Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{7}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 18 }{ \color{blue}{42} })& = & (\frac{-70}{ \color{blue}{42} }x-\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-18& = & -70x-294 \\\Leftrightarrow & 21x \color{red}{-18} \color{blue}{+18} \color{blue}{+70x} & = & \color{red}{-70x} -294 \color{blue}{+70x} \color{blue}{+18} \\\Leftrightarrow & 91x& = & -276 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-276}{91} \\\Leftrightarrow & x = \frac{-276}{91} & & \\ & V = \left\{ \frac{-276}{91} \right\} & \\\end{align}\)
2. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{13}& = & \frac{3}{4}x-7 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{117}{ \color{blue}{156} }x-\frac{1092}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-60& = & 117x-1092 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{-117x} & = & \color{red}{117x} -1092 \color{blue}{-117x} \color{blue}{+60} \\\Leftrightarrow & -65x& = & -1032 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-1032}{-65} \\\Leftrightarrow & x = \frac{1032}{65} & & \\ & V = \left\{ \frac{1032}{65} \right\} & \\\end{align}\)
3. \(\text{210 is het kleinste gemene veelvoud van 7, 15 en 6} \\ \begin{align} & \frac{x}{7}+\frac{4}{15}& = & \frac{5}{6}x-5 \\\Leftrightarrow & \color{blue}{210.} (\frac{30x}{ \color{blue}{210} }+ \frac{ 56 }{ \color{blue}{210} })& = & (\frac{175}{ \color{blue}{210} }x-\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 30x+56& = & 175x-1050 \\\Leftrightarrow & 30x \color{red}{+56} \color{blue}{-56} \color{blue}{-175x} & = & \color{red}{175x} -1050 \color{blue}{-175x} \color{blue}{-56} \\\Leftrightarrow & -145x& = & -1106 \\\Leftrightarrow & \frac{-145x}{ \color{red}{-145} }& = & \frac{-1106}{-145} \\\Leftrightarrow & x = \frac{1106}{145} & & \\ & V = \left\{ \frac{1106}{145} \right\} & \\\end{align}\)
4. \(\text{132 is het kleinste gemene veelvoud van 3, 11 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{11}& = & \frac{3}{4}x+6 \\\Leftrightarrow & \color{blue}{132.} (\frac{44x}{ \color{blue}{132} }- \frac{ 48 }{ \color{blue}{132} })& = & (\frac{99}{ \color{blue}{132} }x+\frac{792}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 44x-48& = & 99x+792 \\\Leftrightarrow & 44x \color{red}{-48} \color{blue}{+48} \color{blue}{-99x} & = & \color{red}{99x} +792 \color{blue}{-99x} \color{blue}{+48} \\\Leftrightarrow & -55x& = & 840 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{840}{-55} \\\Leftrightarrow & x = \frac{-168}{11} & & \\ & V = \left\{ \frac{-168}{11} \right\} & \\\end{align}\)
5. \(\text{84 is het kleinste gemene veelvoud van 4, 14 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{14}& = & \frac{5}{3}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 18 }{ \color{blue}{84} })& = & (\frac{140}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+18& = & 140x-588 \\\Leftrightarrow & 21x \color{red}{+18} \color{blue}{-18} \color{blue}{-140x} & = & \color{red}{140x} -588 \color{blue}{-140x} \color{blue}{-18} \\\Leftrightarrow & -119x& = & -606 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{-606}{-119} \\\Leftrightarrow & x = \frac{606}{119} & & \\ & V = \left\{ \frac{606}{119} \right\} & \\\end{align}\)
6. \(\text{35 is het kleinste gemene veelvoud van 7, 7 en 5} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{4}{5}x+8 \\\Leftrightarrow & \color{blue}{35.} (\frac{5x}{ \color{blue}{35} }+ \frac{ 20 }{ \color{blue}{35} })& = & (\frac{28}{ \color{blue}{35} }x+\frac{280}{ \color{blue}{35} }) \color{blue}{.35} \\\Leftrightarrow & 5x+20& = & 28x+280 \\\Leftrightarrow & 5x \color{red}{+20} \color{blue}{-20} \color{blue}{-28x} & = & \color{red}{28x} +280 \color{blue}{-28x} \color{blue}{-20} \\\Leftrightarrow & -23x& = & 260 \\\Leftrightarrow & \frac{-23x}{ \color{red}{-23} }& = & \frac{260}{-23} \\\Leftrightarrow & x = \frac{-260}{23} & & \\ & V = \left\{ \frac{-260}{23} \right\} & \\\end{align}\)
7. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{3}{2}x+5 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 40 }{ \color{blue}{130} })& = & (\frac{195}{ \color{blue}{130} }x+\frac{650}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+40& = & 195x+650 \\\Leftrightarrow & 26x \color{red}{+40} \color{blue}{-40} \color{blue}{-195x} & = & \color{red}{195x} +650 \color{blue}{-195x} \color{blue}{-40} \\\Leftrightarrow & -169x& = & 610 \\\Leftrightarrow & \frac{-169x}{ \color{red}{-169} }& = & \frac{610}{-169} \\\Leftrightarrow & x = \frac{-610}{169} & & \\ & V = \left\{ \frac{-610}{169} \right\} & \\\end{align}\)
8. \(\text{70 is het kleinste gemene veelvoud van 7, 10 en 2} \\ \begin{align} & \frac{x}{7}+\frac{3}{10}& = & \frac{7}{2}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }+ \frac{ 21 }{ \color{blue}{70} })& = & (\frac{245}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x+21& = & 245x+490 \\\Leftrightarrow & 10x \color{red}{+21} \color{blue}{-21} \color{blue}{-245x} & = & \color{red}{245x} +490 \color{blue}{-245x} \color{blue}{-21} \\\Leftrightarrow & -235x& = & 469 \\\Leftrightarrow & \frac{-235x}{ \color{red}{-235} }& = & \frac{469}{-235} \\\Leftrightarrow & x = \frac{-469}{235} & & \\ & V = \left\{ \frac{-469}{235} \right\} & \\\end{align}\)
9. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{3}{2}x-1 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{99}{ \color{blue}{66} }x-\frac{66}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-12& = & 99x-66 \\\Leftrightarrow & 22x \color{red}{-12} \color{blue}{+12} \color{blue}{-99x} & = & \color{red}{99x} -66 \color{blue}{-99x} \color{blue}{+12} \\\Leftrightarrow & -77x& = & -54 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{-54}{-77} \\\Leftrightarrow & x = \frac{54}{77} & & \\ & V = \left\{ \frac{54}{77} \right\} & \\\end{align}\)
10. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{15}& = & \frac{-2}{5}x+6 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{-24}{ \color{blue}{60} }x+\frac{360}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+16& = & -24x+360 \\\Leftrightarrow & 15x \color{red}{+16} \color{blue}{-16} \color{blue}{+24x} & = & \color{red}{-24x} +360 \color{blue}{+24x} \color{blue}{-16} \\\Leftrightarrow & 39x& = & 344 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{344}{39} \\\Leftrightarrow & x = \frac{344}{39} & & \\ & V = \left\{ \frac{344}{39} \right\} & \\\end{align}\)
11. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{7}& = & \frac{-2}{3}x-6 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 6 }{ \color{blue}{21} })& = & (\frac{-14}{ \color{blue}{21} }x-\frac{126}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+6& = & -14x-126 \\\Leftrightarrow & 3x \color{red}{+6} \color{blue}{-6} \color{blue}{+14x} & = & \color{red}{-14x} -126 \color{blue}{+14x} \color{blue}{-6} \\\Leftrightarrow & 17x& = & -132 \\\Leftrightarrow & \frac{17x}{ \color{red}{17} }& = & \frac{-132}{17} \\\Leftrightarrow & x = \frac{-132}{17} & & \\ & V = \left\{ \frac{-132}{17} \right\} & \\\end{align}\)
12. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x-\frac{180}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & -24x-180 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{+24x} & = & \color{red}{-24x} -180 \color{blue}{+24x} \color{blue}{+4} \\\Leftrightarrow & 39x& = & -176 \\\Leftrightarrow & \frac{39x}{ \color{red}{39} }& = & \frac{-176}{39} \\\Leftrightarrow & x = \frac{-176}{39} & & \\ & V = \left\{ \frac{-176}{39} \right\} & \\\end{align}\)