Vgln. eerste graad (reeks 4)

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1. \(\frac{x}{7}+\frac{5}{7}=\frac{1}{4}x+6\)
2. \(\frac{x}{4}+\frac{2}{11}=\frac{7}{3}x+8\)
3. \(\frac{x}{2}+\frac{4}{15}=\frac{-7}{5}x+5\)
4. \(\frac{x}{3}+\frac{4}{7}=\frac{7}{4}x-7\)
5. \(\frac{x}{4}+\frac{3}{14}=\frac{-2}{5}x-2\)
6. \(\frac{x}{6}+\frac{2}{15}=\frac{-7}{5}x+8\)
7. \(\frac{x}{6}+\frac{3}{8}=\frac{-7}{5}x-8\)
8. \(\frac{x}{2}-\frac{5}{16}=\frac{-8}{3}x+6\)
9. \(\frac{x}{4}+\frac{3}{11}=\frac{7}{3}x-8\)
10. \(\frac{x}{4}-\frac{4}{13}=\frac{-2}{3}x-2\)
11. \(\frac{x}{7}+\frac{4}{11}=\frac{1}{2}x-2\)
12. \(\frac{x}{5}+\frac{4}{13}=\frac{-5}{6}x+6\)

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

Verbetersleutel

1. \(\text{28 is het kleinste gemene veelvoud van 7, 7 en 4} \\ \begin{align} & \frac{x}{7}+\frac{5}{7}& = & \frac{1}{4}x+6 \\\Leftrightarrow & \color{blue}{28.} (\frac{4x}{ \color{blue}{28} }+ \frac{ 20 }{ \color{blue}{28} })& = & (\frac{7}{ \color{blue}{28} }x+\frac{168}{ \color{blue}{28} }) \color{blue}{.28} \\\Leftrightarrow & 4x+20& = & 7x+168 \\\Leftrightarrow & 4x \color{red}{+20} \color{blue}{-20} \color{blue}{-7x} & = & \color{red}{7x} +168 \color{blue}{-7x} \color{blue}{-20} \\\Leftrightarrow & -3x& = & 148 \\\Leftrightarrow & \frac{-3x}{ \color{red}{-3} }& = & \frac{148}{-3} \\\Leftrightarrow & x = \frac{-148}{3} & & \\ & V = \left\{ \frac{-148}{3} \right\} & \\\end{align}\)
2. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{2}{11}& = & \frac{7}{3}x+8 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 24 }{ \color{blue}{132} })& = & (\frac{308}{ \color{blue}{132} }x+\frac{1056}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+24& = & 308x+1056 \\\Leftrightarrow & 33x \color{red}{+24} \color{blue}{-24} \color{blue}{-308x} & = & \color{red}{308x} +1056 \color{blue}{-308x} \color{blue}{-24} \\\Leftrightarrow & -275x& = & 1032 \\\Leftrightarrow & \frac{-275x}{ \color{red}{-275} }& = & \frac{1032}{-275} \\\Leftrightarrow & x = \frac{-1032}{275} & & \\ & V = \left\{ \frac{-1032}{275} \right\} & \\\end{align}\)
3. \(\text{30 is het kleinste gemene veelvoud van 2, 15 en 5} \\ \begin{align} & \frac{x}{2}+\frac{4}{15}& = & \frac{-7}{5}x+5 \\\Leftrightarrow & \color{blue}{30.} (\frac{15x}{ \color{blue}{30} }+ \frac{ 8 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{150}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 15x+8& = & -42x+150 \\\Leftrightarrow & 15x \color{red}{+8} \color{blue}{-8} \color{blue}{+42x} & = & \color{red}{-42x} +150 \color{blue}{+42x} \color{blue}{-8} \\\Leftrightarrow & 57x& = & 142 \\\Leftrightarrow & \frac{57x}{ \color{red}{57} }& = & \frac{142}{57} \\\Leftrightarrow & x = \frac{142}{57} & & \\ & V = \left\{ \frac{142}{57} \right\} & \\\end{align}\)
