Simultaneous Equations Notes

Simultaneous Equations

• Definition: Two equations with two unknowns, usually $x$ and $y$.
• Solving: Finding values for $x$ and $y$ that satisfy both equations simultaneously.

Elimination Method

• Basic Principle: Manipulate the equations to eliminate one variable, allowing you to solve for the other.

Same Number of x's or y's

Different Signs: ADD the Equations

• Example:
  $2x - y = 1$
  $3x + y = 9$
• Adding the equations eliminates $y$:
  $5x = 10$
• Solve for $x$:
  $x = 2$
• Substitute $x = 2$ into equation A:
  $2(2) - y = 1$
  $4 - y = 1$
  $y = 3$
• Answer: $x = 2$, $y = 3$

Same Signs: SUBTRACT the Equations

• Example:
  $5x + y = 17$
  $3x + y = 11$

• Subtracting the equations eliminates $y$:
  $2x = 6$

• Solve for $x$:
  $x = 3$

• Substitute $x = 3$ into equation A:
  $5(3) + y = 17$
  $15 + y = 17$
  $y = 2$

• Answer: $x = 3$, $y = 2$

• Example:
  $2x + 3y = 9$
  $2x + y = 7$

• Subtracting the equations eliminates $x$:
  $2y = 2$

• Solve for $y$:
  $y = 1$

• Substitute $y = 1$ into equation A:
  $2x + 3(1) = 9$
  $2x = 6$
  $x = 3$

• Answer: $x = 3$, $y = 1$

• Example:
  $4x - 3y = 14$
  $2x + 3y = 16$

• Adding the equations eliminates $y$:
  $6x = 30$

• Solve for $x$:
  $x = 5$

• Substitute $x = 5$ into equation A:
  $4(5) - 3y = 14$
  $20 - 3y = 14$
  $3y = 6$
  $y = 2$

• Answer: $x = 5$, $y = 2$

Elimination Method: Basic Steps

• Look at the equations.
• Check if the number of $x$'s or $y$'s is the same in both equations.
• If signs are different, ADD the equations; otherwise, SUBTRACT.
• Solve the resulting equation (with one variable).
• Substitute the obtained value back into one of the original equations to solve for the other variable.
• CHECK by substituting both answers into both original equations.

Elimination Method: Different Number of x's or y's

Multiplying One Equation

• If the number of $x$'s or $y$'s is not the same, multiply one equation to make them the same.

• Example:
  $5x + 2y = 17$
  $3x + y = 10$

• Multiply the second equation (B) by 2:
  $6x + 2y = 20$

• Now the equations are:
  $5x + 2y = 17$
  $6x + 2y = 20$

• Subtract equation A from the modified equation B:
  $x = 3$

• Substitute $x = 3$ into equation A:
  $5(3) + 2y = 17$
  $15 + 2y = 17$
  $2y = 2$
  $y = 1$

• Answer: $x = 3$, $y = 1$

• Example:
  $3x + 6y = 21$
  $4x - 2y = 8$

• Multiply the second equation (B) by 3:
  $12x - 6y = 24$

• Now the equations are:
  $3x + 6y = 21$
  $12x - 6y = 24$

• Add equations A and the modified B:
  $15x = 45$

• Solve for $x$:
  $x = 3$

• Substitute $x = 3$ into equation A:
  $3(3) + 6y = 21$
  $9 + 6y = 21$
  $6y = 12$
  $y = 2$

• Answer: $x = 3$, $y = 2$

Multiplying Both Equations

• If multiplying one equation doesn't directly help, multiply both equations to get the same number of either $x$'s or $y$'s.

• Example:
  $3x + 7y = 26$
  $5x + 2y = 24$

• Multiply equation A by 5 and equation B by 3 to get 15x in each:
  $15x + 35y = 130$
  $15x + 6y = 72$

• Subtract the second equation from the first:
  $29y = 58$

• Solve for $y$:
  $y = 2$

• Substitute $y = 2$ into equation B:
  $5x + 2(2) = 24$
  $5x + 4 = 24$
  $5x = 20$
  $x = 4$

• Answer: $x = 4$, $y = 2$

• Alternatively, multiply equation A by 2 and equation B by 7 to get 14y in each.

• Example:
  $3x - 2y = 7$
  $5x + 3y = 37$

• Multiply equation A by 3 and equation B by 2:
  $9x - 6y = 21$
  $10x + 6y = 74$

• Add the two equations:
  $19x = 95$

• Solve for $x$:
  $x = 5$

• Substitute $x = 5$ into equation B:
  $5(5) + 3y = 37$
  $25 + 3y = 37$
  $3y = 12$
  $y = 4$

• Answer: $x = 5$, $y = 4$

• Alternatively, multiply equation A by 5 and equation B by 3 to get 15x in each. Note the importance of signs when adding or subtracting the equations.