System of linear equations can also be solved using the substitution method.We will show with examples.

Before you learn this lesson, make sure you understand how to solve linear equations

x + y = 20

x − y = 10

You have two equations. Pick either the first or the second equation and solve for either x or y

Since I am the one solving it, I have decided to choose the equation at the bottom (x − y = 10) and I will solve for x

x − y = 10

Add y to both sides

x − y + y = 10 + y

x = 10 + y

Since you used the equation at the bottom to solve for x, you will substitute x into the equation on top (x + y = 20)

Using x + y = 20, erase x and write 10 + y since x = 10 + y

We get 10 + y + y = 20

10 + 2y = 20

Minus 10 from both sides

10 − 10 + 2y = 20 − 10

2y = 10

Divide both sides by 2

y = 5

Now you have y, you can replace its value into either equation to get x

Replacing y into x + y = 20 gives

x + 5 = 20

Minus 5 from both sides

x + 5 − 5 = 20 − 5

x = 15

The solution to the system is x = 15 and y = 5

Indeed 15 + 5 = 20 and 15 − 5 = 10

3x + y = 10

-4x − 2y = 2

You have two equations. Pick either the first or the second equation and solve for either x or y

I have decided to choose the equation on top (3x + y = 10) and I will solve for y

3x + y = 10

Subtract 3x from both sides

3x − 3x + y = 10 − 3x

y = 10 − 3x

Since you used the equation on top to solve for y, you will substitute y into the equation at the bottom (-4x − 2y = 2)

Using -4x − 2y = 2, erase y and write 10 − 3x keeping in mind that there is a multiplication between 2 and y

We get -4x − 2 ×(10 − 3x ) = 2

-4x − 2 ×(10 − 3x ) = 2

-4x − 20 + 6x = 2 (After multiplying -2 by 10 and -2 by -3x)

2x − 20 = 2

Add 20 to both sides

2x − 20 + 20 = 2 + 20

2x = 22

Divide both sides by 2

x = 11

Now you have x, you can replace its value into either equation to get y

Replacing x into 3x + y = 10 gives

3 × 11 + y = 10

33 + y = 10

Minus 33 from both sides

33 − 33 + y = 10 − 33

y = -23

The solution to the system is x = 11 and y = -23

Indeed, 3 × 11 + -23 = 33 + -23 = 10 and -4 × 11 − 2 × -23 = -44 + 46 = 2

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