Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{84 is het kleinste gemene veelvoud van 3, 14 en 4} \\ \begin{align} & \frac{x}{3}+\frac{5}{14}& = & \frac{1}{4}x+1 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 30 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{84}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+30& = & 21x+84 \\\Leftrightarrow & 28x \color{red}{+30} \color{blue}{-30} \color{blue}{-21x} & = & \color{red}{21x} +84 \color{blue}{-21x} \color{blue}{-30} \\\Leftrightarrow & 7x& = & 54 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{54}{7} \\\Leftrightarrow & x = \frac{54}{7} & & \\ & V = \left\{ \frac{54}{7} \right\} & \\\end{align}\)
2. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{1}{2}x-1 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x-\frac{110}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-40& = & 55x-110 \\\Leftrightarrow & 22x \color{red}{-40} \color{blue}{+40} \color{blue}{-55x} & = & \color{red}{55x} -110 \color{blue}{-55x} \color{blue}{+40} \\\Leftrightarrow & -33x& = & -70 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{-70}{-33} \\\Leftrightarrow & x = \frac{70}{33} & & \\ & V = \left\{ \frac{70}{33} \right\} & \\\end{align}\)
3. \(\text{36 is het kleinste gemene veelvoud van 4, 9 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{9}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{36.} (\frac{9x}{ \color{blue}{36} }- \frac{ 20 }{ \color{blue}{36} })& = & (\frac{12}{ \color{blue}{36} }x-\frac{216}{ \color{blue}{36} }) \color{blue}{.36} \\\Leftrightarrow & 9x-20& = & 12x-216 \\\Leftrightarrow & 9x \color{red}{-20} \color{blue}{+20} \color{blue}{-12x} & = & \color{red}{12x} -216 \color{blue}{-12x} \color{blue}{+20} \\\Leftrightarrow & -3x& = & -196 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{-196}{-3} \\\Leftrightarrow & x = \frac{196}{3} & & \\ & V = \left\{ \frac{196}{3} \right\} & \\\end{align}\)
4. \(\text{364 is het kleinste gemene veelvoud van 7, 13 en 4} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{-7}{4}x+4 \\\Leftrightarrow & \color{blue}{364.} (\frac{52x}{ \color{blue}{364} }- \frac{ 112 }{ \color{blue}{364} })& = & (\frac{-637}{ \color{blue}{364} }x+\frac{1456}{ \color{blue}{364} }) \color{blue}{.364} \\\Leftrightarrow & 52x-112& = & -637x+1456 \\\Leftrightarrow & 52x \color{red}{-112} \color{blue}{+112} \color{blue}{+637x} & = & \color{red}{-637x} +1456 \color{blue}{+637x} \color{blue}{+112} \\\Leftrightarrow & 689x& = & 1568 \\\Leftrightarrow & \frac{689x}{ \color{red}{689} }& = & \frac{1568}{689} \\\Leftrightarrow & x = \frac{1568}{689} & & \\ & V = \left\{ \frac{1568}{689} \right\} & \\\end{align}\)
5. \(\text{10 is het kleinste gemene veelvoud van 5, 10 en 2} \\ \begin{align} & \frac{x}{5}-\frac{3}{10}& = & \frac{3}{2}x+5 \\\Leftrightarrow & \color{blue}{10.} (\frac{2x}{ \color{blue}{10} }- \frac{ 3 }{ \color{blue}{10} })& = & (\frac{15}{ \color{blue}{10} }x+\frac{50}{ \color{blue}{10} }) \color{blue}{.10} \\\Leftrightarrow & 2x-3& = & 15x+50 \\\Leftrightarrow & 2x \color{red}{-3} \color{blue}{+3} \color{blue}{-15x} & = & \color{red}{15x} +50 \color{blue}{-15x} \color{blue}{+3} \\\Leftrightarrow & -13x& = & 53 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{53}{-13} \\\Leftrightarrow & x = \frac{-53}{13} & & \\ & V = \left\{ \frac{-53}{13} \right\} & \\\end{align}\)
6. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{7}& = & \frac{-5}{3}x-8 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 24 }{ \color{blue}{84} })& = & (\frac{-140}{ \color{blue}{84} }x-\frac{672}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+24& = & -140x-672 \\\Leftrightarrow & 21x \color{red}{+24} \color{blue}{-24} \color{blue}{+140x} & = & \color{red}{-140x} -672 \color{blue}{+140x} \color{blue}{-24} \\\Leftrightarrow & 161x& = & -696 \\\Leftrightarrow & \frac{161x}{ \color{red}{161} }& = & \frac{-696}{161} \\\Leftrightarrow & x = \frac{-696}{161} & & \\ & V = \left\{ \frac{-696}{161} \right\} & \\\end{align}\)
