Vgln. eerste graad (reeks 4)

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

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

Verbetersleutel

1. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 3} \\ \begin{align} & \frac{x}{2}-\frac{2}{15}& = & \frac{1}{3}x-4 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }- \frac{ 4 }{ \color{blue}{30} })& = & (\frac{10}{ \color{blue}{30} }x-\frac{120}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x-4& = & 10x-120 \\\Leftrightarrow & 15x \color{red}{-4} \color{blue}{+4} \color{blue}{-10x} & = & \color{red}{10x} -120 \color{blue}{-10x} \color{blue}{+4} \\\Leftrightarrow & 5x& = & -116 \\\Leftrightarrow & \frac{5x}{ \color{red}{5} }& = & \frac{-116}{5} \\\Leftrightarrow & x = \frac{-116}{5} & & \\ & V = \left\{ \frac{-116}{5} \right\} & \\\end{align}\)
2. \(\text{78 is het kleinste gemene veelvoud van 3, 13 en 2} \\ \begin{align} & \frac{x}{3}-\frac{3}{13}& = & \frac{1}{2}x-7 \\\Leftrightarrow & \color{blue}{78.} (\frac{26x}{ \color{blue}{78} }- \frac{ 18 }{ \color{blue}{78} })& = & (\frac{39}{ \color{blue}{78} }x-\frac{546}{ \color{blue}{78} }) \color{blue}{.78} \\\Leftrightarrow & 26x-18& = & 39x-546 \\\Leftrightarrow & 26x \color{red}{-18} \color{blue}{+18} \color{blue}{-39x} & = & \color{red}{39x} -546 \color{blue}{-39x} \color{blue}{+18} \\\Leftrightarrow & -13x& = & -528 \\\Leftrightarrow & \frac{-13x}{ \color{red}{-13} }& = & \frac{-528}{-13} \\\Leftrightarrow & x = \frac{528}{13} & & \\ & V = \left\{ \frac{528}{13} \right\} & \\\end{align}\)
3. \(\text{330 is het kleinste gemene veelvoud van 6, 11 en 5} \\ \begin{align} & \frac{x}{6}-\frac{3}{11}& = & \frac{-7}{5}x+6 \\\Leftrightarrow & \color{blue}{330.} (\frac{55x}{ \color{blue}{330} }- \frac{ 90 }{ \color{blue}{330} })& = & (\frac{-462}{ \color{blue}{330} }x+\frac{1980}{ \color{blue}{330} }) \color{blue}{.330} \\\Leftrightarrow & 55x-90& = & -462x+1980 \\\Leftrightarrow & 55x \color{red}{-90} \color{blue}{+90} \color{blue}{+462x} & = & \color{red}{-462x} +1980 \color{blue}{+462x} \color{blue}{+90} \\\Leftrightarrow & 517x& = & 2070 \\\Leftrightarrow & \frac{517x}{ \color{red}{517} }& = & \frac{2070}{517} \\\Leftrightarrow & x = \frac{2070}{517} & & \\ & V = \left\{ \frac{2070}{517} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 4, 7 en 3} \\ \begin{align} & \frac{x}{4}+\frac{4}{7}& = & \frac{7}{3}x+4 \\\Leftrightarrow & \color{blue}{84.} (\frac{21x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{196}{ \color{blue}{84} }x+\frac{336}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 21x+48& = & 196x+336 \\\Leftrightarrow & 21x \color{red}{+48} \color{blue}{-48} \color{blue}{-196x} & = & \color{red}{196x} +336 \color{blue}{-196x} \color{blue}{-48} \\\Leftrightarrow & -175x& = & 288 \\\Leftrightarrow & \frac{-175x}{ \color{red}{-175} }& = & \frac{288}{-175} \\\Leftrightarrow & x = \frac{-288}{175} & & \\ & V = \left\{ \frac{-288}{175} \right\} & \\\end{align}\)
