Solving Systems of Equations

3 Approaches

Ms. NongAdapted from Mrs. N. Newman’s PPT

Click here to begin

Method #1

Graphically

Method #2

Algebraically Using Addition and/or Subtraction

Method #3

Algebraically Using Substitution

POSSIBLE ANSWER:

Answer: (x, y) or (x, y, z)

Answer: No Solution Answer: Identity

In order to solve a system of equations graphically you

typically begin by making sure both equations are in Slope-

Intercept 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

How to Use Graphs to solve Linear Systems.

Looking at the System Graphs:

•If the lines cross once, there will be one solution.

•If the lines are parallel, there will be no solutions.

•If the lines are the same, there will be an infinite number of

solutions.

Check by substitute answers to equations:

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

one 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

Lets look at another Systems solve by Substitution

y = 4x3x + y = -

21Step 5: Check the solution in both equations.

y = 4x

-12 = 4(-3)

-12 = -12

3x + y = -21

3(-3) + (-12) = -21

-9 + (-12) = -21

-21= -21

This concludes my presentation on simultaneous equations.

Please feel free to view it again at your leisure.

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