Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{2}+\frac{5}{8}=\frac{6}{5}x-6\)
2. \(\frac{x}{2}-\frac{5}{7}=\frac{-8}{3}x+4\)
3. \(\frac{x}{7}+\frac{2}{11}=\frac{1}{2}x-7\)
4. \(\frac{x}{2}-\frac{3}{16}=\frac{1}{5}x+7\)
5. \(\frac{x}{2}-\frac{3}{14}=\frac{-4}{5}x+6\)
6. \(\frac{x}{4}+\frac{4}{7}=\frac{-8}{3}x-2\)
7. \(\frac{x}{3}+\frac{2}{7}=\frac{1}{2}x+8\)
8. \(\frac{x}{4}+\frac{5}{6}=\frac{7}{3}x+4\)
9. \(\frac{x}{3}-\frac{3}{11}=\frac{3}{2}x+8\)
10. \(\frac{x}{5}-\frac{3}{7}=\frac{-5}{6}x-7\)
11. \(\frac{x}{4}+\frac{5}{11}=\frac{-7}{5}x+4\)
12. \(\frac{x}{2}+\frac{2}{11}=\frac{1}{3}x+6\)

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

Verbetersleutel

1. \(\text{40 is het kleinste gemene veelvoud van 2, 8 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{8}& = & \frac{6}{5}x-6 \\\Leftrightarrow & \color{blue}{40.} (\frac{20x}{ \color{blue}{40} }+ \frac{ 25 }{ \color{blue}{40} })& = & (\frac{48}{ \color{blue}{40} }x-\frac{240}{ \color{blue}{40} }) \color{blue}{.40} \\\Leftrightarrow & 20x+25& = & 48x-240 \\\Leftrightarrow & 20x \color{red}{+25} \color{blue}{-25} \color{blue}{-48x} & = & \color{red}{48x} -240 \color{blue}{-48x} \color{blue}{-25} \\\Leftrightarrow & -28x& = & -265 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{-265}{-28} \\\Leftrightarrow & x = \frac{265}{28} & & \\ & V = \left\{ \frac{265}{28} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{7}& = & \frac{-8}{3}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{-112}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-30& = & -112x+168 \\\Leftrightarrow & 21x \color{red}{-30} \color{blue}{+30} \color{blue}{+112x} & = & \color{red}{-112x} +168 \color{blue}{+112x} \color{blue}{+30} \\\Leftrightarrow & 133x& = & 198 \\\Leftrightarrow & \frac{133x}{ \color{red}{133} }& = & \frac{198}{133} \\\Leftrightarrow & x = \frac{198}{133} & & \\ & V = \left\{ \frac{198}{133} \right\} & \\\end{align}\)
3. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 28 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x-\frac{1078}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+28& = & 77x-1078 \\\Leftrightarrow & 22x \color{red}{+28} \color{blue}{-28} \color{blue}{-77x} & = & \color{red}{77x} -1078 \color{blue}{-77x} \color{blue}{-28} \\\Leftrightarrow & -55x& = & -1106 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-1106}{-55} \\\Leftrightarrow & x = \frac{1106}{55} & & \\ & V = \left\{ \frac{1106}{55} \right\} & \\\end{align}\)
4. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{1}{5}x+7 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{16}{ \color{blue}{80} }x+\frac{560}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-15& = & 16x+560 \\\Leftrightarrow & 40x \color{red}{-15} \color{blue}{+15} \color{blue}{-16x} & = & \color{red}{16x} +560 \color{blue}{-16x} \color{blue}{+15} \\\Leftrightarrow & 24x& = & 575 \\\Leftrightarrow & \frac{24x}{ \color{red}{24} }& = & \frac{575}{24} \\\Leftrightarrow & x = \frac{575}{24} & & \\ & V = \left\{ \frac{575}{24} \right\} & \\\end{align}\)
5. \(\text{70 is het kleinste gemene veelvoud van 2, 14 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{14}& = & \frac{-4}{5}x+6 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{-56}{ \color{blue}{70} }x+\frac{420}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x-15& = & -56x+420 \\\Leftrightarrow & 35x \color{red}{-15} \color{blue}{+15} \color{blue}{+56x} & = & \color{red}{-56x} +420 \color{blue}{+56x} \color{blue}{+15} \\\Leftrightarrow & 91x& = & 435 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{435}{91} \\\Leftrightarrow & x = \frac{435}{91} & & \\ & V = \left\{ \frac{435}{91} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-8}{3}x-2 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x-\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -224x-168 