91Ó°ÊÓ

The slope of a linear equation is a crucial concept to understand. It indicates how steep a line is and the direction it’s going. In the equation given in the exercise, which is written as \( y = 3x - 3 \), the slope \( m \) is the coefficient of \( x \), which is 3. This means for every increase of 1 in \( x \), the \( y \)-value increases by 3. The slope can be positive, negative, or zero:

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A zero slope means the line is flat.

In our case, the slope is 3, showing a line rising upward at a steady rate.

The constant difference refers to the consistent change in the \( y \)-values when the \( x \)-values are adjusted by a fixed amount. It's directly tied to the slope in a linear equation. Here, the slope \( m = 3 \) tells us about the constant difference:

• When \( x \) increases by 1, \( y \) increases by the slope amount, which is 3.
• From the second part of the exercise, if \( x \) decreases by 2, the \( y \)-value changes by \( 3 \times (-2) = -6 \). Thus, the \( y \)-values decrease by 6.

The constant difference helps us quickly understand how the \( y \)-values react to changes in \( x \). This makes it easy to predict how the line moves.

In a linear equation like \( y = 3x - 3 \), the \( y \)-values are determined by the equation, based on the given \( x \)-values. The \( y \text{-values} \) depend on the slope and the starting point (or y-intercept). Let’s break it down:
1. Starting Point: When \( x = 0 \), we get the y-intercept by solving for \( y \text{-values} \): \( y = 3(0) - 3 = -3 \). So, the line crosses the y-axis at -3.
2. Using the Slope: When you know the slope and move along the \( x \)-axis, you can find the corresponding \( y \)-values. For example, if \( x \text{increased} \) by 1, \( y = 3(1) - 3 = 0 \). If \( x = -2 \), \( y = 3(-2) - 3 = -9 \).
By increasing or decreasing \( x \), keeping the slope in mind will tell you the exact behavior of the \( y \)-values.