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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 4, 15 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{15}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{15x}{ \color{blue}{60} }- \frac{ 16 }{ \color{blue}{60} })& = & (\frac{20}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 15x-16& = & 20x-180 \\\Leftrightarrow & 15x \color{red}{-16} \color{blue}{+16} \color{blue}{-20x} & = & \color{red}{20x} -180 \color{blue}{-20x} \color{blue}{+16} \\\Leftrightarrow & -5x& = & -164 \\\Leftrightarrow & \frac{-5x}{ \color{red}{-5} }& = & \frac{-164}{-5} \\\Leftrightarrow & x = \frac{164}{5} & & \\ & V = \left\{ \frac{164}{5} \right\} & \\\end{align}\)
2. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{4}{3}x+3 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x+\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & 56x+126 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{-56x} & = & \color{red}{56x} +126 \color{blue}{-56x} \color{blue}{-12} \\\Leftrightarrow & -35x& = & 114 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{114}{-35} \\\Leftrightarrow & x = \frac{-114}{35} & & \\ & V = \left\{ \frac{-114}{35} \right\} & \\\end{align}\)
3. \(\text{70 is het kleinste gemene veelvoud van 2, 7 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{7}& = & \frac{-2}{5}x+4 \\\Leftrightarrow & \color{blue}{70.} (\frac{35x}{ \color{blue}{70} }+ \frac{ 50 }{ \color{blue}{70} })& = & (\frac{-28}{ \color{blue}{70} }x+\frac{280}{ \color{blue}{70} }) \color{blue}{.70} \\\Leftrightarrow & 35x+50& = & -28x+280 \\\Leftrightarrow & 35x \color{red}{+50} \color{blue}{-50} \color{blue}{+28x} & = & \color{red}{-28x} +280 \color{blue}{+28x} \color{blue}{-50} \\\Leftrightarrow & 63x& = & 230 \\\Leftrightarrow & \frac{63x}{ \color{red}{63} }& = & \frac{230}{63} \\\Leftrightarrow & x = \frac{230}{63} & & \\ & V = \left\{ \frac{230}{63} \right\} & \\\end{align}\)
4. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{1}{3}x-6 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }+ \frac{ 15 }{ \color{blue}{48} })& = & (\frac{16}{ \color{blue}{48} }x-\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x+15& = & 16x-288 \\\Leftrightarrow & 24x \color{red}{+15} \color{blue}{-15} \color{blue}{-16x} & = & \color{red}{16x} -288 \color{blue}{-16x} \color{blue}{-15} \\\Leftrightarrow & 8x& = & -303 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{-303}{8} \\\Leftrightarrow & x = \frac{-303}{8} & & \\ & V = \left\{ \frac{-303}{8} \right\} & \\\end{align}\)
5. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{1}{5}x+8 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }+ \frac{ 50 }{ \color{blue}{110} })& = & (\frac{22}{ \color{blue}{110} }x+\frac{880}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x+50& = & 22x+880 \\\Leftrightarrow & 55x \color{red}{+50} \color{blue}{-50} \color{blue}{-22x} & = & \color{red}{22x} +880 \color{blue}{-22x} \color{blue}{-50} \\\Leftrightarrow & 33x& = & 830 \\\Leftrightarrow & \frac{33x}{ \color{red}{33} }& = & \frac{830}{33} \\\Leftrightarrow & x = \frac{830}{33} & & \\ & V = \left\{ \frac{830}{33} \right\} & \\\end{align}\)
6. \(\text{240 is het kleinste gemene veelvoud van 5, 16 en 6} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{1}{6}x-1 \\\Leftrightarrow & \color{blue}{240.} (\frac{48x}{ \color{blue}{240} }+ \frac{ 75 }{ \color{blue}{240} })& = & (\frac{40}{ \color{blue}{240} }x-\frac{240}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 48x+75& = & 40x-240 \\\Leftrightarrow & 48x \color{red}{+75} \color{blue}{-75} \color{blue}{-40x} & = & \color{red}{40x} -240 \color{blue}{-40x} \color{blue}{-75} \\\Leftrightarrow & 8x& = & -315 \\\Leftrightarrow & \frac{8x}{ \color{red}{8} }& = & \frac{-315}{8} \\\Leftrightarrow & x = \frac{-315}{8} & & \\ & V = \left\{ \frac{-315}{8} \right\} & \\\end{align}\)
7. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{3}{13}& = & \frac{7}{3}x+2 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 36 }{ \color{blue}{156} })& = & (\frac{364}{ \color{blue}{156} }x+\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-36& = & 364x+312 \\\Leftrightarrow & 39x \color{red}{-36} \color{blue}{+36} \color{blue}{-364x} & = & \color{red}{364x} +312 \color{blue}{-364x} \color{blue}{+36} \\\Leftrightarrow & -325x& = & 348 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{348}{-325} \\\Leftrightarrow & x = \frac{-348}{325} & & \\ & V = \left\{ \frac{-348}{325} \right\} & \\\end{align}\)
8. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}-\frac{5}{11}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }- \frac{ 60 }{ \color{blue}{132} })& = & (\frac{44}{ \color{blue}{132} }x-\frac{396}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x-60& = & 44x-396 \\\Leftrightarrow & 33x \color{red}{-60} \color{blue}{+60} \color{blue}{-44x} & = & \color{red}{44x} -396 \color{blue}{-44x} \color{blue}{+60} \\\Leftrightarrow & -11x& = & -336 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-336}{-11} \\\Leftrightarrow & x = \frac{336}{11} & & \\ & V = \left\{ \frac{336}{11} \right\} & \\\end{align}\)
9. \(\text{90 is het kleinste gemene veelvoud van 5, 9 en 2} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{90.} (\frac{18x}{ \color{blue}{90} }+ \frac{ 40 }{ \color{blue}{90} })& = & (\frac{45}{ \color{blue}{90} }x-\frac{540}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 18x+40& = & 45x-540 \\\Leftrightarrow & 18x \color{red}{+40} \color{blue}{-40} \color{blue}{-45x} & = & \color{red}{45x} -540 \color{blue}{-45x} \color{blue}{-40} \\\Leftrightarrow & -27x& = & -580 \\\Leftrightarrow & \frac{-27x}{ \color{red}{-27} }& = & \frac{-580}{-27} \\\Leftrightarrow & x = \frac{580}{27} & & \\ & V = \left\{ \frac{580}{27} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 5, 15 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{15}& = & \frac{1}{2}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{6x}{ \color{blue}{30} }- \frac{ 8 }{ \color{blue}{30} })& = & (\frac{15}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 6x-8& = & 15x+120 \\\Leftrightarrow & 6x \color{red}{-8} \color{blue}{+8} \color{blue}{-15x} & = & \color{red}{15x} +120 \color{blue}{-15x} \color{blue}{+8} \\\Leftrightarrow & -9x& = & 128 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{128}{-9} \\\Leftrightarrow & x = \frac{-128}{9} & & \\ & V = \left\{ \frac{-128}{9} \right\} & \\\end{align}\)
11. \(\text{110 is het kleinste gemene veelvoud van 5, 11 en 2} \\ \begin{align} & \frac{x}{5}-\frac{4}{11}& = & \frac{1}{2}x+1 \\\Leftrightarrow & \color{blue}{110.} (\frac{22x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{55}{ \color{blue}{110} }x+\frac{110}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 22x-40& = & 55x+110 \\\Leftrightarrow & 22x \color{red}{-40} \color{blue}{+40} \color{blue}{-55x} & = & \color{red}{55x} +110 \color{blue}{-55x} \color{blue}{+40} \\\Leftrightarrow & -33x& = & 150 \\\Leftrightarrow & \frac{-33x}{ \color{red}{-33} }& = & \frac{150}{-33} \\\Leftrightarrow & x = \frac{-50}{11} & & \\ & V = \left\{ \frac{-50}{11} \right\} & \\\end{align}\)
12. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}-\frac{5}{9}& = & \frac{3}{2}x+8 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }- \frac{ 70 }{ \color{blue}{126} })& = & (\frac{189}{ \color{blue}{126} }x+\frac{1008}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x-70& = & 189x+1008 \\\Leftrightarrow & 18x \color{red}{-70} \color{blue}{+70} \color{blue}{-189x} & = & \color{red}{189x} +1008 \color{blue}{-189x} \color{blue}{+70} \\\Leftrightarrow & -171x& = & 1078 \\\Leftrightarrow & \frac{-171x}{ \color{red}{-171} }& = & \frac{1078}{-171} \\\Leftrightarrow & x = \frac{-1078}{171} & & \\ & V = \left\{ \frac{-1078}{171} \right\} & \\\end{align}\)