The substitution method involves expressing one variable in terms of the other, then replacing it in the second equation.

Steps for Substitution Method

1. Choose one equation and solve for one variable in terms of the other
2. Substitute this expression into the second equation
3. Solve the resulting equation for one variable
4. Substitute this value back to find the other variable
5. Write the solution as an ordered pair $(x, y)$

Worked Example 3: Substitution

Solve: $y = 2x + 1$ $x + y = 10$

Solution

The first equation already has $y$ expressed in terms of $x$. Substitute this into the second equation:

$x + (2x + 1) = 10$

$3x + 1 = 10$

$3x = 9$

$x = 3$

Now substitute $x = 3$ into $y = 2x + 1$:

$y = 2(3) + 1 = 7$

Answer: $x = 3$, $y = 7$

Worked Example 4: Rearranging First

Solve: $x + y = 12$ $2x - y = 3$

Solution

From the first equation, express $x$ in terms of $y$:

$x = 12 - y$

Substitute this into the second equation:

$2(12 - y) - y = 3$

$24 - 2y - y = 3$

$24 - 3y = 3$

$-3y = 3 - 24$

$-3y = -21$

$y = 7$

Substitute $y = 7$ into $x = 12 - y$:

$x = 12 - 7 = 5$

Answer: $x = 5$, $y = 7$

Always check your solution by substituting both values into both original equations. If both equations are satisfied, your answer is correct.

• Use elimination when the coefficients of one variable are the same or can easily be made the same.
• Use substitution when one equation already has a variable isolated, or can be easily rearranged to isolate one variable.

Both methods will give you the same correct answer. Practice both to become comfortable with each.

Simultaneous equations are used in everyday life when we need to find two unknown values at the same time. For example, if a student buys 3 exercise books and 2 pens from a shop in Dar es Salaam for TSh 15,000, and another student buys 2 exercise books and 3 pens for the same total amount, we can set up two equations and solve them to find the price of each item. This helps in budgeting and comparing prices when shopping.