Solving Simultaneous Equations: Substitution

Solving simultaneous equations by substitution involves solving one equation for one of the variables and substituting that expression into the other equation. This reduces the number of variables, allowing you to solve for the remaining variable.

Once you have found the value of one variable, you can substitute that value back into one of the original equations to find the value of the other variable.

Steps:

1. Choose one of the equations and solve it for one of the variables in terms of the other variable.

2. Substitute the expression for the variable you solved for into the other equation, replacing all instances of that variable.

3. Solve the resulting equation for the remaining variable.

4. Substitute the value you found for the remaining variable back into one of the original equations to find the value of the first variable.

Some examples should make this clear

1: Solve the following system of equations by substitution:

x + y = 8

x - y = 2

From the second equation, we can solve for x in terms of y:

x = y + 2

Substituting this into the first equation, we get:

(y + 2) + y = 8

2y + 2 = 8

2y = 6

y = 3

Substituting y = 3 back into the second equation, we get:

x - 3 = 2

x = 5

Solution: x = 5, y = 3

2: Solve the following by substitution:

2x - y = 1

x + 3y = 11

From the first equation, we can solve for y in terms of x:

2x - y = 1

-y = 1 – 2x (multiply through by -1)

y = 2x – 1

Substitute into the second equation:

x + 3(2x – 1) = 11

x + 6x – 3 = 11

7x = 14

x = 2

Substitute the value of x into the first equation:

2(2) – y = 1

4 – y = 1

-y = -3

y = 3

Solution: x = 2, y = 3

3: Solve the following system of equations by using substitution:

2x + 3y = 13

x + 5y = −4

From the first equation, we can solve for x in terms of y

x = –4 – 5y

Substituting this into the first equation, we get:

2(–4 – 5y) + 3y = 13

–8 – 10y + 3y = 13

– 7y = 21

y = – 3

Substitute the value of y into the second equation:

x + 5(– 3) = −4

x – 15 = –4

x = – 4 + 15 = 11

The solution then is:

x = 11, y = – 3