Mammoth Memory

Graphical simultaneous equations

We know

Simultaneous equations `=`  two equations with two unknowns

and

Graphical = Draw the lines on the graph.

 

The only additional item you need to know is:

The meeting place of the equations is the solution

When they meet they find the solution.

(The point where the two lines intersect on a graph is the solution).

 Both equations plotted on a graph, the meeting place is the solution

If you find where `x`  and `y`  intersect on a graph that is the solution.

NOTE:

To continue you must understand our section on formula for a straight line.

That is:

  1. Always make `y`  the subject of the formula
  2. Formula of a straight line is `y=mx+c`
  3. Plot `x`  and `y`  for given values of `x`

 

Example 1

Plot the graphs of the following equations and find their solution by finding the point of intersection.

`2x+3y=6`

`4x-6y=-4`

So first plot the graph of the following equation:

`2x+3y=6`

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`

Lets plot some points on the graph

If `x=5` then `y=2-2/3(5)` `=-1.33`
`x=4` then `y=2-2/3(4)` `=-0.66`
`x=3` then `y=2-2/3(3)` `=0`
`x=2` then `y=2-2/3(2)` `=0.67`
`x=1` then `y=2-2/3(1)` `=1.33`
`x=0` then `y=2-2/3(0)` `=2`

First plot the first equation on the graph following the x and y axis

Now lets add to the graph the equation:

`4x-6y=-4`

Subtract `4x`  from both sides to get `y`  on its own.

`-4x+4x-6y=-4-4x`

`-6y=-4-4x`

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

`y=(-4)/(-6)-(4x)/-6`

`y=2/3+(2x)/3`

Lets plot some points on the graph

If `x=5` then `y=2/3+(2times5)/3` `=2/3+10/3=12/3` `=4`
`x=4` then `y=2/3+(2times4)/3` `=2/3+8/3=10/3` `=3\1/3`
`x=3` then `y=2/3+(2times3)/3` `=2/3+6/3=8/3` `=2\2/3`
`x=2` then `y=2/3+(2times2)/3` `=2/3+4/3=6/3` `=2`
`x=1` then `y=2/3+(2times1)/3` `=2/3+2/3=4/3` `=1\1/3`
`x=0` then `y=2/3+(2times0)/3`   `=2/3`

Then plot the second equation

We can see from the graph that the two lines intersect (and therefore the solution) at:

`x=1`           `y=1\1/3`

 

Example 2

Plot the graphs of the following equations and find their solution by finding the point of intersection.

`y=x+3`

`x+y=7`

So first plot the following equation on a graph.

`y=x+3`

Lets put some points on the graph

If `x=5` then `y=5+3` `=8`
`x=4` then `y=4+3` `=7`
`x=3` then `y=3+3` `=6`
`x=2` then `y=2+3` `=5`
`x=1` then `y=1+3` `=4`
`x=0` then `y=0+3` `=3`

Plot the first equation example 2

Now lets add to the graph the equation:

`x+y=7`

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

`x-x+y=7-x`

`y=7-x`

Lets plot some points on the graph

If `x=5` then `y=7-5` `=2`
`x=4` then `y=7-4` `=3`
`x=3` then `y=7-3` `=4`
`x=2` then `y=7-2` `=5`
`x=1` then `y=7-1` `=6`
`x=0` then `y=7-0` `=7`

Plot the second equation and find where both intersect example 2

We can see from the graph that the two line intersect (and therefore the solution) at:

`x=2`         `y=5`

 

Example 3

Plot the graphs of the following equations and find their solution by finding the point of intersection.

`2x+2y=6`

`4x-6y=12`

So first plot the graph of the following equation.

`2x+2y=6`

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`

Lets put some points on the graph

If `x=5`  then  `y=3-5` `=-2`
 `x=4`  then  `y=3-4` `=-1`
 `x=3`  then  `y=3-3` `=0`
 `x=2`  then  `y=3-2` `=1`
 `x=1`  then  `y=3-1` `=2`
 `x=0`  then  `y=3-0` `=3`

Plot the first equation example 3

Now lets add to the graph the equation:

`4x-6y=12`

Subtract `4x`  from both sides to get `y`  on its own

`4x-4x-6y=12-4x`

`-6y=12-4x`

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

`(-6y)/(-6)=12/-6-(4x)/-6`

`y=-2+2/3x`

Lets plot some points on the graph

If  `x=5`  then  `y=-2+2/3times5` `=-2+10/3=-2+3\1/2` `=1\1/2`
           
  `x=4`  then  `y=-2+2/3times4` `=-2+8/3=-2+2\2/3` `=2/3`
           
  `x=3`  then  `y=-2+2/3times3` `=-2+6/3=-2+2` `=0`
           
  `x=2`  then  `y=-2+2/3times2` `=-2+4/3=-2+1\1/3` `=-2/3`
           
  `x=1`  then  `y=-2+2/3times1` `=-2+2/3=-2+2/3` `=-1\1/3`
           
  `x=0`  then  `y=-2+2/3times0`   `=-2`

Plot the second equation and find where both intersect to find the solution example 3

We can see from the graph that the two line intersect (and therefore the solution) at:

`x=3`         `y=0`

 

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