5. \(\text{66 is het kleinste gemene veelvoud van 2, 11 en 3} \\ \begin{align} & \frac{x}{2}+\frac{5}{11}& = & \frac{-5}{3}x-7 \\\Leftrightarrow & \color{blue}{66.} (\frac{33x}{ \color{blue}{66} }+ \frac{ 30 }{ \color{blue}{66} })& = & (\frac{-110}{ \color{blue}{66} }x-\frac{462}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 33x+30& = & -110x-462 \\\Leftrightarrow & 33x \color{red}{+30} \color{blue}{-30} \color{blue}{+110x} & = & \color{red}{-110x} -462 \color{blue}{+110x} \color{blue}{-30} \\\Leftrightarrow & 143x& = & -492 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-492}{143} \\\Leftrightarrow & x = \frac{-492}{143} & & \\ & V = \left\{ \frac{-492}{143} \right\} & \\\end{align}\)
6. \(\text{66 is het kleinste gemene veelvoud van 3, 11 en 2} \\ \begin{align} & \frac{x}{3}-\frac{5}{11}& = & \frac{1}{2}x+8 \\\Leftrightarrow & \color{blue}{66.} (\frac{22x}{ \color{blue}{66} }- \frac{ 30 }{ \color{blue}{66} })& = & (\frac{33}{ \color{blue}{66} }x+\frac{528}{ \color{blue}{66} }) \color{blue}{.66} \\\Leftrightarrow & 22x-30& = & 33x+528 \\\Leftrightarrow & 22x \color{red}{-30} \color{blue}{+30} \color{blue}{-33x} & = & \color{red}{33x} +528 \color{blue}{-33x} \color{blue}{+30} \\\Leftrightarrow & -11x& = & 558 \\\Leftrightarrow & \frac{-11x}{ \color{red}{-11} }& = & \frac{558}{-11} \\\Leftrightarrow & x = \frac{-558}{11} & & \\ & V = \left\{ \frac{-558}{11} \right\} & \\\end{align}\)
7. \(\text{210 is het kleinste gemene veelvoud van 6, 7 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{7}& = & \frac{3}{5}x-1 \\\Leftrightarrow & \color{blue}{210.} (\frac{35x}{ \color{blue}{210} }- \frac{ 60 }{ \color{blue}{210} })& = & (\frac{126}{ \color{blue}{210} }x-\frac{210}{ \color{blue}{210} }) \color{blue}{.210} \\\Leftrightarrow & 35x-60& = & 126x-210 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{-126x} & = & \color{red}{126x} -210 \color{blue}{-126x} \color{blue}{+60} \\\Leftrightarrow & -91x& = & -150 \\\Leftrightarrow & \frac{-91x}{ \color{red}{-91} }& = & \frac{-150}{-91} \\\Leftrightarrow & x = \frac{150}{91} & & \\ & V = \left\{ \frac{150}{91} \right\} & \\\end{align}\)
8. \(\text{14 is het kleinste gemene veelvoud van 7, 7 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{7}& = & \frac{7}{2}x-6 \\\Leftrightarrow & \color{blue}{14.} (\frac{2x}{ \color{blue}{14} }+ \frac{ 8 }{ \color{blue}{14} })& = & (\frac{49}{ \color{blue}{14} }x-\frac{84}{ \color{blue}{14} }) \color{blue}{.14} \\\Leftrightarrow & 2x+8& = & 49x-84 \\\Leftrightarrow & 2x \color{red}{+8} \color{blue}{-8} \color{blue}{-49x} & = & \color{red}{49x} -84 \color{blue}{-49x} \color{blue}{-8} \\\Leftrightarrow & -47x& = & -92 \\\Leftrightarrow & \frac{-47x}{ \color{red}{-47} }& = & \frac{-92}{-47} \\\Leftrightarrow & x = \frac{92}{47} & & \\ & V = \left\{ \frac{92}{47} \right\} & \\\end{align}\)
