Solving Systems of Equations

3 Approaches

Mrs. N. NewmanClick here to begin

Method #1

Graphically

Method #2

Algebraically Using Addition and/or Subtraction

Method #3

Algebraically Using Substitution

Door #1

Door #2

Door #3

In order to solve a system of equations graphically you

typically begin by making sure both equations are in standard

form.

Where m is the slope and b is the y-intercept.

Examples:

y = 3x- 4

y = -2x +6

Slope is 3 and y-intercept is - 4.

Slope is -2 and y-intercept is 6.

bmxy

Graph the line by locating the appropriate

intercept, this your first coordinate. Then move to your next

coordinate using your slope.

Use this same process and graph the second line.

Once both lines have been graphed locate the point of

intersection for the lines. This point is your solution set.

In this example the solution set is [2,2].

In order to solve a system of equations algebraically using addition first you must be sure that both equation are in the same chronological order.

Example: 2

4

yx

xy

2

4

xy

xyCould be

Now select which of the two variables you want to eliminate.

For the example below I decided to remove x.

2

4

xy

xy

The reason I chose to eliminate x is because they are the additive inverse of each other.

That means they will cancel when added together.

Now add the two equations together.

2

4

xy

xy

Your total is:

therefore 3

62

y

y

Now substitute the known value into either one of the original equations.I decided to substitute 3 in for y in the second equation.

1

23

x

x

Now state your solution set always remembering to do so in alphabetical order.

[-1,3]

Lets suppose for a moment that the equations are in the same sequential order. However, you notice that neither coefficients are additive

inverses of the other.

1273

332

yx

yx

Identify the least common multiple of the coefficient you chose to

eliminate. So, the LCM of 2 and 3 in this example would be 6.

Multiply one or both equations by their

respective multiples. Be sure to choose numbers that

will result in additive inverses.

)1273(2

)332(3

yx

yx

24146

996

yx

yxbecomes

Now add the two equations together.

24146

996

yx

yxbecomes 155 y

Therefore 3y

Now substitute the known value into either one of the original equations.

3

62

392

3)3(32

3

x

x

x

x

y

Now state your solution set always remembering to do so in alphabetical

order.

[-3,3]

In order to solve a system equations algebraically using substitution you must have on variable isolated in one of the equations. In other words you will need to solve for y in terms

of x or solve for x in terms of y.

In this example it has been done for you in the first

equation.

2

4

yx

xy

Now lets suppose for a moment that you are given a set of equations like this..

1273

332

yx

yx

Choosing to isolate y in the first equation the result is :

13

2 xy

Now substitute what y equals into the second equation.

2

4

yx

xy

becomes24 xx

Better know as

Therefore 1

22

242

x

x

x

This concludes my presentation on simultaneous equations.

Please feel free to view it again at your leisure.