Vgln. eerste graad (reeks 4)

Hoofdmenu Eentje per keer  🖨

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

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

---

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

Verbetersleutel

1. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{9}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 50 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x+\frac{180}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-50& = & -72x+180 \\\Leftrightarrow & 15x \color{red}{-50} \color{blue}{+50} \color{blue}{+72x} & = & \color{red}{-72x} +180 \color{blue}{+72x} \color{blue}{+50} \\\Leftrightarrow & 87x& = & 230 \\\Leftrightarrow & \frac{87x}{ \color{red}{87} }& = & \frac{230}{87} \\\Leftrightarrow & x = \frac{230}{87} & & \\ & V = \left\{ \frac{230}{87} \right\} & \\\end{align}\)
2. \(\text{48 is het kleinste gemene veelvoud van 3, 16 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{16}& = & \frac{-7}{4}x+6 \\\Leftrightarrow & \color{blue}{48.} (\frac{16x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-84}{ \color{blue}{48} }x+\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 16x-15& = & -84x+288 \\\Leftrightarrow & 16x \color{red}{-15} \color{blue}{+15} \color{blue}{+84x} & = & \color{red}{-84x} +288 \color{blue}{+84x} \color{blue}{+15} \\\Leftrightarrow & 100x& = & 303 \\\Leftrightarrow & \frac{100x}{ \color{red}{100} }& = & \frac{303}{100} \\\Leftrightarrow & x = \frac{303}{100} & & \\ & V = \left\{ \frac{303}{100} \right\} & \\\end{align}\)
3. \(\text{140 is het kleinste gemene veelvoud van 7, 10 en 4} \\ \begin{align} & \frac{x}{7}-\frac{3}{10}& = & \frac{-3}{4}x+8 \\\Leftrightarrow & \color{blue}{140.} (\frac{20x}{ \color{blue}{140} }- \frac{ 42 }{ \color{blue}{140} })& = & (\frac{-105}{ \color{blue}{140} }x+\frac{1120}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 20x-42& = & -105x+1120 \\\Leftrightarrow & 20x \color{red}{-42} \color{blue}{+42} \color{blue}{+105x} & = & \color{red}{-105x} +1120 \color{blue}{+105x} \color{blue}{+42} \\\Leftrightarrow & 125x& = & 1162 \\\Leftrightarrow & \frac{125x}{ \color{red}{125} }& = & \frac{1162}{125} \\\Leftrightarrow & x = \frac{1162}{125} & & \\ & V = \left\{ \frac{1162}{125} \right\} & \\\end{align}\)
4. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{-294}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & -294x+210 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{+294x} & = & \color{red}{-294x} +210 \color{blue}{+294x} \color{blue}{+120} \\\Leftrightarrow & 329x& = & 330 \\\Leftrightarrow & \frac{329x}{ \color{red}{329} }& = & \frac{330}{329} \\\Leftrightarrow & x = \frac{330}{329} & & \\ & V = \left\{ \frac{330}{329} \right\} & \\\end{align}\)
5. \(\text{18 is het kleinste gemene veelvoud van 2, 9 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{9}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{18.} (\frac{9x}{ \color{blue}{18} }- \frac{ 4 }{ \color{blue}{18} })& = & (\frac{6}{ \color{blue}{18} }x+\frac{36}{ \color{blue}{18} }) \color{blue}{.18} \\\Leftrightarrow & 9x-4& = & 6x+36 \\\Leftrightarrow & 9x \color{red}{-4} \color{blue}{+4} \color{blue}{-6x} & = & \color{red}{6x} +36 \color{blue}{-6x} \color{blue}{+4} \\\Leftrightarrow & 3x& = & 40 \\\Leftrightarrow & \frac{3x}{ \color{red}{3} }& = & \frac{40}{3} \\\Leftrightarrow & x = \frac{40}{3} & & \\ & V = \left\{ \frac{40}{3} \right\} & \\\end{align}\)
6. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}+\frac{5}{16}& = & \frac{1}{2}x-6 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x-\frac{480}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x+25& = & 40x-480 \\\Leftrightarrow & 16x \color{red}{+25} \color{blue}{-25} \color{blue}{-40x} & = & \color{red}{40x} -480 \color{blue}{-40x} \color{blue}{-25} \\\Leftrightarrow & -24x& = & -505 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{-505}{-24} \\\Leftrightarrow & x = \frac{505}{24} & & \\ & V = \left\{ \frac{505}{24} \right\} & \\\end{align}\)
