1. Linear Patterns:
A linear pattern is a pattern in which numbers increase or decrease by the same constant value for each time.
A linear pattern is denoted in the form of \(y = ax + b\).
Example:
Observe the table to find the value of y.
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) |
| \(y\) | \(5\) | \(8\) | \(11\) | \(14\) | \(17\) | \(?\) |
When \(x = 1\), \(y = 5 = 3 \times 1 + 2\)
When \(x = 2\), \(y = 8 = 3 \times 2 + 2\)
When \(x = 3\), \(y = 11 = 3 \times 3 + 2\)
When \(x = 4\), \(y = 14 = 3 \times 4 + 2\)
When \(x =5\), \(y = 17 = 3 \times 5 + 2\)
In above term each term has a difference of \(3\)
Using the above pattern we get,
When \(x = 6\), \(y = 3 \times 6 + 2 = 20\)
The linear pattern formed using the above table is,
\(y = 3x + 2\)
2. Linear equations:
When a linear polynomial is equal to \(0\) is called linear equations. And the value which statisfies that equation is called its root or zero.
Example:
\(ax + by + c = 0\) where \(a \neq 0\), \(b \neq 0\)