Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{210}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 21x-210 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-21x} & = & \color{red}{21x} -210 \color{blue}{-21x} \color{blue}{-12} \\\Leftrightarrow & -7x& = & -222 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-222}{-7} \\\Leftrightarrow & x = \frac{222}{7} & & \\ & V = \left\{ \frac{222}{7} \right\} & \\\end{align}\)
2. \(\text{210 is het kleinste gemene veelvoud van 5, 14 en 6} \\ \begin{align} & \frac{x}{5}-\frac{5}{14}& = & \frac{7}{6}x+4 \\\Leftrightarrow & \color{blue}{210.} (\frac{42x}{ \color{blue}{210} }- \frac{ 75 }{ \color{blue}{210} })& = & (\frac{245}{ \color{blue}{210} }x+\frac{840}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 42x-75& = & 245x+840 \\\Leftrightarrow & 42x \color{red}{-75} \color{blue}{+75} \color{blue}{-245x} & = & \color{red}{245x} +840 \color{blue}{-245x} \color{blue}{+75} \\\Leftrightarrow & -203x& = & 915 \\\Leftrightarrow & \frac{-203x}{ \color{red}{-203} }& = & \frac{915}{-203} \\\Leftrightarrow & x = \frac{-915}{203} & & \\ & V = \left\{ \frac{-915}{203} \right\} & \\\end{align}\)
3. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{-2}{3}x-4 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{-44}{ \color{blue}{66} }x-\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+30& = & -44x-264 \\\Leftrightarrow & 33x \color{red}{+30} \color{blue}{-30} \color{blue}{+44x} & = & \color{red}{-44x} -264 \color{blue}{+44x} \color{blue}{-30} \\\Leftrightarrow & 77x& = & -294 \\\Leftrightarrow & \frac{77x}{ \color{red}{77} }& = & \frac{-294}{77} \\\Leftrightarrow & x = \frac{-42}{11} & & \\ & V = \left\{ \frac{-42}{11} \right\} & \\\end{align}\)
4. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{13}& = & \frac{5}{4}x-7 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 60 }{ \color{blue}{156} })& = & (\frac{195}{ \color{blue}{156} }x-\frac{1092}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+60& = & 195x-1092 \\\Leftrightarrow & 52x \color{red}{+60} \color{blue}{-60} \color{blue}{-195x} & = & \color{red}{195x} -1092 \color{blue}{-195x} \color{blue}{-60} \\\Leftrightarrow & -143x& = & -1152 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-1152}{-143} \\\Leftrightarrow & x = \frac{1152}{143} & & \\ & V = \left\{ \frac{1152}{143} \right\} & \\\end{align}\)
5. \(\text{195 is het kleinste gemene veelvoud van 3, 13 en 5} \\ \begin{align} & \frac{x}{3}+\frac{3}{13}& = & \frac{-2}{5}x+3 \\\Leftrightarrow & \color{blue}{195.} (\frac{65x}{ \color{blue}{195} }+ \frac{ 45 }{ \color{blue}{195} })& = & (\frac{-78}{ \color{blue}{195} }x+\frac{585}{ \color{blue}{195} }) \color{blue}{.195} \\\Leftrightarrow & 65x+45& = & -78x+585 \\\Leftrightarrow & 65x \color{red}{+45} \color{blue}{-45} \color{blue}{+78x} & = & \color{red}{-78x} +585 \color{blue}{+78x} \color{blue}{-45} \\\Leftrightarrow & 143x& = & 540 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{540}{143} \\\Leftrightarrow & x = \frac{540}{143} & & \\ & V = \left\{ \frac{540}{143} \right\} & \\\end{align}\)
6. \(\text{130 is het kleinste gemene veelvoud van 5, 13 en 2} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{130.} (\frac{26x}{ \color{blue}{130} }+ \frac{ 30 }{ \color{blue}{130} })& = & (\frac{65}{ \color{blue}{130} }x-\frac{910}{ \color{blue}{130} }) \color{blue}{.130} \\\Leftrightarrow & 26x+30& = & 65x-910 \\\Leftrightarrow & 26x \color{red}{+30} \color{blue}{-30} \color{blue}{-65x} & = & \color{red}{65x} -910 \color{blue}{-65x} \color{blue}{-30} \\\Leftrightarrow & -39x& = & -940 \\\Leftrightarrow & \frac{-39x}{ \color{red}{-39} }& = & \frac{-940}{-39} \\\Leftrightarrow & x = \frac{940}{39} & & \\ & V = \left\{ \frac{940}{39} \right\} & \\\end{align}\)
7. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{11}& = & \frac{-5}{3}x-1 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 48 }{ \color{blue}{132} })& = & (\frac{-220}{ \color{blue}{132} }x-\frac{132}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+48& = & -220x-132 \\\Leftrightarrow & 33x \color{red}{+48} \color{blue}{-48} \color{blue}{+220x} & = & \color{red}{-220x} -132 \color{blue}{+220x} \color{blue}{-48} \\\Leftrightarrow & 253x& = & -180 \\\Leftrightarrow & \frac{253x}{ \color{red}{253} }& = & \frac{-180}{253} \\\Leftrightarrow & x = \frac{-180}{253} & & \\ & V = \left\{ \frac{-180}{253} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x-\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & -72x-180 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{+72x} & = & \color{red}{-72x} -180 \color{blue}{+72x} \color{blue}{-20} \\\Leftrightarrow & 87x& = & -200 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{-200}{87} \\\Leftrightarrow & x = \frac{-200}{87} & & \\ & V = \left\{ \frac{-200}{87} \right\} & \\\end{align}\)
9. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{13}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-130}{ \color{blue}{78} }x+\frac{468}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-18& = & -130x+468 \\\Leftrightarrow & 39x \color{red}{-18} \color{blue}{+18} \color{blue}{+130x} & = & \color{red}{-130x} +468 \color{blue}{+130x} \color{blue}{+18} \\\Leftrightarrow & 169x& = & 486 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{486}{169} \\\Leftrightarrow & x = \frac{486}{169} & & \\ & V = \left\{ \frac{486}{169} \right\} & \\\end{align}\)
10. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{1}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{42}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 42x-210 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-42x} & = & \color{red}{42x} -210 \color{blue}{-42x} \color{blue}{+60} \\\Leftrightarrow & -7x& = & -150 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-150}{-7} \\\Leftrightarrow & x = \frac{150}{7} & & \\ & V = \left\{ \frac{150}{7} \right\} & \\\end{align}\)
11. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{7}& = & \frac{-2}{5}x+5 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 90 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x+\frac{1050}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+90& = & -84x+1050 \\\Leftrightarrow & 35x \color{red}{+90} \color{blue}{-90} \color{blue}{+84x} & = & \color{red}{-84x} +1050 \color{blue}{+84x} \color{blue}{-90} \\\Leftrightarrow & 119x& = & 960 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{960}{119} \\\Leftrightarrow & x = \frac{960}{119} & & \\ & V = \left\{ \frac{960}{119} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}-\frac{3}{16}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }- \frac{ 15 }{ \color{blue}{80} })& = & (\frac{16}{ \color{blue}{80} }x-\frac{320}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x-15& = & 16x-320 \\\Leftrightarrow & 40x \color{red}{-15} \color{blue}{+15} \color{blue}{-16x} & = & \color{red}{16x} -320 \color{blue}{-16x} \color{blue}{+15} \\\Leftrightarrow & 24x& = & -305 \\\Leftrightarrow & \frac{24x}{ \color{red}{24} }& = & \frac{-305}{24} \\\Leftrightarrow & x = \frac{-305}{24} & & \\ & V = \left\{ \frac{-305}{24} \right\} & \\\end{align}\)