Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}+\frac{3}{13}& = & \frac{1}{4}x+4 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }+ \frac{ 60 }{ \color{blue}{260} })& = & (\frac{65}{ \color{blue}{260} }x+\frac{1040}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x+60& = & 65x+1040 \\\Leftrightarrow & 52x \color{red}{+60} \color{blue}{-60} \color{blue}{-65x} & = & \color{red}{65x} +1040 \color{blue}{-65x} \color{blue}{-60} \\\Leftrightarrow & -13x& = & 980 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{980}{-13} \\\Leftrightarrow & x = \frac{-980}{13} & & \\ & V = \left\{ \frac{-980}{13} \right\} & \\\end{align}\)
2. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{13}& = & \frac{6}{5}x-8 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }+ \frac{ 90 }{ \color{blue}{390} })& = & (\frac{468}{ \color{blue}{390} }x-\frac{3120}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x+90& = & 468x-3120 \\\Leftrightarrow & 65x \color{red}{+90} \color{blue}{-90} \color{blue}{-468x} & = & \color{red}{468x} -3120 \color{blue}{-468x} \color{blue}{-90} \\\Leftrightarrow & -403x& = & -3210 \\\Leftrightarrow & \frac{-403x}{ \color{red}{-403} }& = & \frac{-3210}{-403} \\\Leftrightarrow & x = \frac{3210}{403} & & \\ & V = \left\{ \frac{3210}{403} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 6, 10 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{10}& = & \frac{-7}{5}x+4 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-9& = & -42x+120 \\\Leftrightarrow & 5x \color{red}{-9} \color{blue}{+9} \color{blue}{+42x} & = & \color{red}{-42x} +120 \color{blue}{+42x} \color{blue}{+9} \\\Leftrightarrow & 47x& = & 129 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{129}{47} \\\Leftrightarrow & x = \frac{129}{47} & & \\ & V = \left\{ \frac{129}{47} \right\} & \\\end{align}\)
4. \(\text{21 is het kleinste gemene veelvoud van 7, 7 en 3} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{1}{3}x+5 \\\Leftrightarrow & \color{blue}{21.} (\frac{3x}{ \color{blue}{21} }+ \frac{ 12 }{ \color{blue}{21} })& = & (\frac{7}{ \color{blue}{21} }x+\frac{105}{ \color{blue}{21} }) \color{blue}{.21} \\\Leftrightarrow & 3x+12& = & 7x+105 \\\Leftrightarrow & 3x \color{red}{+12} \color{blue}{-12} \color{blue}{-7x} & = & \color{red}{7x} +105 \color{blue}{-7x} \color{blue}{-12} \\\Leftrightarrow & -4x& = & 93 \\\Leftrightarrow & \frac{-4x}{ \color{red}{-4} }& = & \frac{93}{-4} \\\Leftrightarrow & x = \frac{-93}{4} & & \\ & V = \left\{ \frac{-93}{4} \right\} & \\\end{align}\)
5. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{2}{15}& = & \frac{-7}{5}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+4& = & -42x+30 \\\Leftrightarrow & 15x \color{red}{+4} \color{blue}{-4} \color{blue}{+42x} & = & \color{red}{-42x} +30 \color{blue}{+42x} \color{blue}{-4} \\\Leftrightarrow & 57x& = & 26 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{26}{57} \\\Leftrightarrow & x = \frac{26}{57} & & \\ & V = \left\{ \frac{26}{57} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 2, 7 en 3} \\ \begin{align} & \frac{x}{2}-\frac{4}{7}& = & \frac{7}{3}x+7 \\\Leftrightarrow & \color{blue}{42.} (\frac{21x}{ \color{blue}{42} }- \frac{ 24 }{ \color{blue}{42} })& = & (\frac{98}{ \color{blue}{42} }x+\frac{294}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 21x-24& = & 98x+294 \\\Leftrightarrow & 21x \color{red}{-24} \color{blue}{+24} \color{blue}{-98x} & = & \color{red}{98x} +294 \color{blue}{-98x} \color{blue}{+24} \\\Leftrightarrow & -77x& = & 318 \\\Leftrightarrow & \frac{-77x}{ \color{red}{-77} }& = & \frac{318}{-77} \\\Leftrightarrow & x = \frac{-318}{77} & & \\ & V = \left\{ \frac{-318}{77} \right\} & \\\end{align}\)
