Exercises: Solving linear equations over the rationals

Solve each equation.

1. a) $\frac{x}{3} = 5$
2. b) $\frac{2x}{5} = 8$
3. c) $x + \frac{1}{2} = \frac{3}{4}$

Solve each equation.

1. a) $\frac{x}{2} + 3 = 7$
2. b) $\frac{x - 1}{4} = 2$
3. c) $\frac{3x}{4} = \frac{9}{2}$

Solve each equation, giving the answer as a fraction or decimal.

1. a) $2x + 1 = 6$
2. b) $0.5x - 2 = 3$
3. c) $4x - 3 = 2x + 5$

Solve each equation.

1. a) $\frac{x}{2} + \frac{x}{3} = 10$
2. b) $\frac{2x - 1}{3} = \frac{x + 2}{2}$
3. c) $\frac{x + 3}{4} - \frac{x - 1}{6} = 1$

Solve each equation.

1. a) $3(x - \frac{1}{2}) = 2x + 4$
2. b) $\frac{1}{2}(x + 4) = \frac{1}{3}(x + 9)$
3. c) $0.2x + 1.5 = 0.5x - 0.3$

Three friends share a prize. Orla gets $\frac{1}{3}$ of the total, Pádraig gets $\frac{1}{4}$ of the total and Eoin gets the remaining €100.

1. a) Form an equation for the total prize $T$.
2. b) Solve it.
3. c) Find how much Orla and Pádraig each receive.

Solve each equation.

1. a) $\frac{3x - 2}{4} - \frac{2x + 1}{3} = \frac{1}{6}$
2. b) $\frac{5}{2}(x - 1) - \frac{3}{4}(2x + 3) = \frac{1}{2}$

A number is such that one third of it, increased by half of the number that is 4 more than it, equals 11.

1. a) Form an equation.
2. b) Solve it and check your answer.
3. c) Explain how you checked your answer.

A cyclist travels from Athlone to a town $d$ km away. Going there at $20 \text{ km/h}$ and returning at $15 \text{ km/h}$, the total time is $3\frac{1}{2}$ hours.

1. a) Form an equation in $d$ using time = distance ÷ speed.
2. b) Solve it and find $d$.
3. c) Find the average speed for the whole journey.