Linear Equations

Intercepts of a Line

An intercept is the point where a line crosses an axis on a coordinate plane.

1. Y-intercept

   • This is the point where the line crosses the y-axis.

   • At the y-intercept, the x-coordinate is always $0$. Its coordinates are $(0, y)$.

   • To find the y-intercept, substitute $x=0$ into the equation of the line and solve for $y$.

2. X-intercept

   • This is the point where the line crosses the x-axis.

   • At the x-intercept, the y-coordinate is always $0$. Its coordinates are $(x, 0)$.

   • To find the x-intercept, substitute $y=0$ into the equation of the line and solve for $x$.

Slope of a Line

Slope measures the steepness and direction of a line. It is often referred to as "rise over run" and is denoted by the letter $m$.

• Formula for slope: Given two distinct points $(x<em>1, y</em>1)$ and $(x<em>2, y</em>2)$ on a line, the slope $m$ is calculated using the formula:
  $m = \frac{\text{change in y}}{\text{change in x}} = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

• Interpretation of Slope:

  • Positive Slope (m > 0): The line rises as you move from left to right.

  • Negative Slope (m < 0): The line falls as you move from left to right.

  • Zero Slope ($m = 0$): The line is horizontal.

  • Undefined Slope: The line is vertical ($x<em>2 - x</em>1 = 0$).

Linear Equations

These concepts are fundamental to understanding and writing linear equations.

• Slope-intercept form: A common way to express a linear equation is $y = mx + b$, where:

  • $m$ is the slope of the line.

  • $b$ is the y-intercept (the y-coordinate when $x=0$).

• Point-slope form: If you know the slope $m$ and one point $(x<em>1, y</em>1)$ on the line, you can use the point-slope form to write the equation:
  $y - y<em>1 = m(x - x</em>1)$
  This form is particularly useful for finding the equation of a line when given a point and the slope, as it directly incorporates these values.