Simultaneous Equations: Linear and Quadratic

• This chapter focuses on linear and quadratic simultaneous equations, their algebraic solutions, graphical interpretations, and linear and quadratic inequalities

Linear Simultaneous Equations

• Linear simultaneous equations involve two equations with two unknowns
• The equations share a common set of values for the unknowns that satisfy both equations

Solving Linear Simultaneous Equations

• Elimination and substitution are methods used to find the set of values that validate the equations

• Elimination requires making the coefficient of one unknown the same in both equations Example:

• Given equations:

  • 2x + y = 6
  • 6x + 2y = 24

• Multiply the first equation by 2:

  • 4x + 2y = 12

• Subtract the new equation from the second:

  • (6x + 2y) - (4x + 2y) = 24 - 12
  • 2x = 12
  • x = 6

• Substitute x = 6 into the first equation:

  • 2(6) + y = 6
  • y = -6

• Solution: x = 6 and y = -6

Quadratic Simultaneous Equations

• Involve one quadratic and one linear equation
• Require the method of substitution to solve
• May have up to two sets of solutions

Solving Quadratic Simultaneous Equations

• Steps to substitute the linear equation into the quadratic equation:
  • Rewrite linear equation 2x + y + 2 = 0 as x = -y/2 - 1
  • Substitute into quadratic equation y² + 2x = 10
  • y² + 2(-y/2 - 1) = 10
  • Simplify: y² - y - 12 = 0
  • Factorize:(y + 3)(y - 4) = 0, so y = -3 or y = 4
• Substitute y values back to find x values:
  • y = -3: x = -(-3)/2 - 1 = 3/2 - 1 = 1/2
  • y = 4: x = -4/2 - 1 = -3
• Solutions:
  • (x, y) = (1/2, -3)
  • (x, y) = (-3, 4)

Simultaneous Equations on Graphs

• Solutions to a set of simultaneous equations correspond to intersection points on a graph
• Intersection points represent shared coordinate values that satisfy both equations

Graphical Solutions

• Plot both equations on a graph

• Identify intersection points

• Example:

  • y = x² and y = x + 2
  • Point (2,4) satisfy both equations:
  • 4 = (2)² i
  • 4 = 2 + 2
  • Point (-1,1)satisfy both equations:
  • 1 = (-1)²
  • 1 = -1 + 2

• Graphs can solve simultaneous equations by plotting and finding any intersection points

Discriminant and Number of Solutions

• The discriminant of a quadratic equation (ax² + bx + c = 0) can determine the number of solutions after substitution
• This also indicates the number of intersection points on a graph

Linear Inequalities

• Involve finding the set of real numbers that validate an inequality
• Similar methods to solving linear equations apply

Solving Linear Inequalities

• Example:
  • Solve: 2x - 5 < 3x + 8 and 3x + 9 ≤ x - 5
  • 2x - 5 < 3x + 8 simplifies to x > -13
  • 3x + 9 ≤ x - 5 simplifies to x ≤ -7
• Therefore, -13 < x ≤ -7

Plotting Inequalities on a Number Line

• The overlapping region between -13 and -7 represents the solution set

Quadratic Inequalities

• Involves finding solutions to quadratic inequalities
• Need to find critical values by solving the quadratic equation
• Example:
  • Solve: 2x² - 3x - 2 > 0

Finding Critical Values

• Solve the left-hand side quadratic equation:
  • 2x² - 3x - 2 = 0
  • (2x + 1)(x - 2) = 0
  • Critical points: x = -1/2 and x = 2

Interpreting Solutions on a Graph

• Values above the x-axis correspond to 2x² - 3x - 2 > 0, giving x > 2 or x < -1/2
• Values below the x-axis correspond to 2x² - 3x - 2 < 0, giving -1/2 < x < 2

Inequalities on Graphs and Regions

• Involves interpreting functions graphically
• Example:
  • Given plots of y = x² and y = 2x + 3
  • Determine solutions to 2x + 3 > x²

Finding Intersection Points

• Equate the two equations to find the intersection points:
  • x² = 2x + 3
  • x² - 2x - 3 = 0
  • (x - 3)(x + 1) = 0
  • x = 3 and x = -1
• Intersection points are (3,9) and (-1,1)

Determining Solutions

• For 2x + 3 > x², the line y = 2x + 3 is above the curve y = x²
• The solution is -1 < x < 3

Shading Regions on Graphs

• Regions can represent areas that satisfy given linear or quadratic inequalities
• Example:
  • Shade regions satisfying x² - 8x + 15 ≤ y and y - x < 3
• If y > f(x),shade area above the curve or line
• If y ≤ f(x), shade area below the curve or line

Shading to Find Solutions

• For y - x < 3, the region satisfied is below the green dotted line
• For x² - 8x + 15 ≤ y, the region satisfied is above the red curve
• Shared region for both inequalities is illustrated by the grey shaded area

Explore linear and quadratic simultaneous equations. Learn to solve these equations algebraically and understand their graphical interpretations. The lesson also covers linear and quadratic inequalities.