Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 2, 10 en 3} \\ \begin{align} & \frac{x}{2}-\frac{3}{10}& = & \frac{-5}{3}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 9 }{ \color{blue}{30} })& = & (\frac{-50}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-9& = & -50x+150 \\\Leftrightarrow & 15x \color{red}{-9} \color{blue}{+9} \color{blue}{+50x} & = & \color{red}{-50x} +150 \color{blue}{+50x} \color{blue}{+9} \\\Leftrightarrow & 65x& = & 159 \\\Leftrightarrow & \frac{65x}{ \color{red}{65} }& = & \frac{159}{65} \\\Leftrightarrow & x = \frac{159}{65} & & \\ & V = \left\{ \frac{159}{65} \right\} & \\\end{align}\)
2. \(\text{56 is het kleinste gemene veelvoud van 7, 8 en 4} \\ \begin{align} & \frac{x}{7}+\frac{3}{8}& = & \frac{-7}{4}x+4 \\\Leftrightarrow & \color{blue}{56.} (\frac{8x}{ \color{blue}{56} }+ \frac{ 21 }{ \color{blue}{56} })& = & (\frac{-98}{ \color{blue}{56} }x+\frac{224}{ \color{blue}{56} }) \color{blue}{.56} \\\Leftrightarrow & 8x+21& = & -98x+224 \\\Leftrightarrow & 8x \color{red}{+21} \color{blue}{-21} \color{blue}{+98x} & = & \color{red}{-98x} +224 \color{blue}{+98x} \color{blue}{-21} \\\Leftrightarrow & 106x& = & 203 \\\Leftrightarrow & \frac{106x}{ \color{red}{106} }& = & \frac{203}{106} \\\Leftrightarrow & x = \frac{203}{106} & & \\ & V = \left\{ \frac{203}{106} \right\} & \\\end{align}\)
3. \(\text{126 is het kleinste gemene veelvoud van 7, 9 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{9}& = & \frac{3}{2}x+3 \\\Leftrightarrow & \color{blue}{126.} (\frac{18x}{ \color{blue}{126} }+ \frac{ 56 }{ \color{blue}{126} })& = & (\frac{189}{ \color{blue}{126} }x+\frac{378}{ \color{blue}{126} }) \color{blue}{.126} \\\Leftrightarrow & 18x+56& = & 189x+378 \\\Leftrightarrow & 18x \color{red}{+56} \color{blue}{-56} \color{blue}{-189x} & = & \color{red}{189x} +378 \color{blue}{-189x} \color{blue}{-56} \\\Leftrightarrow & -171x& = & 322 \\\Leftrightarrow & \frac{-171x}{ \color{red}{-171} }& = & \frac{322}{-171} \\\Leftrightarrow & x = \frac{-322}{171} & & \\ & V = \left\{ \frac{-322}{171} \right\} & \\\end{align}\)
4. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{9}& = & \frac{1}{5}x+4 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }+ \frac{ 20 }{ \color{blue}{90} })& = & (\frac{18}{ \color{blue}{90} }x+\frac{360}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x+20& = & 18x+360 \\\Leftrightarrow & 15x \color{red}{+20} \color{blue}{-20} \color{blue}{-18x} & = & \color{red}{18x} +360 \color{blue}{-18x} \color{blue}{-20} \\\Leftrightarrow & -3x& = & 340 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{340}{-3} \\\Leftrightarrow & x = \frac{-340}{3} & & \\ & V = \left\{ \frac{-340}{3} \right\} & \\\end{align}\)
5. \(\text{156 is het kleinste gemene veelvoud van 3, 13 en 4} \\ \begin{align} & \frac{x}{3}-\frac{5}{13}& = & \frac{3}{4}x+7 \\\Leftrightarrow & \color{blue}{156.} (\frac{52x}{ \color{blue}{156} }- \frac{ 60 }{ \color{blue}{156} })& = & (\frac{117}{ \color{blue}{156} }x+\frac{1092}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 52x-60& = & 117x+1092 \\\Leftrightarrow & 52x \color{red}{-60} \color{blue}{+60} \color{blue}{-117x} & = & \color{red}{117x} +1092 \color{blue}{-117x} \color{blue}{+60} \\\Leftrightarrow & -65x& = & 1152 \\\Leftrightarrow & \frac{-65x}{ \color{red}{-65} }& = & \frac{1152}{-65} \\\Leftrightarrow & x = \frac{-1152}{65} & & \\ & V = \left\{ \frac{-1152}{65} \right\} & \\\end{align}\)
6. \(\text{42 is het kleinste gemene veelvoud van 3, 7 en 2} \\ \begin{align} & \frac{x}{3}+\frac{2}{7}& = & \frac{7}{2}x-3 \\\Leftrightarrow & \color{blue}{42.} (\frac{14x}{ \color{blue}{42} }+ \frac{ 12 }{ \color{blue}{42} })& = & (\frac{147}{ \color{blue}{42} }x-\frac{126}{ \color{blue}{42} }) \color{blue}{.42} \\\Leftrightarrow & 14x+12& = & 147x-126 \\\Leftrightarrow & 14x \color{red}{+12} \color{blue}{-12} \color{blue}{-147x} & = & \color{red}{147x} -126 \color{blue}{-147x} \color{blue}{-12} \\\Leftrightarrow & -133x& = & -138 \\\Leftrightarrow & \frac{-133x}{ \color{red}{-133} }& = & \frac{-138}{-133} \\\Leftrightarrow & x = \frac{138}{133} & & \\ & V = \left\{ \frac{138}{133} \right\} & \\\end{align}\)
7. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}+\frac{4}{11}& = & \frac{1}{2}x-4 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }+ \frac{ 24 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x-\frac{264}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x+24& = & 33x-264 \\\Leftrightarrow & 22x \color{red}{+24} \color{blue}{-24} \color{blue}{-33x} & = & \color{red}{33x} -264 \color{blue}{-33x} \color{blue}{-24} \\\Leftrightarrow & -11x& = & -288 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{-288}{-11} \\\Leftrightarrow & x = \frac{288}{11} & & \\ & V = \left\{ \frac{288}{11} \right\} & \\\end{align}\)
8. \(\text{90 is het kleinste gemene veelvoud van 2, 9 en 5} \\ \begin{align} & \frac{x}{2}+\frac{5}{9}& = & \frac{-4}{5}x-7 \\\Leftrightarrow & \color{blue}{90.} (\frac{45x}{ \color{blue}{90} }+ \frac{ 50 }{ \color{blue}{90} })& = & (\frac{-72}{ \color{blue}{90} }x-\frac{630}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 45x+50& = & -72x-630 \\\Leftrightarrow & 45x \color{red}{+50} \color{blue}{-50} \color{blue}{+72x} & = & \color{red}{-72x} -630 \color{blue}{+72x} \color{blue}{-50} \\\Leftrightarrow & 117x& = & -680 \\\Leftrightarrow & \frac{117x}{ \color{red}{117} }& = & \frac{-680}{117} \\\Leftrightarrow & x = \frac{-680}{117} & & \\ & V = \left\{ \frac{-680}{117} \right\} & \\\end{align}\)
9. \(\text{390 is het kleinste gemene veelvoud van 6, 13 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{13}& = & \frac{7}{5}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{65x}{ \color{blue}{390} }- \frac{ 60 }{ \color{blue}{390} })& = & (\frac{546}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 65x-60& = & 546x+2340 \\\Leftrightarrow & 65x \color{red}{-60} \color{blue}{+60} \color{blue}{-546x} & = & \color{red}{546x} +2340 \color{blue}{-546x} \color{blue}{+60} \\\Leftrightarrow & -481x& = & 2400 \\\Leftrightarrow & \frac{-481x}{ \color{red}{-481} }& = & \frac{2400}{-481} \\\Leftrightarrow & x = \frac{-2400}{481} & & \\ & V = \left\{ \frac{-2400}{481} \right\} & \\\end{align}\)
10. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{15}& = & \frac{3}{5}x+1 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{18}{ \color{blue}{30} }x+\frac{30}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x-4& = & 18x+30 \\\Leftrightarrow & 5x \color{red}{-4} \color{blue}{+4} \color{blue}{-18x} & = & \color{red}{18x} +30 \color{blue}{-18x} \color{blue}{+4} \\\Leftrightarrow & -13x& = & 34 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{34}{-13} \\\Leftrightarrow & x = \frac{-34}{13} & & \\ & V = \left\{ \frac{-34}{13} \right\} & \\\end{align}\)
11. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{7}{3}x-5 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{308}{ \color{blue}{132} }x-\frac{660}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & 308x-660 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{-308x} & = & \color{red}{308x} -660 \color{blue}{-308x} \color{blue}{-24} \\\Leftrightarrow & -275x& = & -684 \\\Leftrightarrow & \frac{-275x}{ \color{red}{-275} }& = & \frac{-684}{-275} \\\Leftrightarrow & x = \frac{684}{275} & & \\ & V = \left\{ \frac{684}{275} \right\} & \\\end{align}\)
12. \(\text{78 is het kleinste gemene veelvoud van 2, 13 en 3} \\ \begin{align} & \frac{x}{2}+\frac{3}{13}& = & \frac{-2}{3}x-7 \\\Leftrightarrow & \color{blue}{78.} (\frac{39x}{ \color{blue}{78} }+ \frac{ 18 }{ \color{blue}{78} })& = & (\frac{-52}{ \color{blue}{78} }x-\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 39x+18& = & -52x-546 \\\Leftrightarrow & 39x \color{red}{+18} \color{blue}{-18} \color{blue}{+52x} & = & \color{red}{-52x} -546 \color{blue}{+52x} \color{blue}{-18} \\\Leftrightarrow & 91x& = & -564 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-564}{91} \\\Leftrightarrow & x = \frac{-564}{91} & & \\ & V = \left\{ \frac{-564}{91} \right\} & \\\end{align}\)