Question

What can you say about two lines that have equal slopes and equal \(y\) -intercepts?

Short answer

The lines are the same.

Step-by-step solution

Identify the Equation of a Line

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.

Compare Slopes

Two lines with the same slope are parallel. If the slopes \( m_1 \) and \( m_2 \) of the two lines are equal, then: \[ m_1 = m_2 \]

Compare Y-Intercepts

If the two lines also have the same \( y \)-intercept \( b \), then: \[ b_1 = b_2 \]

Determine Line Properties

When two lines have both the same slope and the same \( y \)-intercept, their equations are identical. This means the two lines are not just parallel; they are the same line.

Key concepts

Equation of a Line

To start understanding lines in geometry, we need to grasp the concept of the equation of a line. The slope-intercept form is one of the most common ways to write a line equation. It is written as: \[ y = mx + b \] Where:

• \( y \) is the dependent variable or the output value.
• \( x \) is the independent variable or the input value.
• \( m \) represents the slope, which indicates how steep the line is.
• \( b \) is the y-intercept, showing where the line crosses the y-axis.

Understanding each part of this equation is key, as it allows us to define how a line behaves and looks on a graph.

Parallel Lines

Parallel lines are an interesting concept in geometry. These lines never intersect, no matter how far they extend on the plane. For lines to be parallel, they must have the same slope. This means in their slope-intercept form: \[ y = m_1x + b_1 \] and \[ y = m_2x + b_2 \] We have that \( m_1 = m_2 \). This is the condition for parallelism. The different y-intercepts \( b_1 \) and \( b_2 \) ensure the lines do not overlap.

Identical Lines

Identical lines are essentially the same line on a graph. They overlap completely and are indistinguishable from one another. For two lines to be identical, they need to have not just the same slope but also the same y-intercept. In equation form, if \[ y = m_1x + b_1 \] and \[ y = m_2x + b_2 \], then \( m_1 = m_2 \) and \( b_1 = b_2 \). This means that every corresponding point on one line is also a point on the other line. Identical lines will coincide at all points, effectively making them the same line.