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Simultaneous Equations

To solve two equations algebraically at the same time:

  1. Number the equations (1) and (2).
  2. Choose a method to eliminate a variable:
  3. Solve to find one variable.
  4. Substitute that value into one of the equations. (Often (1) )
  5. Solve to find the other variable.
  6. Check by substituting both values into the other equation. (Often (2) )
  7. Write down the solution.
Always do the check - it will show any errors !

Elimination by Substitution

Here, we eliminate a variable by substituting one equation into the other.

Example
\[ (1)\quad y = 14 - x \] \[ (2)\quad x - y = 8 \]

Substitute equation (1) into equation (2):

\[ x - (14-x) = 8 \] \[ x -14+ x= 8 \] \[ 2x -14 = 18 \] \[ 2x = 22 \] \[ x = 11 \]

Now substitute \(x\) back into equation (1):

\[ y = 14 - x \] \[ y = 14 - 11\] \[ y = 3 \]

Check using equation (2):

\[ \text{Want: } x - y = 8 \] \[ \text{Have : } 11 - 3 = 8 \qquad \checkmark \]

Check works , so \(x = 11\) and \(y = 3\)

Elimination by Addition

Here, we eliminate a variable by adding both equations.

Example
\[ (1)\quad 3x - y = 7 \] \[ (2)\quad 2x + y = 8 \]

Adding the equations eliminates \(y\):

\[ \begin{align*} (1)\quad 3x - y &= 7 \\ (2)\quad 2x + y &= 8 \\ \cdots\cdots\cdots\cdots\cdots\quad\text{(add equations)} \\[4pt] 5x &= 15 \\ x &= 3 \end{align*} \]

Substitute \(x = 3\) into equation (2):

\[ 2(3) + y = 8 \] \[ y = 2 \]

Check using equation (1):

\[ \text{Want: }3x - y = 7 \] \[ \text{Have : } 3(3) -(2) = 9 - 2 =7 \qquad \checkmark \]

Check works , so \(x = 3\) and \(y = 2\)

Elimination by Subtraction

Here, we eliminate a variable by subtracting one equation from the other.

Example
\[ (1)\quad 4x + y = 19 \] \[ (2)\quad 4x - 2y = 10 \]

Subtract equation (2) from equation (1):

\[ \begin{align*} (1)\quad 4x + y &= 19 \\ (2)\quad 4x - 2y &= 10 \\ \cdots\cdots\cdots\cdots\cdots\quad\text{(subtract equations)} \\[4pt] 3y &= 9 \\ y &= 3 \end{align*} \]

Substitute \(y = 3\) into equation (1):

\[ 4x + 3 = 19 \] \[ 4x = 16 \] \[ x = 4 \]

Check using equation (2):

\[ \text{Want: }4x -2y = 10 \] \[ \text{Have : } 4(4) -2(3) = 16 - 6 =10 \qquad \checkmark \]

Check works , so \(x = 4\) and \(y = 3\)

Elimination by Multiplying One Equation

Here, we multiply one equation to create matching coefficients.

Example
\[ (1)\quad x + 2y = 8 \] \[ (2)\quad 3x - y = 17 \]

Multiply equation (2) by 2 to get equation (3) :

\[ (3)\quad 6x - 2y = 34 \]

Add to equation (1):

\[ \begin{align*} (1)\quad x + 2y &= 8 \\ (3)\quad 6x - 2y &= 34 \\ \cdots\cdots\cdots\cdots\cdots\quad\text{(add equations)} \\[4pt] 7x &= 42 \\ x &= 6 \end{align*} \]

Substitute into equation (1):

\[ \begin{align*} x + 2y &= 8 \\ 6 + 2y &= 8 \\ 2y &= 8 -6 \\ 2y &= 2 \\ y &= 1 \end{align*} \]

Check using equation (2):

\[ \text{Want: }3x - y = 17 \] \[ \text{Have : } 3(6) -(1) = 18 - 1 =17 \qquad \checkmark \]

Check works , so \(x = 6\) and \(y = 1\)

Elimination by Multiplying Both Equations

Here, we multiply both equations to create matching coefficients.

Example
\[ (1)\quad 2x + 3y = 7 \] \[ (2)\quad 3x - 2y = 4 \]

Multiply equation (1) by 2 to get equation (3) :

Multiply equation (2) by 3 to get equation (4)

\[ (3)\quad4x + 6y = 14 \] \[ (4)\quad9x - 6y = 12 \]

Add the equations:

\[ \begin{align*} 4x + 6y &= 14 \\ 9x - 6y &= 12 \\ \cdots\cdots\cdots\cdots\cdots\quad\text{(add equations)} \\[4pt] 13x &= 26 \\ x &= 2 \end{align*} \]

Substitute into equation (1):

\[ 2(2) + 3y = 7 \] \[ 4 + 3y = 7 \] \[ 3y = 3 \] \[ y = 1 \]

Check using equation (2):

\[ \text{Want: }3x - 2y = 4 \] \[ \text{Have : } 3(2) - 2(1) = 6 -2 = 4 \qquad \checkmark \]

Check works , so \(x = 2\) and \(y = 1\)

For solving graphically, click here

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