a) Perform the multiplications, then combine like terms.
\latex{ 3 \times (2x - 1) \;}\latex{+}\latex{\; (3 - x) \times 5 = 3 \times 2x - 3 \times 1 \;}\latex{+}\latex{\; 3 \times 5\;}\latex{ -}\latex{\;x \times 5 =} \latex{= 6x - 3 + 15 - 5x = 6x - 5x - 3 + 15 = x + 12 }
b) \latex{ y \times (y + 1)\;}\latex{ -}\latex{\; y \times (y - 1) = y \times y + y \times 1\;}\latex{ -}\latex{\; y \times y\;}\latex{ +}\latex{\; y \times 1 =}
\latex{= y^2 + y - y^2 + y = y^2 - y^2 + y + y = 2y }
c) Express the fractions with a common denominator.
\latex{ \frac{x}{3} + \frac{x}{4}-\frac{x}{5} = \frac{20x}{60} + \frac{15x}{60}-\frac{12x}{60} = \frac{20x+15x-12x}{60} = \frac{23x}{60}}
d) Express the fractions with a common denominator.
\latex{ \frac{x}{2}+\frac{x-1}{3}=\frac{3x+2\times (x-1)}{6}=\frac{3x+2x-2}{6}=\frac{5x-2}{6} }
When rewriting the fraction with a common denominator, the sum in the numerator should be placed in parentheses.
e) Rewrite the fractions with a common denominator, then combine like terms.
\latex{ \frac{5a-b}{6} - \frac{a-2b}{3} = \frac{5a-b}{6} - \frac{2\times(a-2b)}{2\times3} = \frac{5a-b-2\times(a-2b)}{6} =}
\latex{= \frac{5a-b-2a+4b}{6} = \frac{3a+3b}{6} = \frac{\overset{{1}}{\bcancel3} \times (a+b)}{\underset{2}{\bcancel{6}} } = \frac{a+b}{2} }
