91Ó°ÊÓ

The slope of a linear function is a core component that defines its steepness. It's represented by the variable \( m \) in the linear equation \( y = mx + b \). The slope tells us how much \( y \) changes for a unit change in \( x \).

To understand slope better:

• If \( m > 0 \), the line rises as it moves from left to right.
• If \( m < 0 \), the line falls as it moves from left to right.
• A larger absolute value of \( m \) indicates a steeper slope.

To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line, use the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This will calculate the change in \( y \) over the change in \( x \), also known as "rise over run."

In our problem, having different slopes \( m_1 \) and \( m_2 \) means the two lines tilt differently on the graph. This difference is a key feature, even though they share the same \( x \)-intercept.

The \( x \)-intercept of a linear function is where the line crosses the \( x \)-axis. This point is significant because at the \( x \)-intercept, the value of \( y \) is always zero. You can find the \( x \)-intercept by setting \( y = 0 \) in the line equation and solving for \( x \).

For instance, consider the equation \( y = mx + b \). To find the \( x \)-intercept, set \( y = 0 \), which gives us:\[ 0 = mx + b \] Solving for \( x \) yields:\[ x = -\frac{b}{m} \] This formula shows the \( x \)-intercept is directly influenced by both the slope \( m \) and the \( y \)-intercept \( b \).

In the exercise, both linear functions have the same \( x \)-intercept. Thus, when the lines are set at \( y = 0 \), the same \( x \) value satisfies both equations. This shared \( x \)-coordinate value becomes a crucial piece in analyzing the relationship further.

The \( y \)-intercept in a linear function is the point at which the line crosses the \( y \)-axis. It is represented by \( b \) in the equation of the line \( y = mx + b \). The \( y \)-intercept is the value of \( y \) when \( x = 0 \).

To find the \( y \)-intercept, you simply look at the constant term \( b \) in the equation. For instance, in the equation \( y = 2x + 3 \), the \( y \)-intercept is 3. This means regardless of the slope, the line will cross the \( y \)-axis at \( y = 3 \).

In our problem, despite having different slopes, the two linear functions can only share the same \( y \)-intercept if they meet the condition of also having an \( x \)-intercept of zero. This condition is crucial because if \( x = 0 \), both lines can begin at the same point on the \( y \)-axis, making \( b_1 = b_2 \) possible.