# Simultaneous equations - the substitution method

To solve two equations with two unknowns. Substitute the y  in one equation into the y  of the other.

Substitute the y

Example 1

Solve             x+y=24  ..................... equation 1

2x-y=-6  ................ equation 2

Take the first equation

x+y=24  ...................... equation 1

Subtract x  from both sides to get y  on its own.

-x+x+y=24-x

y=24-x

Now substitute this into equation 2:

2x-y=-6

2x-1times(24-x)=-6

2x-24+x=-6

3x-24=-6

Add 24 to both sides to get x  on its own.

3x-24+24=-6+24

3x=18

Divide both sides by 3 to get x  on its own

(3x)/3=18/3

x=6

Now that x=6  we can plug this into either equation 1 or 2 to get the value of y

Try                x+y=24

6+y=24

Subtract 6 from both sides to get the value of y

-6+6+y=24-6

y=18

Solution is x=6     y=18

Example 2

Solve            2x+3y=6 ...................... equation 1

4x-6y=-4 ................ equation 2

NOTE:

This is the same question as simultaneous equation graphical.

Take the first equation

2x+3y=6 ...................... equation 1

Subtract 2x  from both sides to get y  on its own.

2x-2x+3y=6-2x

3y=6-2x

Divide both sides by 3 to get y  on its own.

y=6/3-(2x)/3

y=2-2/3x

Now substitute this into equation 2:

4x-6y=-4

4x-6(2-2/3x)=-4

4x-6times2-6times(-2/3)x=-4

4x-12+12/3x=-4

4x-12+4x=-4

8x-12=-4

Add 12 to both sides to get x  on its own.

12+8x-12=-4+12

8x=8

Divide both sides by 8 to get x  on its own

x=8/8

x=1

Now that x=1  we can plug this into either original equation 1 or 2 to get the value of y

Try                4x-6y=-4

4times1-6y=-4

4-6y=-4

Subtract 4 from both sides to get the value of y

4-4-6y=-4-4

-6y=-8

Divide both sides by -6 to get the value of y

(-6y)/-6=(-8)/-6

y=8/6

y=4/3

y=1\1/3

Solution is x=1     y=1\1/3

Example 3

Solve            y=x+3  ............... equation 1

x+y=7 ................ equation 2

NOTE:

This is the same question as simultaneous equation graphical.

Take the first equation

y=x+3 ................ equation 1

and substitute it into equation 2.

x+y=7

x+(x+3)=7

2x+3=7

Subtract 3  from both sides to get y  on its own.

-3+2x+3=7-3

2x=7-3

2x=4

Divide both sides by 2 to get x  on its own.

(2x)/2=4/2

x=4/2

x=2

Now that x=2  we can plug this into either equation 1 or 2 to get the value of y

Try                x+y=7

2+y=7

Subtract 2 from both sides to get a value of y

2-2+y=7-2

y=7-2

y=5

Solution is x=2     y=5

Example 4

Solve            2x+2y=6  .................. equation 1

4x-6y=12  ................ equation 2

NOTE:

This is the same question as simultaneous equation graphical.

Take the first equation

2x+2y=6  .................. equation 1

Subtract 2x from both sides to get y on its own

-2x+2x+2y=6-2x

2y=6-2x

Divide both sides by 2 to get y  on its own.

(2y)/2=6/2-(2x)/2

y=3-x

Now substitute this into equation 2

4x-6y=12

4x-6(3-x)=12

4x-6times3-(6)times(-x)=12

4x-6times3-(-6x)=12

4x-18+6x=12

10x-18=12

Add 18 to both sides to get x  on its own.

10x-18+18=12+18

10x=12+18

10x=30

Divide both sides by 10 to get x  on its own.

(10x)/10=30/10

x=30/10

x=3

Now that x=3  we can plug this into either original equation 1 or 2 to get the value of y

Try                4x-6y=12

4times(3)-6y=12

12-6y=12

Subtract 12 from both sides to get a value of y

12-12-6y=12-12

-6y=12-12

-6y=0

Divide both sides by -6  to get y  on its own.

(-6y)/-6=0/-6

y=0/-6

y=0

Solution is x=3     y=0