Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{3}-\frac{5}{7}=\frac{7}{2}x-6\)
2. \(\frac{x}{4}+\frac{3}{13}=\frac{7}{3}x+5\)
3. \(\frac{x}{6}-\frac{5}{14}=\frac{1}{5}x-1\)
4. \(\frac{x}{7}+\frac{4}{15}=\frac{-7}{4}x+7\)
5. \(\frac{x}{2}-\frac{3}{16}=\frac{-2}{5}x+4\)
6. \(\frac{x}{3}+\frac{3}{8}=\frac{1}{2}x-3\)
7. \(\frac{x}{7}+\frac{5}{6}=\frac{1}{6}x+5\)
8. \(\frac{x}{4}+\frac{2}{11}=\frac{6}{5}x-4\)
9. \(\frac{x}{7}+\frac{2}{15}=\frac{3}{4}x+3\)
10. \(\frac{x}{6}-\frac{2}{7}=\frac{4}{5}x-5\)
11. \(\frac{x}{3}-\frac{5}{14}=\frac{1}{2}x+4\)
12. \(\frac{x}{4}-\frac{4}{7}=\frac{-7}{5}x+7\)

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{7}& = & \frac{7}{2}x-6 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 30 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-30& = & 147x-252 \\\Leftrightarrow & 14x \color{red}{-30} \color{blue}{+30} \color{blue}{-147x} & = & \color{red}{147x} -252 \color{blue}{-147x} \color{blue}{+30} \\\Leftrightarrow & -133x& = & -222 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-222}{-133} \\\Leftrightarrow & x = \frac{222}{133} & & \\ & V = \left\{ \frac{222}{133} \right\} & \\\end{align}\)
2. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{13}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }+ \frac{ 36 }{ \color{blue}{156} })& = & (\frac{364}{ \color{blue}{156} }x+\frac{780}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x+36& = & 364x+780 \\\Leftrightarrow & 39x \color{red}{+36} \color{blue}{-36} \color{blue}{-364x} & = & \color{red}{364x} +780 \color{blue}{-364x} \color{blue}{-36} \\\Leftrightarrow & -325x& = & 744 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{744}{-325} \\\Leftrightarrow & x = \frac{-744}{325} & & \\ & V = \left\{ \frac{-744}{325} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 6, 14 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{14}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-75& = & 42x-210 \\\Leftrightarrow & 35x \color{red}{-75} \color{blue}{+75} \color{blue}{-42x} & = & \color{red}{42x} -210 \color{blue}{-42x} \color{blue}{+75} \\\Leftrightarrow & -7x& = & -135 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-135}{-7} \\\Leftrightarrow & x = \frac{135}{7} & & \\ & V = \left\{ \frac{135}{7} \right\} & \\\end{align}\)
4. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{-7}{4}x+7 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }+ \frac{ 112 }{ \color{blue}{420} })& = & (\frac{-735}{ \color{blue}{420} }x+\frac{2940}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x+112& = & -735x+2940 \\\Leftrightarrow & 60x \color{red}{+112} \color{blue}{-112} \color{blue}{+735x} & = & \color{red}{-735x} +2940 \color{blue}{+735x} \color{blue}{-112} \\\Leftrightarrow & 795x& = & 2828 \\\Leftrightarrow & \frac{795x}{ \color{red}{795} }& = & \frac{2828}{795} \\\Leftrightarrow & x = \frac{2828}{795} & & \\ & V = \left\{ \frac{2828}{795} \right\} & \\\end{align}\)
5. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{-32}{ \color{blue}{80} }x+\frac{320}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-15& = & -32x+320 \\\Leftrightarrow & 40x \color{red}{-15} \color{blue}{+15} \color{blue}{+32x} & = & \color{red}{-32x} +320 \color{blue}{+32x} \color{blue}{+15} \\\Leftrightarrow & 72x& = & 335 \\\Leftrightarrow & \frac{72x}{ \color{red}{72} }& = & \frac{335}{72} \\\Leftrightarrow & x = \frac{335}{72} & & \\ & V = \left\{ \frac{335}{72} \right\} & \\\end{align}\)
6. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{8}& = & \frac{1}{2}x-3 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }+ \frac{ 9 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x-\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x+9& = & 12x-72 \\\Leftrightarrow & 8x \color{red}{+9} \color{blue}{-9} \color{blue}{-12x} & = & \color{red}{12x} -72 \color{blue}{-12x} \color{blue}{-9} \\\Leftrightarrow & -4x& = & -81 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{-81}{-4} \\\Leftrightarrow & x = \frac{81}{4} & & \\ & V = \left\{ \frac{81}{4} \right\} & \\\end{align}\)
