Solve Simultaneous Equations By Substitution

Before viewing this page, it would be helpful to learn how to Solve Simultaneous Equations By Graphing.

The purpose of solving simultaneous equations is to find the same x-value and the same y-value that satisfies both equations. To solve, one term from one equation is substituted into the other equation.


Example One

Solve these two equations by substitution:
y = x + 6
x = –2y

Answer:
The x value and the y value are the same in both equations.
In the second equation, x is equal to –2y, so we will substitute –2y for x into the first equation.

y = x + 6
y = –2y + 6
y + 2y = 6
3y = 6
y = 2

Now, we'll find the x value by substituting y = 2 into either equation.

y = x + 6
2 = x + 6
x = 2 – 6
x = –4

The simultaneous solution for both equations is x = –4 and y = 2.


Example Two

Solve these two equations by substitution:
y = 3x – 4
x = y + 2

Answer:
The x value and the y value are the same in both equations.
In the second equation, x is equal to (y + 2), so we will substitute (y + 2) for x into the first equation.
Be careful to use brackets.

y = 3x – 4
y = 3 (y + 2) – 4
y = 3y + 6 – 4
y = 3y + 2
y – 3y = 2
–2y = 2
y = –1

Now, we'll find the x value by substituting y = –1 into either equation.
The second equation looks the easiest.

x = y + 2
x = –1 + 2
x = 1

The simultaneous solution for both equations is x = 1 and y = –1.


Example Three

Solve these two equations by substitution:
y = 2x + 1
y = x + 3

Answer:
The x value and the y value are the same in both equations.
In the first equation, y is equal to 2x + 1. In the second equation, y is equal to x + 3.
Since both are equal to y, they are equal to each other.

2x + 1 = x + 3
2x – x = 3 – 1
x = 2

Now, we'll find the y value by substituting x = 2 into either equation.
The second equation looks the easiest.

y = x + 3
y = 2 + 3
y = 5

The simultaneous solution for both equations is x = 2 and y = 5.

Question - Solve by Substitution

y = 4x + 1
x = y + 2

Answer
x = –1
y = –3