91Ó°ÊÓ

In mathematics, a linear relationship is one where two variables, such as \(x\) and \(y\), change at a constant rate. If you plot these variables on a graph, you will get a straight line. This means that for every increment in \(x\), \(y\) will change by the same amount.
When working with linear relationships, you might come across tables that list pairs of \(x\) and \(y\) values. The key to identifying a linear relationship in such a table is to look for consistent differences. This means checking if both \(x\) and \(y\) change by fixed amounts.
Let's take a look at an example:
\begin{array}{|c|c|c|c|c|c|}\text{x} & {0} & {1} & {2} & {3} & {4} \ \text{y} & {4.1} & {13.1} & {22.1} & {31.1} & {40.1} \ \text{-} & \text{-} & \text{-} & \text{-} & \text{-} \ \text{-} & \text{-} & \text{-} & \text{-} & \text{-} \texamine differences} \ \text{x differences} & {1} & {1} & {1} & {1} \ \text{y differences} & {9} & {9} & {9} & {9} \ end{array} As shown, the differences in \(x\) are always 1, and the differences in \(y\) are a consistent 9. This regular pattern indicates a linear relationship.

The slope-intercept form is a way of writing linear equations. It is written as
\[\begin{equation} y = mx + b \end{equation}\]
Here, \(m\) is the slope (the rate of change), and \(b\) is the y-intercept (the value of \(y\) when \(x = 0\)).
In the example table, the differences in \(x\) and \(y\) tell us that \(m = 9\). To find \(b\), we look at the point where \(x = 0\):

\[\begin{equation} y = 9x + b \end{equation}\]\,
Substitution,\ \( y = 4.1\) when \( x = 0 \) leads to:

\[\begin{equation} 4.1 = 9 \cdot 0 + b \ \rightarrow b = 4.1 \end{equation}\]. This gives the line equation:

\[\begin{equation} y = 9x + 4.1 \end{equation}\]

Detecting consistent differences is crucial in identifying linear relationships. When examining a table of values:

• Check the differences between consecutive \( x \) values. They should be constant.


• Compute the differences between consecutive \( y \) values. If these differences are also constant, the relationship likely exhibits a linear pattern.

Observing these consistent differences reveals that the rate of change (slope) between \( x \) and \( y \) remains unchanged. This uniformity is a telltale sign of a linear relationship. Hence, a table with these traits represents a linear equation.
In the given example:

\begin{\tabular{ |c|c|c|c|c|} x & { 0 \} & 1 & 2 & { 3 \} & { 4 \} \ & & & & y & 4.1 & \text { 13.1} & \text { 22.1} & 31.1 & 40.1 \ \text { x-differences} & 1 & \text { --} & \text { --} & 1 & \text { --} & 1 & \text { --} & 1 & \text { --} & 1 & \text { --} & 1 & \text { --} y-differences} & 9 & \text { --} & \text { --} & 9 & \text { --} 9 \text { 9 & 9. }\text{ & \text { 9 \}}

This confirms the consistent differences, signifying that the data in the table satisfies this crucial aspect of linear relationships.