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

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

Verbetersleutel

1. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{6}{5}x-1 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 80 }{ \color{blue}{140} })& = & (\frac{168}{ \color{blue}{140} }x-\frac{140}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+80& = & 168x-140 \\\Leftrightarrow & 35x \color{red}{+80} \color{blue}{-80} \color{blue}{-168x} & = & \color{red}{168x} -140 \color{blue}{-168x} \color{blue}{-80} \\\Leftrightarrow & -133x& = & -220 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-220}{-133} \\\Leftrightarrow & x = \frac{220}{133} & & \\ & V = \left\{ \frac{220}{133} \right\} & \\\end{align}\)
2. \(\text{240 is het kleinste gemene veelvoud van 6, 16 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{16}& = & \frac{1}{5}x+4 \\\Leftrightarrow & \color{blue}{240.} (\frac{40x}{ \color{blue}{240} }- \frac{ 75 }{ \color{blue}{240} })& = & (\frac{48}{ \color{blue}{240} }x+\frac{960}{ \color{blue}{240} }) \color{blue}{.240} \\\Leftrightarrow & 40x-75& = & 48x+960 \\\Leftrightarrow & 40x \color{red}{-75} \color{blue}{+75} \color{blue}{-48x} & = & \color{red}{48x} +960 \color{blue}{-48x} \color{blue}{+75} \\\Leftrightarrow & -8x& = & 1035 \\\Leftrightarrow & \frac{-8x}{ \color{red}{-8} }& = & \frac{1035}{-8} \\\Leftrightarrow & x = \frac{-1035}{8} & & \\ & V = \left\{ \frac{-1035}{8} \right\} & \\\end{align}\)
3. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{7}& = & \frac{1}{3}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 36 }{ \color{blue}{84} })& = & (\frac{28}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+36& = & 28x-252 \\\Leftrightarrow & 21x \color{red}{+36} \color{blue}{-36} \color{blue}{-28x} & = & \color{red}{28x} -252 \color{blue}{-28x} \color{blue}{-36} \\\Leftrightarrow & -7x& = & -288 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-288}{-7} \\\Leftrightarrow & x = \frac{288}{7} & & \\ & V = \left\{ \frac{288}{7} \right\} & \\\end{align}\)
4. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{-4}{5}x+2 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-24}{ \color{blue}{30} }x+\frac{60}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & -24x+60 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{+24x} & = & \color{red}{-24x} +60 \color{blue}{+24x} \color{blue}{+9} \\\Leftrightarrow & 29x& = & 69 \\\Leftrightarrow & \frac{29x}{ \color{red}{29} }& = & \frac{69}{29} \\\Leftrightarrow & x = \frac{69}{29} & & \\ & V = \left\{ \frac{69}{29} \right\} & \\\end{align}\)
5. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}+\frac{2}{7}& = & \frac{4}{3}x+6 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{56}{ \color{blue}{42} }x+\frac{252}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x+12& = & 56x+252 \\\Leftrightarrow & 21x \color{red}{+12} \color{blue}{-12} \color{blue}{-56x} & = & \color{red}{56x} +252 \color{blue}{-56x} \color{blue}{-12} \\\Leftrightarrow & -35x& = & 240 \\\Leftrightarrow & \frac{-35x}{ \color{red}{-35} }& = & \frac{240}{-35} \\\Leftrightarrow & x = \frac{-48}{7} & & \\ & V = \left\{ \frac{-48}{7} \right\} & \\\end{align}\)
6. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{-7}{5}x-4 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{-546}{ \color{blue}{390} }x-\frac{1560}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & -546x-1560 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{+546x} & = & \color{red}{-546x} -1560 \color{blue}{+546x} \color{blue}{-90} \\\Leftrightarrow & 611x& = & -1650 \\\Leftrightarrow & \frac{611x}{ \color{red}{611} }& = & \frac{-1650}{611} \\\Leftrightarrow & x = \frac{-1650}{611} & & \\ & V = \left\{ \frac{-1650}{611} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{6}{5}x+1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 252x+210 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-252x} & = & \color{red}{252x} +210 \color{blue}{-252x} \color{blue}{+120} \\\Leftrightarrow & -217x& = & 330 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{330}{-217} \\\Leftrightarrow & x = \frac{-330}{217} & & \\ & V = \left\{ \frac{-330}{217} \right\} & \\\end{align}\)
8. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}+\frac{2}{13}& = & \frac{1}{4}x-8 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }+ \frac{ 24 }{ \color{blue}{156} })& = & (\frac{39}{ \color{blue}{156} }x-\frac{1248}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x+24& = & 39x-1248 \\\Leftrightarrow & 52x \color{red}{+24} \color{blue}{-24} \color{blue}{-39x} & = & \color{red}{39x} -1248 \color{blue}{-39x} \color{blue}{-24} \\\Leftrightarrow & 13x& = & -1272 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{-1272}{13} \\\Leftrightarrow & x = \frac{-1272}{13} & & \\ & V = \left\{ \frac{-1272}{13} \right\} & \\\end{align}\)
9. \(\text{180 is het kleinste gemene veelvoud van 4, 9 en 5} \\ \begin{align} & \frac{x}{4}-\frac{2}{9}& = & \frac{-4}{5}x-6 \\\Leftrightarrow & \color{blue}{180.} (\frac{45x}{ \color{blue}{180} }- \frac{ 40 }{ \color{blue}{180} })& = & (\frac{-144}{ \color{blue}{180} }x-\frac{1080}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 45x-40& = & -144x-1080 \\\Leftrightarrow & 45x \color{red}{-40} \color{blue}{+40} \color{blue}{+144x} & = & \color{red}{-144x} -1080 \color{blue}{+144x} \color{blue}{+40} \\\Leftrightarrow & 189x& = & -1040 \\\Leftrightarrow & \frac{189x}{ \color{red}{189} }& = & \frac{-1040}{189} \\\Leftrightarrow & x = \frac{-1040}{189} & & \\ & V = \left\{ \frac{-1040}{189} \right\} & \\\end{align}\)
10. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{11}& = & \frac{6}{5}x+3 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 60 }{ \color{blue}{330} })& = & (\frac{396}{ \color{blue}{330} }x+\frac{990}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-60& = & 396x+990 \\\Leftrightarrow & 55x \color{red}{-60} \color{blue}{+60} \color{blue}{-396x} & = & \color{red}{396x} +990 \color{blue}{-396x} \color{blue}{+60} \\\Leftrightarrow & -341x& = & 1050 \\\Leftrightarrow & \frac{-341x}{ \color{red}{-341} }& = & \frac{1050}{-341} \\\Leftrightarrow & x = \frac{-1050}{341} & & \\ & V = \left\{ \frac{-1050}{341} \right\} & \\\end{align}\)
11. \(\text{105 is het kleinste gemene veelvoud van 3, 7 en 5} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{-4}{5}x-5 \\\Leftrightarrow & \color{blue}{105.} (\frac{35x}{ \color{blue}{105} }+ \frac{ 60 }{ \color{blue}{105} })& = & (\frac{-84}{ \color{blue}{105} }x-\frac{525}{ \color{blue}{105} }) \color{blue}{.105} \\\Leftrightarrow & 35x+60& = & -84x-525 \\\Leftrightarrow & 35x \color{red}{+60} \color{blue}{-60} \color{blue}{+84x} & = & \color{red}{-84x} -525 \color{blue}{+84x} \color{blue}{-60} \\\Leftrightarrow & 119x& = & -585 \\\Leftrightarrow & \frac{119x}{ \color{red}{119} }& = & \frac{-585}{119} \\\Leftrightarrow & x = \frac{-585}{119} & & \\ & V = \left\{ \frac{-585}{119} \right\} & \\\end{align}\)
12. \(\text{252 is het kleinste gemene veelvoud van 7, 9 en 4} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{7}{4}x-2 \\\Leftrightarrow & \color{blue}{252.} (\frac{36x}{ \color{blue}{252} }+ \frac{ 112 }{ \color{blue}{252} })& = & (\frac{441}{ \color{blue}{252} }x-\frac{504}{ \color{blue}{252} }) \color{blue}{.252} \\\Leftrightarrow & 36x+112& = & 441x-504 \\\Leftrightarrow & 36x \color{red}{+112} \color{blue}{-112} \color{blue}{-441x} & = & \color{red}{441x} -504 \color{blue}{-441x} \color{blue}{-112} \\\Leftrightarrow & -405x& = & -616 \\\Leftrightarrow & \frac{-405x}{ \color{red}{-405} }& = & \frac{-616}{-405} \\\Leftrightarrow & x = \frac{616}{405} & & \\ & V = \left\{ \frac{616}{405} \right\} & \\\end{align}\)