4. \(\text{84 is het kleinste gemene veelvoud van 3, 7 en 4} \\ \begin{align} & \frac{x}{3}+\frac{4}{7}& = & \frac{7}{4}x-7 \\\Leftrightarrow & \color{blue}{84.} (\frac{28x}{ \color{blue}{84} }+ \frac{ 48 }{ \color{blue}{84} })& = & (\frac{147}{ \color{blue}{84} }x-\frac{588}{ \color{blue}{84} }) \color{blue}{.84} \\\Leftrightarrow & 28x+48& = & 147x-588 \\\Leftrightarrow & 28x \color{red}{+48} \color{blue}{-48} \color{blue}{-147x} & = & \color{red}{147x} -588 \color{blue}{-147x} \color{blue}{-48} \\\Leftrightarrow & -119x& = & -636 \\\Leftrightarrow & \frac{-119x}{ \color{red}{-119} }& = & \frac{-636}{-119} \\\Leftrightarrow & x = \frac{636}{119} & & \\ & V = \left\{ \frac{636}{119} \right\} & \\\end{align}\)
5. \(\text{140 is het kleinste gemene veelvoud van 4, 14 en 5} \\ \begin{align} & \frac{x}{4}+\frac{3}{14}& = & \frac{-2}{5}x-2 \\\Leftrightarrow & \color{blue}{140.} (\frac{35x}{ \color{blue}{140} }+ \frac{ 30 }{ \color{blue}{140} })& = & (\frac{-56}{ \color{blue}{140} }x-\frac{280}{ \color{blue}{140} }) \color{blue}{.140} \\\Leftrightarrow & 35x+30& = & -56x-280 \\\Leftrightarrow & 35x \color{red}{+30} \color{blue}{-30} \color{blue}{+56x} & = & \color{red}{-56x} -280 \color{blue}{+56x} \color{blue}{-30} \\\Leftrightarrow & 91x& = & -310 \\\Leftrightarrow & \frac{91x}{ \color{red}{91} }& = & \frac{-310}{91} \\\Leftrightarrow & x = \frac{-310}{91} & & \\ & V = \left\{ \frac{-310}{91} \right\} & \\\end{align}\)
6. \(\text{30 is het kleinste gemene veelvoud van 6, 15 en 5} \\ \begin{align} & \frac{x}{6}+\frac{2}{15}& = & \frac{-7}{5}x+8 \\\Leftrightarrow & \color{blue}{30.} (\frac{5x}{ \color{blue}{30} }+ \frac{ 4 }{ \color{blue}{30} })& = & (\frac{-42}{ \color{blue}{30} }x+\frac{240}{ \color{blue}{30} }) \color{blue}{.30} \\\Leftrightarrow & 5x+4& = & -42x+240 \\\Leftrightarrow & 5x \color{red}{+4} \color{blue}{-4} \color{blue}{+42x} & = & \color{red}{-42x} +240 \color{blue}{+42x} \color{blue}{-4} \\\Leftrightarrow & 47x& = & 236 \\\Leftrightarrow & \frac{47x}{ \color{red}{47} }& = & \frac{236}{47} \\\Leftrightarrow & x = \frac{236}{47} & & \\ & V = \left\{ \frac{236}{47} \right\} & \\\end{align}\)
7. \(\text{120 is het kleinste gemene veelvoud van 6, 8 en 5} \\ \begin{align} & \frac{x}{6}+\frac{3}{8}& = & \frac{-7}{5}x-8 \\\Leftrightarrow & \color{blue}{120.} (\frac{20x}{ \color{blue}{120} }+ \frac{ 45 }{ \color{blue}{120} })& = & (\frac{-168}{ \color{blue}{120} }x-\frac{960}{ \color{blue}{120} }) \color{blue}{.120} \\\Leftrightarrow & 20x+45& = & -168x-960 \\\Leftrightarrow & 20x \color{red}{+45} \color{blue}{-45} \color{blue}{+168x} & = & \color{red}{-168x} -960 \color{blue}{+168x} \color{blue}{-45} \\\Leftrightarrow & 188x& = & -1005 \\\Leftrightarrow & \frac{188x}{ \color{red}{188} }& = & \frac{-1005}{188} \\\Leftrightarrow & x = \frac{-1005}{188} & & \\ & V = \left\{ \frac{-1005}{188} \right\} & \\\end{align}\)
8. \(\text{48 is het kleinste gemene veelvoud van 2, 16 en 3} \\ \begin{align} & \frac{x}{2}-\frac{5}{16}& = & \frac{-8}{3}x+6 \\\Leftrightarrow & \color{blue}{48.} (\frac{24x}{ \color{blue}{48} }- \frac{ 15 }{ \color{blue}{48} })& = & (\frac{-128}{ \color{blue}{48} }x+\frac{288}{ \color{blue}{48} }) \color{blue}{.48} \\\Leftrightarrow & 24x-15& = & -128x+288 \\\Leftrightarrow & 24x \color{red}{-15} \color{blue}{+15} \color{blue}{+128x} & = & \color{red}{-128x} +288 \color{blue}{+128x} \color{blue}{+15} \\\Leftrightarrow & 152x& = & 303 \\\Leftrightarrow & \frac{152x}{ \color{red}{152} }& = & \frac{303}{152} \\\Leftrightarrow & x = \frac{303}{152} & & \\ & V = \left\{ \frac{303}{152} \right\} & \\\end{align}\)
