Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{5}x+3 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 75 }{ \color{blue}{165} })& = & (\frac{33}{ \color{blue}{165} }x+\frac{495}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-75& = & 33x+495 \\\Leftrightarrow & 55x \color{red}{-75} \color{blue}{+75} \color{blue}{-33x} & = & \color{red}{33x} +495 \color{blue}{-33x} \color{blue}{+75} \\\Leftrightarrow & 22x& = & 570 \\\Leftrightarrow & \frac{22x}{ \color{red}{22} }& = & \frac{570}{22} \\\Leftrightarrow & x = \frac{285}{11} & & \\ & V = \left\{ \frac{285}{11} \right\} & \\\end{align}\)
2. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}-\frac{2}{7}& = & \frac{4}{3}x-8 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }- \frac{ 6 }{ \color{blue}{21} })& = & (\frac{28}{ \color{blue}{21} }x-\frac{168}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x-6& = & 28x-168 \\\Leftrightarrow & 3x \color{red}{-6} \color{blue}{+6} \color{blue}{-28x} & = & \color{red}{28x} -168 \color{blue}{-28x} \color{blue}{+6} \\\Leftrightarrow & -25x& = & -162 \\\Leftrightarrow & \frac{-25x}{ \color{red}{-25} }& = & \frac{-162}{-25} \\\Leftrightarrow & x = \frac{162}{25} & & \\ & V = \left\{ \frac{162}{25} \right\} & \\\end{align}\)
3. \(\text{105 is het kleinste gemene veelvoud van 7, 15 en 5} \\ \begin{align} & \frac{x}{7}-\frac{4}{15}& = & \frac{-7}{5}x-2 \\\Leftrightarrow & \color{blue}{105.} (\frac{15x}{ \color{blue}{105} }- \frac{ 28 }{ \color{blue}{105} })& = & (\frac{-147}{ \color{blue}{105} }x-\frac{210}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 15x-28& = & -147x-210 \\\Leftrightarrow & 15x \color{red}{-28} \color{blue}{+28} \color{blue}{+147x} & = & \color{red}{-147x} -210 \color{blue}{+147x} \color{blue}{+28} \\\Leftrightarrow & 162x& = & -182 \\\Leftrightarrow & \frac{162x}{ \color{red}{162} }& = & \frac{-182}{162} \\\Leftrightarrow & x = \frac{-91}{81} & & \\ & V = \left\{ \frac{-91}{81} \right\} & \\\end{align}\)
4. \(\text{385 is het kleinste gemene veelvoud van 7, 11 en 5} \\ \begin{align} & \frac{x}{7}+\frac{2}{11}& = & \frac{1}{5}x-2 \\\Leftrightarrow & \color{blue}{385.} (\frac{55x}{ \color{blue}{385} }+ \frac{ 70 }{ \color{blue}{385} })& = & (\frac{77}{ \color{blue}{385} }x-\frac{770}{ \color{blue}{385} }) \color{blue}{.385} \\\Leftrightarrow & 55x+70& = & 77x-770 \\\Leftrightarrow & 55x \color{red}{+70} \color{blue}{-70} \color{blue}{-77x} & = & \color{red}{77x} -770 \color{blue}{-77x} \color{blue}{-70} \\\Leftrightarrow & -22x& = & -840 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{-840}{-22} \\\Leftrightarrow & x = \frac{420}{11} & & \\ & V = \left\{ \frac{420}{11} \right\} & \\\end{align}\)
5. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}-\frac{4}{9}& = & \frac{-2}{5}x+1 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }- \frac{ 80 }{ \color{blue}{180} })& = & (\frac{-72}{ \color{blue}{180} }x+\frac{180}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x-80& = & -72x+180 \\\Leftrightarrow & 45x \color{red}{-80} \color{blue}{+80} \color{blue}{+72x} & = & \color{red}{-72x} +180 \color{blue}{+72x} \color{blue}{+80} \\\Leftrightarrow & 117x& = & 260 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{260}{117} \\\Leftrightarrow & x = \frac{20}{9} & & \\ & V = \left\{ \frac{20}{9} \right\} & \\\end{align}\)
6. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{7}& = & \frac{3}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 150 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-150& = & 126x+420 \\\Leftrightarrow & 35x \color{red}{-150} \color{blue}{+150} \color{blue}{-126x} & = & \color{red}{126x} +420 \color{blue}{-126x} \color{blue}{+150} \\\Leftrightarrow & -91x& = & 570 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{570}{-91} \\\Leftrightarrow & x = \frac{-570}{91} & & \\ & V = \left\{ \frac{-570}{91} \right\} & \\\end{align}\)