7. \(\text{330 is het kleinste gemene veelvoud van 5, 11 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{11}& = & \frac{1}{6}x-1 \\\Leftrightarrow & \color{blue}{330.} (\frac{66x}{ \color{blue}{330} }+ \frac{ 150 }{ \color{blue}{330} })& = & (\frac{55}{ \color{blue}{330} }x-\frac{330}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 66x+150& = & 55x-330 \\\Leftrightarrow & 66x \color{red}{+150} \color{blue}{-150} \color{blue}{-55x} & = & \color{red}{55x} -330 \color{blue}{-55x} \color{blue}{-150} \\\Leftrightarrow & 11x& = & -480 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{-480}{11} \\\Leftrightarrow & x = \frac{-480}{11} & & \\ & V = \left\{ \frac{-480}{11} \right\} & \\\end{align}\)
8. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{1}{5}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{6}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & 6x-120 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-6x} & = & \color{red}{6x} -120 \color{blue}{-6x} \color{blue}{+4} \\\Leftrightarrow & -x& = & -116 \\\Leftrightarrow & \frac{-x}{ \color{red}{-1} }& = & \frac{-116}{-1} \\\Leftrightarrow & x = 116 & & \\ & V = \left\{ 116 \right\} & \\\end{align}\)
9. \(\text{70 is het kleinste gemene veelvoud van 7, 14 en 5} \\ \begin{align} & \frac{x}{7}-\frac{3}{14}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{70.} (\frac{10x}{ \color{blue}{70} }- \frac{ 15 }{ \color{blue}{70} })& = & (\frac{14}{ \color{blue}{70} }x+\frac{70}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 10x-15& = & 14x+70 \\\Leftrightarrow & 10x \color{red}{-15} \color{blue}{+15} \color{blue}{-14x} & = & \color{red}{14x} +70 \color{blue}{-14x} \color{blue}{+15} \\\Leftrightarrow & -4x& = & 85 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{85}{-4} \\\Leftrightarrow & x = \frac{-85}{4} & & \\ & V = \left\{ \frac{-85}{4} \right\} & \\\end{align}\)
10. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{4}{13}& = & \frac{1}{2}x-5 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{390}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-24& = & 39x-390 \\\Leftrightarrow & 26x \color{red}{-24} \color{blue}{+24} \color{blue}{-39x} & = & \color{red}{39x} -390 \color{blue}{-39x} \color{blue}{+24} \\\Leftrightarrow & -13x& = & -366 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-366}{-13} \\\Leftrightarrow & x = \frac{366}{13} & & \\ & V = \left\{ \frac{366}{13} \right\} & \\\end{align}\)
11. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{9}& = & \frac{1}{2}x-8 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x-\frac{720}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x-50& = & 45x-720 \\\Leftrightarrow & 18x \color{red}{-50} \color{blue}{+50} \color{blue}{-45x} & = & \color{red}{45x} -720 \color{blue}{-45x} \color{blue}{+50} \\\Leftrightarrow & -27x& = & -670 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-670}{-27} \\\Leftrightarrow & x = \frac{670}{27} & & \\ & V = \left\{ \frac{670}{27} \right\} & \\\end{align}\)
12. \(\text{455 is het kleinste gemene veelvoud van 7, 13 en 5} \\ \begin{align} & \frac{x}{7}-\frac{2}{13}& = & \frac{6}{5}x-8 \\\Leftrightarrow & \color{blue}{455.} (\frac{65x}{ \color{blue}{455} }- \frac{ 70 }{ \color{blue}{455} })& = & (\frac{546}{ \color{blue}{455} }x-\frac{3640}{ \color{blue}{455} }) \color{blue}{.455} \\\Leftrightarrow & 65x-70& = & 546x-3640 \\\Leftrightarrow & 65x \color{red}{-70} \color{blue}{+70} \color{blue}{-546x} & = & \color{red}{546x} -3640 \color{blue}{-546x} \color{blue}{+70} \\\Leftrightarrow & -481x& = & -3570 \\\Leftrightarrow & \frac{-481x}{ \color{red}{-481} }& = & \frac{-3570}{-481} \\\Leftrightarrow & x = \frac{3570}{481} & & \\ & V = \left\{ \frac{3570}{481} \right\} & \\\end{align}\)