math

Slope–Intercept Form and Solving for yy

  • Core form: y=mx+by = m x + b where
    • mm = slope (rate of change)
    • bb = yy-intercept (value when x=0x = 0)
  • To “solve for yy” in any linear equation:
    1. Isolate the yy-term on one side.
    2. Move all other terms to the opposite side using inverse operations (add/subtract).
    3. Divide every term by the coefficient of yy to leave yy alone.
  • Example walkthrough (from pizza/water problem):
    • Original: 2x+3y=7.252x + 3y = 7.25
    • Subtract 2x2x: 3y=−2x+7.253y = -2x + 7.25
    • Divide by 33: y=−23x+7.253y = -\tfrac{2}{3}x + \tfrac{7.25}{3}
    • Numeric intercept: y≈−0.667x+2.4167y \approx -0.667x + 2.4167
  • Purpose: slope-intercept form is preferred by handheld TI calculators; Desmos or similar graphing tools will graph any linear form without rearranging.

Graphing vs. Algebraic Methods

  • Graphing (Desmos/TI):
    • Enter each original equation exactly as written.
    • Let software find intersection point(s) → simultaneous solution (x,y)(x, y).
    • Advantage: no algebraic rearrangement needed; visual verification.
  • Algebraic (by hand or on non-graphing portions of tests):
    • Substitution: solve one equation for a variable, plug into the other.
    • Elimination: add/subtract entire equations to cancel one variable, solve for the other.
    • Useful for SAT/ACT when graphing tech is restricted.

Example 1 – Pizza Slices & Bottled Water

  • Scenario: Two friends buy different combinations, but slices and bottles cost the same in both trips.
    • Abby