# 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`

**Answer:**

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`

**Answer:**

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`

**Answer:**

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`

**Answer:**

Solution is `x=3` `y=0`

# 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`

**Answer:**

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`

**Answer:**

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`

**Answer:**

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`

**Answer:**

Solution is `x=3` `y=0`