7. \(\text{260 is het kleinste gemene veelvoud van 5, 13 en 4} \\ \begin{align} & \frac{x}{5}-\frac{4}{13}& = & \frac{5}{4}x+4 \\\Leftrightarrow & \color{blue}{260.} (\frac{52x}{ \color{blue}{260} }- \frac{ 80 }{ \color{blue}{260} })& = & (\frac{325}{ \color{blue}{260} }x+\frac{1040}{ \color{blue}{260} }) \color{blue}{.260} \\\Leftrightarrow & 52x-80& = & 325x+1040 \\\Leftrightarrow & 52x \color{red}{-80} \color{blue}{+80} \color{blue}{-325x} & = & \color{red}{325x} +1040 \color{blue}{-325x} \color{blue}{+80} \\\Leftrightarrow & -273x& = & 1120 \\\Leftrightarrow & \frac{-273x}{ \color{red}{-273} }& = & \frac{1120}{-273} \\\Leftrightarrow & x = \frac{-160}{39} & & \\ & V = \left\{ \frac{-160}{39} \right\} & \\\end{align}\)
8. \(\text{24 is het kleinste gemene veelvoud van 2, 8 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{8}& = & \frac{7}{3}x+3 \\\Leftrightarrow & \color{blue}{24.} (\frac{12x}{ \color{blue}{24} }- \frac{ 15 }{ \color{blue}{24} })& = & (\frac{56}{ \color{blue}{24} }x+\frac{72}{ \color{blue}{24} }) \color{blue}{.24} \\\Leftrightarrow & 12x-15& = & 56x+72 \\\Leftrightarrow & 12x \color{red}{-15} \color{blue}{+15} \color{blue}{-56x} & = & \color{red}{56x} +72 \color{blue}{-56x} \color{blue}{+15} \\\Leftrightarrow & -44x& = & 87 \\\Leftrightarrow & \frac{-44x}{ \color{red}{-44} }& = & \frac{87}{-44} \\\Leftrightarrow & x = \frac{-87}{44} & & \\ & V = \left\{ \frac{-87}{44} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{2}{13}& = & \frac{1}{6}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 60 }{ \color{blue}{390} })& = & (\frac{65}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+60& = & 65x+2340 \\\Leftrightarrow & 78x \color{red}{+60} \color{blue}{-60} \color{blue}{-65x} & = & \color{red}{65x} +2340 \color{blue}{-65x} \color{blue}{-60} \\\Leftrightarrow & 13x& = & 2280 \\\Leftrightarrow & \frac{13x}{ \color{red}{13} }& = & \frac{2280}{13} \\\Leftrightarrow & x = \frac{2280}{13} & & \\ & V = \left\{ \frac{2280}{13} \right\} & \\\end{align}\)
10. \(\text{80 is het kleinste gemene veelvoud van 5, 16 en 2} \\ \begin{align} & \frac{x}{5}-\frac{5}{16}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{80.} (\frac{16x}{ \color{blue}{80} }- \frac{ 25 }{ \color{blue}{80} })& = & (\frac{40}{ \color{blue}{80} }x-\frac{160}{ \color{blue}{80} }) \color{blue}{.80} \\\Leftrightarrow & 16x-25& = & 40x-160 \\\Leftrightarrow & 16x \color{red}{-25} \color{blue}{+25} \color{blue}{-40x} & = & \color{red}{40x} -160 \color{blue}{-40x} \color{blue}{+25} \\\Leftrightarrow & -24x& = & -135 \\\Leftrightarrow & \frac{-24x}{ \color{red}{-24} }& = & \frac{-135}{-24} \\\Leftrightarrow & x = \frac{45}{8} & & \\ & V = \left\{ \frac{45}{8} \right\} & \\\end{align}\)
11. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}-\frac{2}{11}& = & \frac{5}{2}x+6 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }- \frac{ 28 }{ \color{blue}{154} })& = & (\frac{385}{ \color{blue}{154} }x+\frac{924}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x-28& = & 385x+924 \\\Leftrightarrow & 22x \color{red}{-28} \color{blue}{+28} \color{blue}{-385x} & = & \color{red}{385x} +924 \color{blue}{-385x} \color{blue}{+28} \\\Leftrightarrow & -363x& = & 952 \\\Leftrightarrow & \frac{-363x}{ \color{red}{-363} }& = & \frac{952}{-363} \\\Leftrightarrow & x = \frac{-952}{363} & & \\ & V = \left\{ \frac{-952}{363} \right\} & \\\end{align}\)
12. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}-\frac{4}{7}& = & \frac{1}{4}x+2 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }- \frac{ 48 }{ \color{blue}{84} })& = & (\frac{21}{ \color{blue}{84} }x+\frac{168}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x-48& = & 21x+168 \\\Leftrightarrow & 28x \color{red}{-48} \color{blue}{+48} \color{blue}{-21x} & = & \color{red}{21x} +168 \color{blue}{-21x} \color{blue}{+48} \\\Leftrightarrow & 7x& = & 216 \\\Leftrightarrow & \frac{7x}{ \color{red}{7} }& = & \frac{216}{7} \\\Leftrightarrow & x = \frac{216}{7} & & \\ & V = \left\{ \frac{216}{7} \right\} & \\\end{align}\)