9. \(\text{90 is het kleinste gemene veelvoud van 6, 9 en 5} \\ \begin{align} & \frac{x}{6}-\frac{2}{9}& = & \frac{7}{5}x+5 \\\Leftrightarrow & \color{blue}{90.} (\frac{15x}{ \color{blue}{90} }- \frac{ 20 }{ \color{blue}{90} })& = & (\frac{126}{ \color{blue}{90} }x+\frac{450}{ \color{blue}{90} }) \color{blue}{.90} \\\Leftrightarrow & 15x-20& = & 126x+450 \\\Leftrightarrow & 15x \color{red}{-20} \color{blue}{+20} \color{blue}{-126x} & = & \color{red}{126x} +450 \color{blue}{-126x} \color{blue}{+20} \\\Leftrightarrow & -111x& = & 470 \\\Leftrightarrow & \frac{-111x}{ \color{red}{-111} }& = & \frac{470}{-111} \\\Leftrightarrow & x = \frac{-470}{111} & & \\ & V = \left\{ \frac{-470}{111} \right\} & \\\end{align}\)
10. \(\text{140 is het kleinste gemene veelvoud van 4, 7 en 5} \\ \begin{align} & \frac{x}{4}-\frac{3}{7}& = & \frac{-4}{5}x+3 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }- \frac{ 60 }{ \color{blue}{140} })& = & (\frac{-112}{ \color{blue}{140} }x+\frac{420}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x-60& = & -112x+420 \\\Leftrightarrow & 35x \color{red}{-60} \color{blue}{+60} \color{blue}{+112x} & = & \color{red}{-112x} +420 \color{blue}{+112x} \color{blue}{+60} \\\Leftrightarrow & 147x& = & 480 \\\Leftrightarrow & \frac{147x}{ \color{red}{147} }& = & \frac{480}{147} \\\Leftrightarrow & x = \frac{160}{49} & & \\ & V = \left\{ \frac{160}{49} \right\} & \\\end{align}\)
11. \(\text{180 is het kleinste gemene veelvoud van 5, 9 en 4} \\ \begin{align} & \frac{x}{5}+\frac{4}{9}& = & \frac{1}{4}x-8 \\\Leftrightarrow & \color{blue}{180.} (\frac{36x}{ \color{blue}{180} }+ \frac{ 80 }{ \color{blue}{180} })& = & (\frac{45}{ \color{blue}{180} }x-\frac{1440}{ \color{blue}{180} }) \color{blue}{.180} \\\Leftrightarrow & 36x+80& = & 45x-1440 \\\Leftrightarrow & 36x \color{red}{+80} \color{blue}{-80} \color{blue}{-45x} & = & \color{red}{45x} -1440 \color{blue}{-45x} \color{blue}{-80} \\\Leftrightarrow & -9x& = & -1520 \\\Leftrightarrow & \frac{-9x}{ \color{red}{-9} }& = & \frac{-1520}{-9} \\\Leftrightarrow & x = \frac{1520}{9} & & \\ & V = \left\{ \frac{1520}{9} \right\} & \\\end{align}\)
12. \(\text{110 is het kleinste gemene veelvoud van 2, 11 en 5} \\ \begin{align} & \frac{x}{2}-\frac{4}{11}& = & \frac{2}{5}x+3 \\\Leftrightarrow & \color{blue}{110.} (\frac{55x}{ \color{blue}{110} }- \frac{ 40 }{ \color{blue}{110} })& = & (\frac{44}{ \color{blue}{110} }x+\frac{330}{ \color{blue}{110} }) \color{blue}{.110} \\\Leftrightarrow & 55x-40& = & 44x+330 \\\Leftrightarrow & 55x \color{red}{-40} \color{blue}{+40} \color{blue}{-44x} & = & \color{red}{44x} +330 \color{blue}{-44x} \color{blue}{+40} \\\Leftrightarrow & 11x& = & 370 \\\Leftrightarrow & \frac{11x}{ \color{red}{11} }& = & \frac{370}{11} \\\Leftrightarrow & x = \frac{370}{11} & & \\ & V = \left\{ \frac{370}{11} \right\} & \\\end{align}\)