2.4 Linear equations
- Syllabus
- 2017
- Topic
- 2.4
- Level
- Higher
Learning objectives
2.4A Solving linear equationssolve linear equations with integer or fractional coefficients in one unknown, where the unknown appears on either or both sides2.4B Forming linear equationsset up simple linear equations from given data The three angles of a triangle are a°, (a + 10)°, (a + 20)°. Find the value of a
Solve linear equations systematically
A linear equation states that two expressions have equal value. Solving finds the value of the one unknown that preserves this equality.
| Structure | Reliable move |
|---|---|
| fractions present | multiply every term by a common denominator |
| brackets present | expand accurately, or divide a common factor when valid |
| unknown on both sides | collect all unknown terms on one side |
| constants on both sides | collect constants on the other side |
| ax=b | divide both sides by a |
Solve (8−2x)/3−(2x−3)/2=4. Multiply every term by 6: 2(8−2x)−3(2x−3)=24. Then 25−10x=24, so x=1/10.
Substitute the solution into both sides of the original equation, not only the simplified line. Equal results check signs, brackets and denominators.
An operation applied to only one side breaks equality. When clearing a denominator, multiply every term and preserve brackets around a multi-term numerator.
Form and solve a linear equation from data
Forming an equation translates a condition about one unknown into two equal expressions. The equality comes from a total, shared measurement or other stated relationship.
| Stage | Action |
|---|---|
| choose | define one unknown with its unit |
| express | write every related quantity in terms of it |
| connect | use the stated total or equality to form one equation |
| solve | apply a valid linear-equation method |
| interpret | calculate the requested quantity and check context |
A regular hexagon has side (x−1) cm. An isosceles triangle has equal sides (x+5) cm and base (2x−3) cm. Equal perimeters give 6(x−1)=2(x+5)+(2x−3). Hence x=6.5, so each hexagon side is 5.5 cm.
For triangle angles a, a+10 and a+20, use their total: a+(a+10)+(a+20)=180. Solve for a, then check all three angles are valid.
The solution for the chosen unknown is not always the requested answer. Return to the context, calculate the named length, count or cost, and include its unit.