7. \(\text{24 is het kleinste gemene veelvoud van 3, 8 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{8}& = & \frac{1}{2}x+3 \\\Leftrightarrow & \color{blue}{24.} (\frac{8x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{12}{ \color{blue}{24} }x+\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 8x-15& = & 12x+72 \\\Leftrightarrow & 8x \color{red}{-15} \color{blue}{+15} \color{blue}{-12x} & = & \color{red}{12x} +72 \color{blue}{-12x} \color{blue}{+15} \\\Leftrightarrow & -4x& = & 87 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{87}{-4} \\\Leftrightarrow & x = \frac{-87}{4} & & \\ & V = \left\{ \frac{-87}{4} \right\} & \\\end{align}\)
8. \(\text{45 is het kleinste gemene veelvoud van 3, 9 en 5} \\ \begin{align} & \frac{x}{3}+\frac{2}{9}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{45.} (\frac{15x}{ \color{blue}{45} }+ \frac{ 10 }{ \color{blue}{45} })& = & (\frac{-36}{ \color{blue}{45} }x-\frac{360}{ \color{blue}{45} }) \color{blue}{.45} \\\Leftrightarrow & 15x+10& = & -36x-360 \\\Leftrightarrow & 15x \color{red}{+10} \color{blue}{-10} \color{blue}{+36x} & = & \color{red}{-36x} -360 \color{blue}{+36x} \color{blue}{-10} \\\Leftrightarrow & 51x& = & -370 \\\Leftrightarrow & \frac{51x}{ \color{red}{51} }& = & \frac{-370}{51} \\\Leftrightarrow & x = \frac{-370}{51} & & \\ & V = \left\{ \frac{-370}{51} \right\} & \\\end{align}\)
9. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{1}{3}x-5 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{26}{ \color{blue}{78} }x-\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-24& = & 26x-390 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{-26x} & = & \color{red}{26x} -390 \color{blue}{-26x} \color{blue}{+24} \\\Leftrightarrow & 13x& = & -366 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-366}{13} \\\Leftrightarrow & x = \frac{-366}{13} & & \\ & V = \left\{ \frac{-366}{13} \right\} & \\\end{align}\)
10. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{3}{13}& = & \frac{-7}{4}x+4 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 60 }{ \color{blue}{260} })& = & (\frac{-455}{ \color{blue}{260} }x+\frac{1040}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-60& = & -455x+1040 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{+455x} & = & \color{red}{-455x} +1040 \color{blue}{+455x} \color{blue}{+60} \\\Leftrightarrow & 507x& = & 1100 \\\Leftrightarrow & \frac{507x}{ \color{red}{507} }& = & \frac{1100}{507} \\\Leftrightarrow & x = \frac{1100}{507} & & \\ & V = \left\{ \frac{1100}{507} \right\} & \\\end{align}\)
11. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{8}& = & \frac{-2}{3}x+7 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{-16}{ \color{blue}{24} }x+\frac{168}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-15& = & -16x+168 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{+16x} & = & \color{red}{-16x} +168 \color{blue}{+16x} \color{blue}{+15} \\\Leftrightarrow & 28x& = & 183 \\\Leftrightarrow & \frac{28x}{ \color{red}{28} }& = & \frac{183}{28} \\\Leftrightarrow & x = \frac{183}{28} & & \\ & V = \left\{ \frac{183}{28} \right\} & \\\end{align}\)
12. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{4}{7}& = & \frac{4}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 120 }{ \color{blue}{210} })& = & (\frac{168}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+120& = & 168x+420 \\\Leftrightarrow & 35x \color{red}{+120} \color{blue}{-120} \color{blue}{-168x} & = & \color{red}{168x} +420 \color{blue}{-168x} \color{blue}{-120} \\\Leftrightarrow & -133x& = & 300 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{300}{-133} \\\Leftrightarrow & x = \frac{-300}{133} & & \\ & V = \left\{ \frac{-300}{133} \right\} & \\\end{align}\)