Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{60 is het kleinste gemene veelvoud van 3, 12 en 5} \\ \begin{align} & \frac{x}{3}+\frac{5}{12}& = & \frac{4}{5}x+7 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 25 }{ \color{blue}{60} })& = & (\frac{48}{ \color{blue}{60} }x+\frac{420}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+25& = & 48x+420 \\\Leftrightarrow & 20x \color{red}{+25} \color{blue}{-25} \color{blue}{-48x} & = & \color{red}{48x} +420 \color{blue}{-48x} \color{blue}{-25} \\\Leftrightarrow & -28x& = & 395 \\\Leftrightarrow & \frac{-28x}{ \color{red}{-28} }& = & \frac{395}{-28} \\\Leftrightarrow & x = \frac{-395}{28} & & \\ & V = \left\{ \frac{-395}{28} \right\} & \\\end{align}\)
2. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{7}& = & \frac{1}{4}x-3 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 24 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x-\frac{252}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-24& = & 21x-252 \\\Leftrightarrow & 28x \color{red}{-24} \color{blue}{+24} \color{blue}{-21x} & = & \color{red}{21x} -252 \color{blue}{-21x} \color{blue}{+24} \\\Leftrightarrow & 7x& = & -228 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{-228}{7} \\\Leftrightarrow & x = \frac{-228}{7} & & \\ & V = \left\{ \frac{-228}{7} \right\} & \\\end{align}\)
3. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{11}& = & \frac{-8}{3}x-6 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }- \frac{ 12 }{ \color{blue}{66} })& = & (\frac{-176}{ \color{blue}{66} }x-\frac{396}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x-12& = & -176x-396 \\\Leftrightarrow & 33x \color{red}{-12} \color{blue}{+12} \color{blue}{+176x} & = & \color{red}{-176x} -396 \color{blue}{+176x} \color{blue}{+12} \\\Leftrightarrow & 209x& = & -384 \\\Leftrightarrow & \frac{209x}{ \color{red}{209} }& = & \frac{-384}{209} \\\Leftrightarrow & x = \frac{-384}{209} & & \\ & V = \left\{ \frac{-384}{209} \right\} & \\\end{align}\)
4. \(\text{273 is het kleinste gemene veelvoud van 7, 13 en 3} \\ \begin{align} & \frac{x}{7}+\frac{2}{13}& = & \frac{7}{3}x+5 \\\Leftrightarrow & \color{blue}{273.} (\frac{39x}{ \color{blue}{273} }+ \frac{ 42 }{ \color{blue}{273} })& = & (\frac{637}{ \color{blue}{273} }x+\frac{1365}{ \color{blue}{273} }) \color{blue}{.273} \\\Leftrightarrow & 39x+42& = & 637x+1365 \\\Leftrightarrow & 39x \color{red}{+42} \color{blue}{-42} \color{blue}{-637x} & = & \color{red}{637x} +1365 \color{blue}{-637x} \color{blue}{-42} \\\Leftrightarrow & -598x& = & 1323 \\\Leftrightarrow & \frac{-598x}{ \color{red}{-598} }& = & \frac{1323}{-598} \\\Leftrightarrow & x = \frac{-1323}{598} & & \\ & V = \left\{ \frac{-1323}{598} \right\} & \\\end{align}\)
5. \(\text{60 is het kleinste gemene veelvoud van 3, 15 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{15}& = & \frac{5}{4}x-3 \\\Leftrightarrow & \color{blue}{60.} (\frac{20x}{ \color{blue}{60} }+ \frac{ 16 }{ \color{blue}{60} })& = & (\frac{75}{ \color{blue}{60} }x-\frac{180}{ \color{blue}{60} }) \color{blue}{.60} \\\Leftrightarrow & 20x+16& = & 75x-180 \\\Leftrightarrow & 20x \color{red}{+16} \color{blue}{-16} \color{blue}{-75x} & = & \color{red}{75x} -180 \color{blue}{-75x} \color{blue}{-16} \\\Leftrightarrow & -55x& = & -196 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-196}{-55} \\\Leftrightarrow & x = \frac{196}{55} & & \\ & V = \left\{ \frac{196}{55} \right\} & \\\end{align}\)
6. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{5}{11}& = & \frac{4}{5}x-8 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 150 }{ \color{blue}{330} })& = & (\frac{264}{ \color{blue}{330} }x-\frac{2640}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-150& = & 264x-2640 \\\Leftrightarrow & 55x \color{red}{-150} \color{blue}{+150} \color{blue}{-264x} & = & \color{red}{264x} -2640 \color{blue}{-264x} \color{blue}{+150} \\\Leftrightarrow & -209x& = & -2490 \\\Leftrightarrow & \frac{-209x}{ \color{red}{-209} }& = & \frac{-2490}{-209} \\\Leftrightarrow & x = \frac{2490}{209} & & \\ & V = \left\{ \frac{2490}{209} \right\} & \\\end{align}\)
7. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{13}& = & \frac{7}{3}x-1 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }- \frac{ 24 }{ \color{blue}{78} })& = & (\frac{182}{ \color{blue}{78} }x-\frac{78}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x-24& = & 182x-78 \\\Leftrightarrow & 39x \color{red}{-24} \color{blue}{+24} \color{blue}{-182x} & = & \color{red}{182x} -78 \color{blue}{-182x} \color{blue}{+24} \\\Leftrightarrow & -143x& = & -54 \\\Leftrightarrow & \frac{-143x}{ \color{red}{-143} }& = & \frac{-54}{-143} \\\Leftrightarrow & x = \frac{54}{143} & & \\ & V = \left\{ \frac{54}{143} \right\} & \\\end{align}\)
8. \(\text{231 is het kleinste gemene veelvoud van 7, 11 en 3} \\ \begin{align} & \frac{x}{7}-\frac{4}{11}& = & \frac{-5}{3}x+6 \\\Leftrightarrow & \color{blue}{231.} (\frac{33x}{ \color{blue}{231} }- \frac{ 84 }{ \color{blue}{231} })& = & (\frac{-385}{ \color{blue}{231} }x+\frac{1386}{ \color{blue}{231} }) \color{blue}{.231} \\\Leftrightarrow & 33x-84& = & -385x+1386 \\\Leftrightarrow & 33x \color{red}{-84} \color{blue}{+84} \color{blue}{+385x} & = & \color{red}{-385x} +1386 \color{blue}{+385x} \color{blue}{+84} \\\Leftrightarrow & 418x& = & 1470 \\\Leftrightarrow & \frac{418x}{ \color{red}{418} }& = & \frac{1470}{418} \\\Leftrightarrow & x = \frac{735}{209} & & \\ & V = \left\{ \frac{735}{209} \right\} & \\\end{align}\)
9. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{4}{7}& = & \frac{6}{5}x+2 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 120 }{ \color{blue}{210} })& = & (\frac{252}{ \color{blue}{210} }x+\frac{420}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-120& = & 252x+420 \\\Leftrightarrow & 35x \color{red}{-120} \color{blue}{+120} \color{blue}{-252x} & = & \color{red}{252x} +420 \color{blue}{-252x} \color{blue}{+120} \\\Leftrightarrow & -217x& = & 540 \\\Leftrightarrow & \frac{-217x}{ \color{red}{-217} }& = & \frac{540}{-217} \\\Leftrightarrow & x = \frac{-540}{217} & & \\ & V = \left\{ \frac{-540}{217} \right\} & \\\end{align}\)
10. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{2}{13}& = & \frac{-3}{4}x-8 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 24 }{ \color{blue}{156} })& = & (\frac{-117}{ \color{blue}{156} }x-\frac{1248}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-24& = & -117x-1248 \\\Leftrightarrow & 52x \color{red}{-24} \color{blue}{+24} \color{blue}{+117x} & = & \color{red}{-117x} -1248 \color{blue}{+117x} \color{blue}{+24} \\\Leftrightarrow & 169x& = & -1224 \\\Leftrightarrow & \frac{169x}{ \color{red}{169} }& = & \frac{-1224}{169} \\\Leftrightarrow & x = \frac{-1224}{169} & & \\ & V = \left\{ \frac{-1224}{169} \right\} & \\\end{align}\)
11. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{3}{7}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 18 }{ \color{blue}{42} })& = & (\frac{21}{ \color{blue}{42} }x-\frac{168}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+18& = & 21x-168 \\\Leftrightarrow & 14x \color{red}{+18} \color{blue}{-18} \color{blue}{-21x} & = & \color{red}{21x} -168 \color{blue}{-21x} \color{blue}{-18} \\\Leftrightarrow & -7x& = & -186 \\\Leftrightarrow & \frac{-7x}{ \color{red}{-7} }& = & \frac{-186}{-7} \\\Leftrightarrow & x = \frac{186}{7} & & \\ & V = \left\{ \frac{186}{7} \right\} & \\\end{align}\)
12. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{5}{2}x+3 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{200}{ \color{blue}{80} }x+\frac{240}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-25& = & 200x+240 \\\Leftrightarrow & 16x \color{red}{-25} \color{blue}{+25} \color{blue}{-200x} & = & \color{red}{200x} +240 \color{blue}{-200x} \color{blue}{+25} \\\Leftrightarrow & -184x& = & 265 \\\Leftrightarrow & \frac{-184x}{ \color{red}{-184} }& = & \frac{265}{-184} \\\Leftrightarrow & x = \frac{-265}{184} & & \\ & V = \left\{ \frac{-265}{184} \right\} & \\\end{align}\)