Systems of Equations: Slopes and Intercepts

Systems of Equations: Slopes and y-intercepts

Comparing Slopes and y-intercepts

• Same Slope, Different y-intercepts: Parallel lines, no solution.
• Same Slope, Same y-intercept: Same line, infinitely many solutions.
• Different Slopes, Different y-intercepts: Intersecting lines, one solution.
• Different Slopes, Same y-intercept: Intersecting line, one solution.

Slope-intercept form

• The slope-intercept form of a linear equation is given by: $y = mx + b$, where $m$ is the slope, and $b$ is the y-intercept.

Slope Formula

• The formula to find the slope $m$ between two points $(x<em>1, y</em>1)$ and $(x<em>2, y</em>2)$ is: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$.

Determining the Number of Solutions

• No Solution: Lines have the same slope but different y-intercepts.
• One Solution: Lines have different slopes.
• Infinitely Many Solutions: Lines have the same slope and the same y-intercept.