Vgln. eerste graad (reeks 4)

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

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

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

Verbetersleutel

1. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{16}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 9 }{ \color{blue}{48} })& = & (\frac{24}{ \color{blue}{48} }x-\frac{48}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+9& = & 24x-48 \\\Leftrightarrow & 16x \color{red}{+9} \color{blue}{-9} \color{blue}{-24x} & = & \color{red}{24x} -48 \color{blue}{-24x} \color{blue}{-9} \\\Leftrightarrow & -8x& = & -57 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{-57}{-8} \\\Leftrightarrow & x = \frac{57}{8} & & \\ & V = \left\{ \frac{57}{8} \right\} & \\\end{align}\)
2. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{11}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 50 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{440}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-50& = & 55x-440 \\\Leftrightarrow & 22x \color{red}{-50} \color{blue}{+50} \color{blue}{-55x} & = & \color{red}{55x} -440 \color{blue}{-55x} \color{blue}{+50} \\\Leftrightarrow & -33x& = & -390 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-390}{-33} \\\Leftrightarrow & x = \frac{130}{11} & & \\ & V = \left\{ \frac{130}{11} \right\} & \\\end{align}\)
3. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{16}& = & \frac{3}{4}x+7 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{36}{ \color{blue}{48} }x+\frac{336}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x+15& = & 36x+336 \\\Leftrightarrow & 16x \color{red}{+15} \color{blue}{-15} \color{blue}{-36x} & = & \color{red}{36x} +336 \color{blue}{-36x} \color{blue}{-15} \\\Leftrightarrow & -20x& = & 321 \\\Leftrightarrow & \frac{-20x}{ \color{red}{-20} }& = & \frac{321}{-20} \\\Leftrightarrow & x = \frac{-321}{20} & & \\ & V = \left\{ \frac{-321}{20} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{7}& = & \frac{-2}{5}x+6 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }+ \frac{ 60 }{ \color{blue}{210} })& = & (\frac{-84}{ \color{blue}{210} }x+\frac{1260}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x+60& = & -84x+1260 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+84x} & = & \color{red}{-84x} +1260 \color{blue}{+84x} \color{blue}{-60} \\\Leftrightarrow & 119x& = & 1200 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{1200}{119} \\\Leftrightarrow & x = \frac{1200}{119} & & \\ & V = \left\{ \frac{1200}{119} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 3, 10 en 5} \\ \begin{align} & \frac{x}{3}-\frac{3}{10}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{10x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 10x-9& = & 6x+30 \\\Leftrightarrow & 10x \color{red}{-9} \color{blue}{+9} \color{blue}{-6x} & = & \color{red}{6x} +30 \color{blue}{-6x} \color{blue}{+9} \\\Leftrightarrow & 4x& = & 39 \\\Leftrightarrow & \frac{4x}{ \color{red}{4} }& = & \frac{39}{4} \\\Leftrightarrow & x = \frac{39}{4} & & \\ & V = \left\{ \frac{39}{4} \right\} & \\\end{align}\)
6. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{6}{5}x+7 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 20 }{ \color{blue}{70} })& = & (\frac{84}{ \color{blue}{70} }x+\frac{490}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+20& = & 84x+490 \\\Leftrightarrow & 35x \color{red}{+20} \color{blue}{-20} \color{blue}{-84x} & = & \color{red}{84x} +490 \color{blue}{-84x} \color{blue}{-20} \\\Leftrightarrow & -49x& = & 470 \\\Leftrightarrow & \frac{-49x}{ \color{red}{-49} }& = & \frac{470}{-49} \\\Leftrightarrow & x = \frac{-470}{49} & & \\ & V = \left\{ \frac{-470}{49} \right\} & \\\end{align}\)
7. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{9}& = & \frac{-4}{5}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-50& = & -72x-720 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{+72x} & = & \color{red}{-72x} -720 \color{blue}{+72x} \color{blue}{+50} \\\Leftrightarrow & 87x& = & -670 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{-670}{87} \\\Leftrightarrow & x = \frac{-670}{87} & & \\ & V = \left\{ \frac{-670}{87} \right\} & \\\end{align}\)
8. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{15}& = & \frac{7}{3}x-4 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }+ \frac{ 8 }{ \color{blue}{60} })& = & (\frac{140}{ \color{blue}{60} }x-\frac{240}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x+8& = & 140x-240 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{-140x} & = & \color{red}{140x} -240 \color{blue}{-140x} \color{blue}{-8} \\\Leftrightarrow & -125x& = & -248 \\\Leftrightarrow & \frac{-125x}{ \color{red}{-125} }& = & \frac{-248}{-125} \\\Leftrightarrow & x = \frac{248}{125} & & \\ & V = \left\{ \frac{248}{125} \right\} & \\\end{align}\)
9. \(\text{165 is het kleinste gemene veelvoud van 5, 11 en 3} \\ \begin{align} & \frac{x}{5}+\frac{3}{11}& = & \frac{1}{3}x+7 \\\Leftrightarrow & \color{blue}{165.} (\frac{33x}{ \color{blue}{165} }+ \frac{ 45 }{ \color{blue}{165} })& = & (\frac{55}{ \color{blue}{165} }x+\frac{1155}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 33x+45& = & 55x+1155 \\\Leftrightarrow & 33x \color{red}{+45} \color{blue}{-45} \color{blue}{-55x} & = & \color{red}{55x} +1155 \color{blue}{-55x} \color{blue}{-45} \\\Leftrightarrow & -22x& = & 1110 \\\Leftrightarrow & \frac{-22x}{ \color{red}{-22} }& = & \frac{1110}{-22} \\\Leftrightarrow & x = \frac{-555}{11} & & \\ & V = \left\{ \frac{-555}{11} \right\} & \\\end{align}\)
10. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-8}{3}x-1 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{-224}{ \color{blue}{84} }x-\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & -224x-84 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{+224x} & = & \color{red}{-224x} -84 \color{blue}{+224x} \color{blue}{-48} \\\Leftrightarrow & 245x& = & -132 \\\Leftrightarrow & \frac{245x}{ \color{red}{245} }& = & \frac{-132}{245} \\\Leftrightarrow & x = \frac{-132}{245} & & \\ & V = \left\{ \frac{-132}{245} \right\} & \\\end{align}\)
11. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{-7}{5}x-1 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }- \frac{ 30 }{ \color{blue}{105} })& = & (\frac{-147}{ \color{blue}{105} }x-\frac{105}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x-30& = & -147x-105 \\\Leftrightarrow & 35x \color{red}{-30} \color{blue}{+30} \color{blue}{+147x} & = & \color{red}{-147x} -105 \color{blue}{+147x} \color{blue}{+30} \\\Leftrightarrow & 182x& = & -75 \\\Leftrightarrow & \frac{182x}{ \color{red}{182} }& = & \frac{-75}{182} \\\Leftrightarrow & x = \frac{-75}{182} & & \\ & V = \left\{ \frac{-75}{182} \right\} & \\\end{align}\)
12. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{3}{4}x-1 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 48 }{ \color{blue}{156} })& = & (\frac{117}{ \color{blue}{156} }x-\frac{156}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-48& = & 117x-156 \\\Leftrightarrow & 52x \color{red}{-48} \color{blue}{+48} \color{blue}{-117x} & = & \color{red}{117x} -156 \color{blue}{-117x} \color{blue}{+48} \\\Leftrightarrow & -65x& = & -108 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{-108}{-65} \\\Leftrightarrow & x = \frac{108}{65} & & \\ & V = \left\{ \frac{108}{65} \right\} & \\\end{align}\)