OBJECTIVE To write linear equations in the form y = ax + b and hence fi nd the x and y intercepts
A linear equation is an equation that can be written in the form ax 1 + b = 0, where a ≠ 0, b are fi xed numbers. a is called the coeffi cient of x. b is called the constant term.
An equation is linear if the highest power of the variable in the equation is 1. Method of solution
ax + b = 0 ax = –b x = – b
__ a
There is one, and only one, solution (root) of a linear equation. Every linear equation has one root only.
3x + 2 = 0 3x = – 2 x = – 2
WORKED EXAMPLE
_ 3
Graphical solution of a linear equation
(a) y = f (x) = ax + b is the equation of a straight line. (b) y = 0 is the equation of the x-axis. The solution of ax + b = 0 is the value of x at which the
straight line crosses the x-axis. Its value is x = – b __
Find where y = 4x – 8 crosses the x-axis:
4x – 8 = 0 4x = 8 x = 8
0
_ 4 = 2
y = 4x – 8 crosses the x-axis at (2, 0).
In general, fi nding the points at which a curve y = f (x) crosses the x-axis means exactly the same thing as solving the equation f (x) = 0 or fi nding the roots of the equation f (x) = 0.
Steps for solving linear equations
1. Clear all fractions by multiplying all terms on each side by a common denominator of all terms.
2. Multiply out all brackets. 3. Tidy up each side by combining like terms. 4. Get the variable (usually x) terms on one side and the numbers on the other side. 5. Solve for the variable.