7. \(\text{6 is het kleinste gemene veelvoud van 2, 6 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{6}& = & \frac{1}{3}x+2 \\\Leftrightarrow & \color{blue}{6.} (\frac{3x}{ \color{blue}{6} }- \frac{ 5 }{ \color{blue}{6} })& = & (\frac{2}{ \color{blue}{6} }x+\frac{12}{ \color{blue}{6} }) \color{blue}{.6} \\\Leftrightarrow & 3x-5& = & 2x+12 \\\Leftrightarrow & 3x \color{red}{-5} \color{blue}{+5} \color{blue}{-2x} & = & \color{red}{2x} +12 \color{blue}{-2x} \color{blue}{+5} \\\Leftrightarrow & x& = & 17 \\ & V = \left\{ 17 \right\} & \\\end{align}\)
8. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{13}& = & \frac{4}{3}x+1 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }- \frac{ 84 }{ \color{blue}{273} })& = & (\frac{364}{ \color{blue}{273} }x+\frac{273}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x-84& = & 364x+273 \\\Leftrightarrow & 39x \color{red}{-84} \color{blue}{+84} \color{blue}{-364x} & = & \color{red}{364x} +273 \color{blue}{-364x} \color{blue}{+84} \\\Leftrightarrow & -325x& = & 357 \\\Leftrightarrow & \frac{-325x}{ \color{red}{-325} }& = & \frac{357}{-325} \\\Leftrightarrow & x = \frac{-357}{325} & & \\ & V = \left\{ \frac{-357}{325} \right\} & \\\end{align}\)
9. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{-4}{5}x-2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x-\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & -112x-280 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{+112x} & = & \color{red}{-112x} -280 \color{blue}{+112x} \color{blue}{-80} \\\Leftrightarrow & 147x& = & -360 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{-360}{147} \\\Leftrightarrow & x = \frac{-120}{49} & & \\ & V = \left\{ \frac{-120}{49} \right\} & \\\end{align}\)
10. \(\text{165 is het kleinste gemene veelvoud van 3, 11 en 5} \\ \begin{align} & \frac{x}{3}-\frac{2}{11}& = & \frac{-2}{5}x+8 \\\Leftrightarrow & \color{blue}{165.} (\frac{55x}{ \color{blue}{165} }- \frac{ 30 }{ \color{blue}{165} })& = & (\frac{-66}{ \color{blue}{165} }x+\frac{1320}{ \color{blue}{165} }) \color{blue}{.165} \\\Leftrightarrow & 55x-30& = & -66x+1320 \\\Leftrightarrow & 55x \color{red}{-30} \color{blue}{+30} \color{blue}{+66x} & = & \color{red}{-66x} +1320 \color{blue}{+66x} \color{blue}{+30} \\\Leftrightarrow & 121x& = & 1350 \\\Leftrightarrow & \frac{121x}{ \color{red}{121} }& = & \frac{1350}{121} \\\Leftrightarrow & x = \frac{1350}{121} & & \\ & V = \left\{ \frac{1350}{121} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 3, 14 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{14}& = & \frac{5}{2}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 9 }{ \color{blue}{42} })& = & (\frac{105}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+9& = & 105x-126 \\\Leftrightarrow & 14x \color{red}{+9} \color{blue}{-9} \color{blue}{-105x} & = & \color{red}{105x} -126 \color{blue}{-105x} \color{blue}{-9} \\\Leftrightarrow & -91x& = & -135 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-135}{-91} \\\Leftrightarrow & x = \frac{135}{91} & & \\ & V = \left\{ \frac{135}{91} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 2, 16 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{16}& = & \frac{1}{5}x+1 \\\Leftrightarrow & \color{blue}{80.} (\frac{40x}{ \color{blue}{80} }+ \frac{ 25 }{ \color{blue}{80} })& = & (\frac{16}{ \color{blue}{80} }x+\frac{80}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 40x+25& = & 16x+80 \\\Leftrightarrow & 40x \color{red}{+25} \color{blue}{-25} \color{blue}{-16x} & = & \color{red}{16x} +80 \color{blue}{-16x} \color{blue}{-25} \\\Leftrightarrow & 24x& = & 55 \\\Leftrightarrow & \frac{24x}{ \color{red}{24} }& = & \frac{55}{24} \\\Leftrightarrow & x = \frac{55}{24} & & \\ & V = \left\{ \frac{55}{24} \right\} & \\\end{align}\)