Problem 37 - Algebra Form and Function

Question

The graphs of two linear functions have the same \(x\) intercept, but different slopes. Can they have the same \(y\) -intercept?

Step-by-Step Solution

Answer

Answer: Yes, two linear functions with the same x-intercept but different slopes can have the same y-intercept if their x-intercept is zero.

Step 1: Understand the problem

We have two linear functions with the same x-intercept but different slopes. We need to determine if they can have the same y-intercept.

Step 2: Write the equation of a line in slope-intercept form

The equation of a line in slope-intercept form is given by: \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.

Step 3: Analyze the equation for the x-intercept

The x-intercept is a point where the line crosses the x-axis. At this point, \(y = 0\). So, we can rewrite the equation as: \(0 = mx + b\)

Step 4: Write the equation for both linear functions

Let the two linear functions be: Function 1: \(y = m_1x + b_1\) Function 2: \(y = m_2x + b_2\) where \(m_1\) and \(m_2\) are the slopes, and \(b_1\) and \(b_2\) are the y-intercepts of the two functions respectively. Given that the functions have different slopes, \(m_1 ≠ m_2\).

Step 5: Determine the x-intercept for both functions

For the x-intercept of Function 1, we have: \(0 = m_1x + b_1\) Similarly, for the x-intercept of Function 2, we have: \(0 = m_2x + b_2\) Since both functions have the same x-intercept, their corresponding x-values are the same, so we can equate the two: \(m_1x + b_1 = m_2x + b_2\)

Step 6: Determine if the y-intercepts can be the same

Rearrange the equation to find the relationship between the y-intercepts: \(m_1x - m_2x = b_2 - b_1\) \((m_1 - m_2) x = b_2 - b_1\) Since \({m_1} ≠ {m_2}\), the term \((m_1 - m_2)\) cannot equal zero. This indicates that the only way for the y-intercepts to be the same (\(b_1 = b_2\)) is if the x-intercept is zero (\(x = 0\)). In this scenario, both functions would have the same y-intercept. So, the answer is yes, two linear functions with the same x-intercept but different slopes can have the same y-intercept if their x-intercept is zero.

Key Concepts

Slope

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.

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.

Y-intercept

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.