Linear Equations

Worked example: an equation with mixed fractions

Solve $x - 2\frac{1}{3} = 5\frac{1}{2}$ .

Solution: Isolate $x$ by adding $2\frac{1}{3}$ to both sides:

x = 5\tfrac{1}{2} + 2\tfrac{1}{3} = \left(5+2\right) + \left(\tfrac{1}{2}+\tfrac{1}{3}\right) = 7 + \tfrac{3+2}{6} = 7\tfrac{5}{6}

One fourth of a number, added to one sixth of it, is 15. Find the number.

Solution: Let the number be $n$ :

\dfrac{n}{4} + \dfrac{n}{6} = 15

Using a common denominator of 12: $\dfrac{3n}{12} + \dfrac{2n}{12} = \dfrac{5n}{12} = 15$ , so:

n = \dfrac{15 \times 12}{5} = 36

Linear Equations

Solve $x - 2\frac{1}{3} = 5\frac{1}{2}$ .

Solution: Isolate $x$ by adding $2\frac{1}{3}$ to both sides:

x = 5\tfrac{1}{2} + 2\tfrac{1}{3} = \left(5+2\right) + \left(\tfrac{1}{2}+\tfrac{1}{3}\right) = 7 + \tfrac{3+2}{6} = 7\tfrac{5}{6}

One fourth of a number, added to one sixth of it, is 15. Find the number.

Solution: Let the number be $n$ :

\dfrac{n}{4} + \dfrac{n}{6} = 15

Using a common denominator of 12: $\dfrac{3n}{12} + \dfrac{2n}{12} = \dfrac{5n}{12} = 15$ , so:

n = \dfrac{15 \times 12}{5} = 36