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+224x} & = & \color{red}{-224x} -168 \color{blue}{+224x} \color{blue}{-48} \\\Leftrightarrow & 245x& = & -216 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{-216}{245} \\\Leftrightarrow & x = \frac{-216}{245} & & \\ & V = \left\{ \frac{-216}{245} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{336}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 21x+336 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} +336 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -7x& = & 324 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{324}{-7} \\\Leftrightarrow & x = \frac{-324}{7} & & \\ & V = \left\{ \frac{-324}{7} \right\} & \\\end{align}\)
8. \(\text{12 is het kleinste gemene veelvoud van 4, 6 en 3} \\ \begin{align} & \frac{x}{4}+\frac{5}{6}& = & \frac{7}{3}x+4 \\\Leftrightarrow & \color{blue}{12.} (\frac{3x}{ \color{blue}{12} }+ \frac{ 10 }{ \color{blue}{12} })& = & (\frac{28}{ \color{blue}{12} }x+\frac{48}{ \color{blue}{12} }) \color{blue}{.12} \\\Leftrightarrow & 3x+10& = & 28x+48 \\\Leftrightarrow & 3x \color{red}{+10} \color{blue}{-10} \color{blue}{-28x} & = & \color{red}{28x} +48 \color{blue}{-28x} \color{blue}{-10} \\\Leftrightarrow & -25x& = & 38 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{38}{-25} \\\Leftrightarrow & x = \frac{-38}{25} & & \\ & V = \left\{ \frac{-38}{25} \right\} & \\\end{align}\)
9. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{11}& = & \frac{3}{2}x+8 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 18 }{ \color{blue}{66} })& = & (\frac{99}{ \color{blue}{66} }x+\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-18& = & 99x+528 \\\Leftrightarrow & 22x \color{red}{-18} \color{blue}{+18} \color{blue}{-99x} & = & \color{red}{99x} +528 \color{blue}{-99x} \color{blue}{+18} \\\Leftrightarrow & -77x& = & 546 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{546}{-77} \\\Leftrightarrow & x = \frac{-78}{11} & & \\ & V = \left\{ \frac{-78}{11} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 5, 7 en 6} \\ \begin{align} & \frac{x}{5}-\frac{3}{7}& = & \frac{-5}{6}x-7 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-175}{ \color{blue}{210} }x-\frac{1470}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-90& = & -175x-1470 \\\Leftrightarrow & 42x \color{red}{-90} \color{blue}{+90} \color{blue}{+175x} & = & \color{red}{-175x} -1470 \color{blue}{+175x} \color{blue}{+90} \\\Leftrightarrow & 217x& = & -1380 \\\Leftrightarrow & \frac{217x}{ \color{red}{217} }& = & \frac{-1380}{217} \\\Leftrightarrow & x = \frac{-1380}{217} & & \\ & V = \left\{ \frac{-1380}{217} \right\} & \\\end{align}\)
11. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{5}{11}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 100 }{ \color{blue}{220} })& = & (\frac{-308}{ \color{blue}{220} }x+\frac{880}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+100& = & -308x+880 \\\Leftrightarrow & 55x \color{red}{+100} \color{blue}{-100} \color{blue}{+308x} & = & \color{red}{-308x} +880 \color{blue}{+308x} \color{blue}{-100} \\\Leftrightarrow & 363x& = & 780 \\\Leftrightarrow & \frac{363x}{ \color{red}{363} }& = & \frac{780}{363} \\\Leftrightarrow & x = \frac{260}{121} & & \\ & V = \left\{ \frac{260}{121} \right\} & \\\end{align}\)
12. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{11}& = & \frac{1}{3}x+6 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 12 }{ \color{blue}{66} })& = & (\frac{22}{ \color{blue}{66} }x+\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+12& = & 22x+396 \\\Leftrightarrow & 33x \color{red}{+12} \color{blue}{-12} \color{blue}{-22x} & = & \color{red}{22x} +396 \color{blue}{-22x} \color{blue}{-12} \\\Leftrightarrow & 11x& = & 384 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{384}{11} \\\Leftrightarrow & x = \frac{384}{11} & & \\ & V = \left\{ \frac{384}{11} \right\} & \\\end{align}\)