Algebra 10: Line Equations from Points

Finding a Line Equation from Two Points

• A line's equation is typically written as (y = mx + b), where:
  • (m) is the slope.
  • (b) is the y-intercept (value of (y) when (x = 0)).

Calculating the Slope

• Given two points ((x_1, y_1)) and ((x_2, y_2)), the slope (m) is calculated using the formula: [ m = \frac{y_2 - y_1}{x_2 - x_1} ]
  • This represents the ratio of the change in (y) to the change in (x) between the two points.

Point-Slope Formula

• The point-slope form of a line equation is: [ y - y_1 = m(x - x_1) ]
  • Substitute the calculated slope ((m)) and the coordinates of one given point ((x_1, y_1)) into this equation.

Converting to Slope-Intercept Form

• Simplify the point-slope equation to isolate (y) and express it in the (y = mx + b) form.

Finding the Y-Intercept

• If needed, solve for (b) (the y-intercept) by substituting the calculated slope ((m)) and the coordinates of a given point into the equation: [ b = y - mx ]

Example

• Given points ((2, 3)) and ((4, 7)), calculate the slope: [ m = \frac{7 - 3}{4 - 2} = 2 ]
• Use the point-slope formula with point ((2, 3)): [ y - 3 = 2(x - 2) ]
• Simplify to slope-intercept form: [ y = 2x - 1 ]
  • Therefore, the line's equation is (y = 2x - 1).

Other Important Forms

• Point-Slope Form: (y - y_1 = m(x - x_1))
• Standard Form: (Ax + By = C), where A, B, and C are integers.

This quiz covers the concepts of finding a line equation from two points in algebra. You will learn how to calculate the slope, use the point-slope formula, and convert equations into the slope-intercept form. Test your understanding of these fundamental algebraic principles!