# SOLVING SYSTEMS OF LINEAR EQUATIONS BY THE METHOD

**Definitions:**

• System of linear equations: consists of two or more linear equations with
the same variables.

•Consistent: The system is consistent if there is exactly one solution .

•Inconsistent: The system is inconsistent if there is no solution. This happens
when the two equations represent parallel lines.

•Dependent: The system is dependent if there is an infinite number of ordered
pairs as solutions.

This occurs when the two equations represent the same line.

** Steps for the Method of Elimination :**

1. Align the variables.

2. Multiply one or both equations by the appropriate constant so that the coe±cients
of one of the

variables are opposites of each other.

3. Add the two equations together. (If you have done everything correctly, this
will produce one equation

with one variable).

4. Solve the equation from Step 3.

5. Substitute the solution from Step 4 into either of the original equations.
This will give the value of

the other variable.

•If the equation in Step 3 above is a false statement (such as 7 = 2), then
the system is inconsistent.

•If the equation in Step 3 above is a true statement (such as 0 = 0), then the
system is dependent.

** Common Mistakes to Avoid:**

•Remember that a system of linear equations is not completely solved until
values for both x and y

are found. To avoid this mistake, write all answers as ordered pairs .

•Remember that all ordered pairs are stated with the x-variable first and the
y-variable second;

namely, (x; y).

1. Solve

6x - 5y = 25

4x + 15y = 13

If we multiply the first equation by 3 and

leave the second equation alone, we will eliminate y .

3(6x - 5y = 25)

4x + 15y = 13

We now have:

18x - 15y = 75

4x + 15y = 13

Adding these two equations together we get:

22x = 88

x = 4

Now, we must find the value for y by substituting x = 4 into one of the two original

equations. Substituting into the first equation gives

2. Solve

5x - 2y = 16

2x + y = 8

Multiplying the second equation by 2 will

eliminate the y variable.

5x - 2y = -16

2(2x + y = 8)

This gives us:

5x - 2y = -16

4x + 2y = 16

When we add these two equations together

we get:

9x = 0

x = 0

Now, we must find the value of y by substituting x = 0 into one of the two
original

equations. When we substitute x = 0 into the second equation, we get

2(0) + y = 8

y = 8

3. Solve

8x + 9y = 13

6x - 5y = 45

To eliminate the x variable we will multiply the first equation by -3 and the
second

equation by 4.

-3(8x + 9y = 13)

4(6x - 5y = 45)

This gives us:

-24x - 27y = -39

24x - 20y = 180

Adding these two equations together yields:

-47y = 141

y = -3

Next, we need to find the value for x by substituting y = -3 into either of the two original equations. If we substitute y = -3 into

the first equation, we get

4. Solve

4x - 5y = 35

3x - 4y = 24

If we multiply the first equation by 3 and the

second equation by -4 we will eliminate the

x variable.

3(4x - 5y = 35)

-4(3x - 4y = 24)

This yields:

12x - 15y = 105

-12x + 16y = -96

When we add these two equations together,

we get:

y = 9

Next, to solve for the x variable we need to

substitute y = 9 into one of the two original equations. Substituting y = 9 into the

second equation, we get:

5. Solve

-6x + 9y = 12

2x - 3y = -4

If we multiply the second equation by 3 we

will eliminate the x variable.

-6x + 9y = 12

3(2x - 3y = -4)

This gives us:

-6x + 9y = 12

6x - 9y = -12

When we add these two equations together,

we get:

0 = 0

Since this is a true statement, we know that

the system is dependent. Therefore, there

are an infinite number of solutions.

**Answer: dependent system**

6. Solve

-5x + 4y = 1

15x - 12y = 4

If we multiply the first equation by 3 we will

eliminate the x variable.

3(-5x + 4y = 1)

15x - 12y = 4

This gives us:

-15x + 12y = 3

15x - 12y = 4

Adding these two equations together gives

us:

0 = 7

Since this is a false statement, we know that

the system is inconsistent. Therefore, there

is no solution.

**Answer: No Solution**

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