blue}{-12x} \color{blue}{+8} \\\Leftrightarrow & 3x& = & 248 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{248}{3} \\\Leftrightarrow & x = \frac{248}{3} & & \\ & V = \left\{ \frac{248}{3} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{7}& = & \frac{1}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-90& = & 42x+420 \\\Leftrightarrow & 35x \color{red}{-90} \color{blue}{+90} \color{blue}{-42x} & = & \color{red}{42x} +420 \color{blue}{-42x} \color{blue}{+90} \\\Leftrightarrow & -7x& = & 510 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{510}{-7} \\\Leftrightarrow & x = \frac{-510}{7} & & \\ & V = \left\{ \frac{-510}{7} \right\} & \\\end{align}\)
5. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}+\frac{5}{16}& = & \frac{6}{5}x-5 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{288}{ \color{blue}{240} }x-\frac{1200}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x+75& = & 288x-1200 \\\Leftrightarrow & 40x \color{red}{+75} \color{blue}{-75} \color{blue}{-288x} & = & \color{red}{288x} -1200 \color{blue}{-288x} \color{blue}{-75} \\\Leftrightarrow & -248x& = & -1275 \\\Leftrightarrow & \frac{-248x}{ \color{red}{-248} }& = & \frac{-1275}{-248} \\\Leftrightarrow & x = \frac{1275}{248} & & \\ & V = \left\{ \frac{1275}{248} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{1}{3}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{10}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & 10x+120 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{-10x} & = & \color{red}{10x} +120 \color{blue}{-10x} \color{blue}{+4} \\\Leftrightarrow & 5x& = & 124 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{124}{5} \\\Leftrightarrow & x = \frac{124}{5} & & \\ & V = \left\{ \frac{124}{5} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{7}& = & \frac{1}{6}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 150 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+150& = & 35x+1050 \\\Leftrightarrow & 42x \color{red}{+150} \color{blue}{-150} \color{blue}{-35x} & = & \color{red}{35x} +1050 \color{blue}{-35x} \color{blue}{-150} \\\Leftrightarrow & 7x& = & 900 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{900}{7} \\\Leftrightarrow & x = \frac{900}{7} & & \\ & V = \left\{ \frac{900}{7} \right\} & \\\end{align}\)
8. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{-5}{3}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & -70x+336 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{+70x} & = & \color{red}{-70x} +336 \color{blue}{+70x} \color{blue}{+24} \\\Leftrightarrow & 91x& = & 360 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{360}{91} \\\Leftrightarrow & x = \frac{360}{91} & & \\ & V = \left\{ \frac{360}{91} \right\} & \\\end{align}\)
9. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{7}{3}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & 196x+168 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{-196x} & = & \color{red}{196x} +168 \color{blue}{-196x} \color{blue}{-24} \\\Leftrightarrow & -175x& = & 144 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{144}{-175} \\\Leftrightarrow & x = \frac{-144}{175} & & \\ & V = \left\{ \frac{-144}{175} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{11}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }+ \frac{ 120 }{ \color{blue}{330} })& = & (\frac{-132}{ \color{blue}{330} }x-\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x+120& = & -132x-1980 \\\Leftrightarrow & 55x \color{red}{+120} \color{blue}{-120} \color{blue}{+132x} & = & \color{red}{-132x} -1980 \color{blue}{+132x} \color{blue}{-120} \\\Leftrightarrow & 187x& = & -2100 \\\Leftrightarrow & \frac{187x}{ \color{red}{187} }& = & \frac{-2100}{187} \\\Leftrightarrow & x = \frac{-2100}{187} & & \\ & V = \left\{ \frac{-2100}{187} \right\} & \\\end{align}\)
11. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{9}& = & \frac{1}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 40 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-40& = & 18x-630 \\\Leftrightarrow & 15x \color{red}{-40} \color{blue}{+40} \color{blue}{-18x} & = & \color{red}{18x} -630 \color{blue}{-18x} \color{blue}{+40} \\\Leftrightarrow & -3x& = & -590 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-590}{-3} \\\Leftrightarrow & x = \frac{590}{3} & & \\ & V = \left\{ \frac{590}{3} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{13}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-120& = & -546x+2340 \\\Leftrightarrow & 65x \color{red}{-120} \color{blue}{+120} \color{blue}{+546x} & = & \color{red}{-546x} +2340 \color{blue}{+546x} \color{blue}{+120} \\\Leftrightarrow & 611x& = & 2460 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{2460}{611} \\\Leftrightarrow & x = \frac{2460}{611} & & \\ & V = \left\{ \frac{2460}{611} \right\} & \\\end{align}\)