Simultaneous equations are two equations that are both true at the same time. Solving them means finding the values of the variables that satisfy both equations together.

 

This topic focuses on solving two linear equations with whole number coefficients using algebraic methods.

 

 

Understanding Simultaneous Equations

A pair of simultaneous equations might look like:

$$
x + y = 7
$$

$$
2x - y = 5
$$

 

The solution is the pair of values for \( x \) and \( y \) that make both equations true.

 

 

Solving Simultaneous Equations by Elimination

The elimination method involves adding or subtracting the equations to eliminate one variable.

 

Consider:

$$
x + y = 7
$$

$$
2x - y = 5
$$

 

Add the two equations to eliminate \( y \):

$$
(x + y) + (2x - y) = 7 + 5
$$

$$
3x = 12
$$

 

Divide by \( 3 \):

$$
x = 4
$$

 

Substitute \( x = 4 \) into one of the original equations:

$$
4 + y = 7
$$

$$
y = 3
$$

 

So the solution is \( x = 4 \), \( y = 3 \).

 

 

Elimination When Coefficients Are Different

Sometimes one variable must be made the same in both equations first.

 

For example:

$$
2x + y = 11
$$

$$
3x + y = 16
$$

 

Subtract the first equation from the second:

$$
(3x + y) - (2x + y) = 16 - 11
$$

$$
x = 5
$$

 

Substitute \( x = 5 \) into the first equation:

$$
2(5) + y = 11
$$

$$
10 + y = 11
$$

$$
y = 1
$$

 

So the solution is \( x = 5 \), \( y = 1 \).

 

 

Solving Simultaneous Equations by Substitution

The substitution method involves making one variable the subject and substituting it into the other equation.

 

For example:

$$
y = x + 2
$$

$$
2x + y = 11
$$

 

Substitute \( y = x + 2 \) into the second equation:

$$
2x + (x + 2) = 11
$$

$$
3x + 2 = 11
$$

$$
3x = 9
$$

$$
x = 3
$$

 

Substitute \( x = 3 \) back:

$$
y = 3 + 2
$$

$$
y = 5
$$

 

So the solution is \( x = 3 \), \( y = 5 \).

 

 

Checking the Solution

Always check the solution by substituting the values into both original equations to ensure they are satisfied.

 

 
Key Points to Remember

Simultaneous equations must be solved together.
Use elimination or substitution to remove one variable.
Work step by step and keep equations balanced.
Always check the final solution.

 

Solving simultaneous linear equations is an important algebraic skill used widely in mathematics and real world problem solving.