Solve Simultaneous Equations By Substitution

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.

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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.

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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.

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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.