7. \(\text{42 is het kleinste gemene veelvoud van 7, 6 en 6} \\ \begin{align} & \frac{x}{7}+\frac{5}{6}& = & \frac{1}{6}x+5 \\\Leftrightarrow & \color{blue}{42.} (\frac{6x}{ \color{blue}{42} }+ \frac{ 35 }{ \color{blue}{42} })& = & (\frac{7}{ \color{blue}{42} }x+\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 6x+35& = & 7x+210 \\\Leftrightarrow & 6x \color{red}{+35} \color{blue}{-35} \color{blue}{-7x} & = & \color{red}{7x} +210 \color{blue}{-7x} \color{blue}{-35} \\\Leftrightarrow & -x& = & 175 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{175}{-1} \\\Leftrightarrow & x = -175 & & \\ & V = \left\{ -175 \right\} & \\\end{align}\)
8. \(\text{220 is het kleinste gemene veelvoud van 4, 11 en 5} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{6}{5}x-4 \\\Leftrightarrow & \color{blue}{220.} (\frac{55x}{ \color{blue}{220} }+ \frac{ 40 }{ \color{blue}{220} })& = & (\frac{264}{ \color{blue}{220} }x-\frac{880}{ \color{blue}{220} }) \color{blue}{.220} \\\Leftrightarrow & 55x+40& = & 264x-880 \\\Leftrightarrow & 55x \color{red}{+40} \color{blue}{-40} \color{blue}{-264x} & = & \color{red}{264x} -880 \color{blue}{-264x} \color{blue}{-40} \\\Leftrightarrow & -209x& = & -920 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-920}{-209} \\\Leftrightarrow & x = \frac{920}{209} & & \\ & V = \left\{ \frac{920}{209} \right\} & \\\end{align}\)
9. \(\text{420 is het kleinste gemene veelvoud van 7, 15 en 4} \\ \begin{align} & \frac{x}{7}+\frac{2}{15}& = & \frac{3}{4}x+3 \\\Leftrightarrow & \color{blue}{420.} (\frac{60x}{ \color{blue}{420} }+ \frac{ 56 }{ \color{blue}{420} })& = & (\frac{315}{ \color{blue}{420} }x+\frac{1260}{ \color{blue}{420} }) \color{blue}{.420} \\\Leftrightarrow & 60x+56& = & 315x+1260 \\\Leftrightarrow & 60x \color{red}{+56} \color{blue}{-56} \color{blue}{-315x} & = & \color{red}{315x} +1260 \color{blue}{-315x} \color{blue}{-56} \\\Leftrightarrow & -255x& = & 1204 \\\Leftrightarrow & \frac{-255x}{ \color{red}{-255} }& = & \frac{1204}{-255} \\\Leftrightarrow & x = \frac{-1204}{255} & & \\ & V = \left\{ \frac{-1204}{255} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{4}{5}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 168x-1050 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-168x} & = & \color{red}{168x} -1050 \color{blue}{-168x} \color{blue}{+60} \\\Leftrightarrow & -133x& = & -990 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-990}{-133} \\\Leftrightarrow & x = \frac{990}{133} & & \\ & V = \left\{ \frac{990}{133} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{14}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }- \frac{ 15 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x+\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x-15& = & 21x+168 \\\Leftrightarrow & 14x \color{red}{-15} \color{blue}{+15} \color{blue}{-21x} & = & \color{red}{21x} +168 \color{blue}{-21x} \color{blue}{+15} \\\Leftrightarrow & -7x& = & 183 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{183}{-7} \\\Leftrightarrow & x = \frac{-183}{7} & & \\ & V = \left\{ \frac{-183}{7} \right\} & \\\end{align}\)
12. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{7}& = & \frac{-7}{5}x+7 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-196}{ \color{blue}{140} }x+\frac{980}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-80& = & -196x+980 \\\Leftrightarrow & 35x \color{red}{-80} \color{blue}{+80} \color{blue}{+196x} & = & \color{red}{-196x} +980 \color{blue}{+196x} \color{blue}{+80} \\\Leftrightarrow & 231x& = & 1060 \\\Leftrightarrow & \frac{231x}{ \color{red}{231} }& = & \frac{1060}{231} \\\Leftrightarrow & x = \frac{1060}{231} & & \\ & V = \left\{ \frac{1060}{231} \right\} & \\\end{align}\)