9. \(\text{132 is het kleinste gemene veelvoud van 4, 11 en 3} \\ \begin{align} & \frac{x}{4}+\frac{3}{11}& = & \frac{7}{3}x-8 \\\Leftrightarrow & \color{blue}{132.} (\frac{33x}{ \color{blue}{132} }+ \frac{ 36 }{ \color{blue}{132} })& = & (\frac{308}{ \color{blue}{132} }x-\frac{1056}{ \color{blue}{132} }) \color{blue}{.132} \\\Leftrightarrow & 33x+36& = & 308x-1056 \\\Leftrightarrow & 33x \color{red}{+36} \color{blue}{-36} \color{blue}{-308x} & = & \color{red}{308x} -1056 \color{blue}{-308x} \color{blue}{-36} \\\Leftrightarrow & -275x& = & -1092 \\\Leftrightarrow & \frac{-275x}{ \color{red}{-275} }& = & \frac{-1092}{-275} \\\Leftrightarrow & x = \frac{1092}{275} & & \\ & V = \left\{ \frac{1092}{275} \right\} & \\\end{align}\)
10. \(\text{156 is het kleinste gemene veelvoud van 4, 13 en 3} \\ \begin{align} & \frac{x}{4}-\frac{4}{13}& = & \frac{-2}{3}x-2 \\\Leftrightarrow & \color{blue}{156.} (\frac{39x}{ \color{blue}{156} }- \frac{ 48 }{ \color{blue}{156} })& = & (\frac{-104}{ \color{blue}{156} }x-\frac{312}{ \color{blue}{156} }) \color{blue}{.156} \\\Leftrightarrow & 39x-48& = & -104x-312 \\\Leftrightarrow & 39x \color{red}{-48} \color{blue}{+48} \color{blue}{+104x} & = & \color{red}{-104x} -312 \color{blue}{+104x} \color{blue}{+48} \\\Leftrightarrow & 143x& = & -264 \\\Leftrightarrow & \frac{143x}{ \color{red}{143} }& = & \frac{-264}{143} \\\Leftrightarrow & x = \frac{-24}{13} & & \\ & V = \left\{ \frac{-24}{13} \right\} & \\\end{align}\)
11. \(\text{154 is het kleinste gemene veelvoud van 7, 11 en 2} \\ \begin{align} & \frac{x}{7}+\frac{4}{11}& = & \frac{1}{2}x-2 \\\Leftrightarrow & \color{blue}{154.} (\frac{22x}{ \color{blue}{154} }+ \frac{ 56 }{ \color{blue}{154} })& = & (\frac{77}{ \color{blue}{154} }x-\frac{308}{ \color{blue}{154} }) \color{blue}{.154} \\\Leftrightarrow & 22x+56& = & 77x-308 \\\Leftrightarrow & 22x \color{red}{+56} \color{blue}{-56} \color{blue}{-77x} & = & \color{red}{77x} -308 \color{blue}{-77x} \color{blue}{-56} \\\Leftrightarrow & -55x& = & -364 \\\Leftrightarrow & \frac{-55x}{ \color{red}{-55} }& = & \frac{-364}{-55} \\\Leftrightarrow & x = \frac{364}{55} & & \\ & V = \left\{ \frac{364}{55} \right\} & \\\end{align}\)
12. \(\text{390 is het kleinste gemene veelvoud van 5, 13 en 6} \\ \begin{align} & \frac{x}{5}+\frac{4}{13}& = & \frac{-5}{6}x+6 \\\Leftrightarrow & \color{blue}{390.} (\frac{78x}{ \color{blue}{390} }+ \frac{ 120 }{ \color{blue}{390} })& = & (\frac{-325}{ \color{blue}{390} }x+\frac{2340}{ \color{blue}{390} }) \color{blue}{.390} \\\Leftrightarrow & 78x+120& = & -325x+2340 \\\Leftrightarrow & 78x \color{red}{+120} \color{blue}{-120} \color{blue}{+325x} & = & \color{red}{-325x} +2340 \color{blue}{+325x} \color{blue}{-120} \\\Leftrightarrow & 403x& = & 2220 \\\Leftrightarrow & \frac{403x}{ \color{red}{403} }& = & \frac{2220}{403} \\\Leftrightarrow & x = \frac{2220}{403} & & \\ & V = \left\{ \frac{2220}{403} \right\} & \\\end{align}\)