Finding and Analyzing Linear Equations

Finding the Equation of a Straight Line

• Slope-intercept form: y = mx + c, where m is the slope and c is the y-intercept.
• Slope: (change in y) / (change in x) = (6 - 3) / (8 - 2) = 3/6 = 1/2.
• Using point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the line.
  • Choosing point A (2,3): y - 3 = (1/2)(x - 2)
  • Simplifying to slope-intercept form: y = (1/2)x + 2.

Determining the Intersection Point of Two Linear Equations

• Substitution method:
  • Solve one equation for either x or y.
  • Substitute the expression into the other equation.
  • Solve for the remaining variable.
  • Substitute the value back into one of the original equations to find the other variable.
• Method used for the given equations:
  • Solving the first equation for x: 4x = -8 - 2y => x = -2 - (1/2)y.
  • Substituting this expression for x into the second equation: 4(-2 - (1/2)y) - 3y = 4.
  • Simplifying and solving for y: -8 - 2y - 3y = 4 => -5y = 12 => y = -12/5.
  • Substituting y = -12/5 back into the equation x = -2 - (1/2)y: x = -2 - (1/2)(-12/5) => x = -2 + 6/5 => x = -4/5.
• Intersection point: (-4/5, -12/5).