Simultaneous equations

• What are linear simultaneous equations?

  They are equations with two unknowns that require two equations to solve.

  
• Why do we need two equations to find two unknowns in linear simultaneous equations?

  Because each equation provides a constraint that helps determine the values of the unknowns.

  
• What is an example of linear simultaneous equations?


  3x + 2y = 11 and 2x - y = 5

  
• What are the solutions for the equations 3x + 2y = 11 and 2x - y = 5?


  x = 3 and y = 1

  
• What defines linear simultaneous equations in terms of variables?

  They only contain the variables x and y without any powers or products of these variables.

  
• What does the elimination method do in solving simultaneous equations?


  It completely removes one of the variables from the equations.
• How do you eliminate x's from the equations 3x + 2y = 11 and 2x - y = 5?

  Multiply the first equation by 2 and the second by 3 to make the coefficients of x equal.

  
• What is the result of multiplying the first equation 3x + 2y = 11 by 2?

  6x + 4y = 22

  
• What do you get when you multiply the second equation 2x - y = 5 by 3?

  6x - 3y = 15

  
• What do you do after obtaining 6x + 4y = 22 and 6x - 3y = 15?

  Subtract the second equation from the first to eliminate x.

  
• What is the equation after subtracting 6x - 3y = 15 from 6x + 4y = 22?

  7y = 7

  
• What is the value of y after solving 7y = 7?

  y = 1

  
• How do you find the value of x after finding y = 1?

  Substitute y = 1 back into either original equation.

  
• What is the alternative method to solve linear simultaneous equations besides elimination?

  Substitution
• How do you use substitution to solve the equations 3x + 2y = 11 and 2x - y = 5?


  Rearrange one equation to express y in terms of x and substitute it into the other equation.

  
• What does the second equation 2x - y = 5 become when rearranged into y = ...?

  y = 2x - 5

  
• What is the first step after substituting y = 2x - 5 into the first equation 3x + 2y = 11?

  Replace all y's in the first equation with (2x - 5).

  
• What equation do you get after substituting y = 2x - 5 into 3x + 2y = 11?

  3x + 2(2x - 5) = 11

  
• What is the solution for x after solving the equation 3x + 2(2x - 5) = 11?

  x = 3

  
• What is the value of y after substituting x = 3 into y = 2x - 5?

  y = 1

  
• How do you use graphs to solve linear simultaneous equations?

  Plot both equations on the same axes and find the point of intersection.
• What do you need to find when plotting the equations on a graph?

  The point of intersection of the two lines.
• What is an example of finding the intersection of two equations graphically?

  Finding the intersection of 2x - y = 3 and 3x + y = 7.

  
• What is the solution for the equations 2x - y = 3 and 3x + y = 7?

  x = 2 and y = 1

  
• What should you always do after finding solutions to simultaneous equations?

  Check that the final solutions satisfy the original equations.
• How do you solve the simultaneous equations 5x + 2y = 11 and 4x - 3y = 18?


  Number the equations and make the y terms equal by multiplying.

  
• What do you do after numbering the equations 5x + 2y = 11 and 4x - 3y = 18?


  Make the y terms equal by multiplying the first equation by 3 and the second by 2.

  
• What happens when you add the two equations after making the y terms equal?

  The 6y terms can be eliminated.

  
• How do you find the value of x after eliminating the y terms?

  Divide both sides of the resulting equation by the coefficient of x.

  
• What defines quadratic simultaneous equations?


  They include at least one variable raised to a power or a product of the variables.
• How do you solve quadratic simultaneous equations?

  Use substitution by replacing one variable in the quadratic equation with the linear equation.
• What is the first step in solving the equations x + y = 25 and y - 2x = 5?

  Rearrange the linear equation into y = 2x + 5.

  
• What do you do after rearranging y - 2x = 5 into y = 2x + 5?

  Substitute this into the quadratic equation x + (2x + 5) = 25.

  
• What is the resulting equation after substituting y = 2x + 5 into x + y = 25?

  x + (2x + 5) = 25

  
• What do you do after obtaining the equation x + (2x + 5) = 25?

  Expand and solve the quadratic equation.

  
• What are the solutions for the quadratic equation x + (2x + 5) = 25?

  x = 0 and x = -4

  
• How do you find the corresponding y values for each x value after solving?

  Substitute each x value into the linear equation y = 2x + 5.

  
• What are the final solutions for the quadratic simultaneous equations x + y = 25 and y - 2x = 5?

  x = 0, y = 5 or x = -4, y = -3

  
• What should you do if the resulting quadratic has a repeated root?

  Indicate that the line is a tangent to the curve.
• What does it mean if the resulting quadratic has no roots?

  It means the line does not intersect with the curve or there was a mistake.