Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 6} \\ \begin{align} & \frac{x}{5}+\frac{3}{14}& = & \frac{5}{6}x-3 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 45 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x-\frac{630}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+45& = & 175x-630 \\\Leftrightarrow & 42x \color{red}{+45} \color{blue}{-45} \color{blue}{-175x} & = & \color{red}{175x} -630 \color{blue}{-175x} \color{blue}{-45} \\\Leftrightarrow & -133x& = & -675 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-675}{-133} \\\Leftrightarrow & x = \frac{675}{133} & & \\ & V = \left\{ \frac{675}{133} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{7}& = & \frac{1}{6}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{35}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x+120& = & 35x-1050 \\\Leftrightarrow & 42x \color{red}{+120} \color{blue}{-120} \color{blue}{-35x} & = & \color{red}{35x} -1050 \color{blue}{-35x} \color{blue}{-120} \\\Leftrightarrow & 7x& = & -1170 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-1170}{7} \\\Leftrightarrow & x = \frac{-1170}{7} & & \\ & V = \left\{ \frac{-1170}{7} \right\} & \\\end{align}\)
3. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{15}& = & \frac{1}{4}x+3 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }- \frac{ 8 }{ \color{blue}{60} })& = & (\frac{15}{ \color{blue}{60} }x+\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x-8& = & 15x+180 \\\Leftrightarrow & 20x \color{red}{-8} \color{blue}{+8} \color{blue}{-15x} & = & \color{red}{15x} +180 \color{blue}{-15x} \color{blue}{+8} \\\Leftrightarrow & 5x& = & 188 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{188}{5} \\\Leftrightarrow & x = \frac{188}{5} & & \\ & V = \left\{ \frac{188}{5} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{15}& = & \frac{1}{6}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{5}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x+4& = & 5x+30 \\\Leftrightarrow & 6x \color{red}{+4} \color{blue}{-4} \color{blue}{-5x} & = & \color{red}{5x} +30 \color{blue}{-5x} \color{blue}{-4} \\\Leftrightarrow & x& = & 26 \\ & V = \left\{ 26 \right\} & \\\end{align}\)
5. \(\text{24 is het kleinste gemene veelvoud van 4, 8 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{8}& = & \frac{-2}{3}x+3 \\\Leftrightarrow & \color{blue}{24.} (\frac{6x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{-16}{ \color{blue}{24} }x+\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 6x+9& = & -16x+72 \\\Leftrightarrow & 6x \color{red}{+9} \color{blue}{-9} \color{blue}{+16x} & = & \color{red}{-16x} +72 \color{blue}{+16x} \color{blue}{-9} \\\Leftrightarrow & 22x& = & 63 \\\Leftrightarrow & \frac{22x}{ \color{red}{22} }& = & \frac{63}{22} \\\Leftrightarrow & x = \frac{63}{22} & & \\ & V = \left\{ \frac{63}{22} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 5, 10 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{7}{6}x-3 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{35}{ \color{blue}{30} }x-\frac{90}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-9& = & 35x-90 \\\Leftrightarrow & 6x \color{red}{-9} \color{blue}{+9} \color{blue}{-35x} & = & \color{red}{35x} -90 \color{blue}{-35x} \color{blue}{+9} \\\Leftrightarrow & -29x& = & -81 \\\Leftrightarrow & \frac{-29x}{ \color{red}{-29} }& = & \frac{-81}{-29} \\\Leftrightarrow & x = \frac{81}{29} & & \\ & V = \left\{ \frac{81}{29} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{2}{9}& = & \frac{1}{2}x+2 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x+\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-20& = & 45x+180 \\\Leftrightarrow & 18x \color{red}{-20} \color{blue}{+20} \color{blue}{-45x} & = & \color{red}{45x} +180 \color{blue}{-45x} \color{blue}{+20} \\\Leftrightarrow & -27x& = & 200 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{200}{-27} \\\Leftrightarrow & x = \frac{-200}{27} & & \\ & V = \left\{ \frac{-200}{27} \right\} & \\\end{align}\)
8. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{1}{3}x+8 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x+\frac{144}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-4& = & 6x+144 \\\Leftrightarrow & 9x \color{red}{-4} \color{blue}{+4} \color{blue}{-6x} & = & \color{red}{6x} +144 \color{blue}{-6x} \color{blue}{+4} \\\Leftrightarrow & 3x& = & 148 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{148}{3} \\\Leftrightarrow & x = \frac{148}{3} & & \\ & V = \left\{ \frac{148}{3} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{13}& = & \frac{7}{5}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{546}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+60& = & 546x+2340 \\\Leftrightarrow & 65x \color{red}{+60} \color{blue}{-60} \color{blue}{-546x} & = & \color{red}{546x} +2340 \color{blue}{-546x} \color{blue}{-60} \\\Leftrightarrow & -481x& = & 2280 \\\Leftrightarrow & \frac{-481x}{ \color{red}{-481} }& = & \frac{2280}{-481} \\\Leftrightarrow & x = \frac{-2280}{481} & & \\ & V = \left\{ \frac{-2280}{481} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-168}{ \color{blue}{210} }x-\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & -168x-420 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+168x} & = & \color{red}{-168x} -420 \color{blue}{+168x} \color{blue}{+60} \\\Leftrightarrow & 203x& = & -360 \\\Leftrightarrow & \frac{203x}{ \color{red}{203} }& = & \frac{-360}{203} \\\Leftrightarrow & x = \frac{-360}{203} & & \\ & V = \left\{ \frac{-360}{203} \right\} & \\\end{align}\)
11. \(\text{60 is het kleinste gemene veelvoud van 2, 12 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{12}& = & \frac{-2}{5}x-6 \\\Leftrightarrow & \color{blue}{60.} (\frac{30x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{-24}{ \color{blue}{60} }x-\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 30x+25& = & -24x-360 \\\Leftrightarrow & 30x \color{red}{+25} \color{blue}{-25} \color{blue}{+24x} & = & \color{red}{-24x} -360 \color{blue}{+24x} \color{blue}{-25} \\\Leftrightarrow & 54x& = & -385 \\\Leftrightarrow & \frac{54x}{ \color{red}{54} }& = & \frac{-385}{54} \\\Leftrightarrow & x = \frac{-385}{54} & & \\ & V = \left\{ \frac{-385}{54} \right\} & \\\end{align}\)
12. \(\text{40 is het kleinste gemene veelvoud van 5, 8 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{8}& = & \frac{3}{4}x-7 \\\Leftrightarrow & \color{blue}{40.} (\frac{8x}{ \color{blue}{40} }- \frac{ 15 }{ \color{blue}{40} })& = & (\frac{30}{ \color{blue}{40} }x-\frac{280}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 8x-15& = & 30x-280 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{-30x} & = & \color{red}{30x} -280 \color{blue}{-30x} \color{blue}{+15} \\\Leftrightarrow & -22x& = & -265 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{-265}{-22} \\\Leftrightarrow & x = \frac{265}{22} & & \\ & V = \left\{ \frac{265}{22} \right\} & \\\end{align}\)