Elimination Method - Part 1 - Add or Subtract

Solve System of Equations Use Elimination Method - Part 1 - How it Works - Video

Example 1

Example 1:

When dealing with a system of equations, there are several ways to solve for it. Here we are going to use the elimination method. So we are either going to add or subtract the two equations together so we can eliminate one variable.

We can add the two equations together when the coefficients of the same variable are opposite like 5 and -5.

We can subtract the two equations together when the coefficients of the same variable are the same like 4 and 4.

In this case, we added the two equations together since we have 2 and -2 in front of x.

0x - 3y = 9

-3y = 9

y = -3

Add the two equations

Drop the 0

Divide both sides by -3

Next we can substitute the y = -3 into either equation to solve for x. We chose 2x + 2y = 6 because it is the first one.

2x + 2 * (-3) = 6

2x - 6 = 6

2x = 12

x = 6

Substitute -3 for y

Multiply

Add 6 to both sides

Divide both sides by 2

So the answer to the system is (6, -3)

We could have also graphed the two linear equations to find our solution. Another good aspect of the graph is that it tells how many solutions there are. Since we only have one intersection point, we only have one answer.

Example 2

Example 2:

When dealing with a system of equations, there are several ways to solve for it. Here we are going to use the elimination method. So we are either going to add or subtract the two equations together so we can eliminate one variable.

We can add the two equations together when the coefficients of the same variable are opposite like 5 and -5.

We can subtract the two equations together when the coefficients of the same variable are the same like 4 and 4.

In this case, we subtracted the two equations together since we have 3 and 3 in front of x.

0x + 0y = 9

0 = 9

not true

Subtract the two equations

Add zero


Here we have 0 = 9, which is not true. Did we do something wrong? What does this mean? Since the variables in front of y are 5 and 5, we can subtract to eliminate that variable. Let's do that.

0x + 0y = 9

0 = 9

not true

Subtract the two equations

Add zero


We get the same result, 0 = 9, which is not true. Since we verified both sides and both give not true results, the answer is no solution. 0 never equals 9, since that is impossible we have no solution as our answer. What do you think the graph looks like?

We could have also graphed the two linear equations to find our solution. Another good aspect of the graph is that it tells how many solutions there are. Since our two lines are parallel lines or each equation has the same slope, the lines never touch. Since the lines never touch, there is not any point of intersection so there is not any answer.

Example 3

Example 2:

When dealing with a system of equations, there are several ways to solve for it. Here we are going to use the elimination method. So we are either going to add or subtract the two equations together so we can eliminate one variable.

We can add the two equations together when the coefficients of the same variable are opposite like 5 and -5.

We can subtract the two equations together when the coefficients of the same variable are the same like 4 and 4.

In this case, we added the two equations together since we have 2 and -2 in front of x.

0x + 0y = 0

0 = 0

always true

Add the two equations

Add zero


Here we have 0 = 0, which is always true. Did we do something wrong? What does this mean? Since the variables in front of y are 3 and -3, we can add to eliminate that variable. Let's do that.

0x + 0y = 0

0 = 0

always true

Add the two equations

Add zero


We get the same result, as before, which is always true. Since we verified both sides and both give always true results, the answer is all real numbers. 0 always equals 0, since that is always true we have all real numbers as our answer. What do you think the graph looks like?

We could have also graphed the two linear equations to find our solution. Another good aspect of the graph is that it tells how many solutions there are. Since our two lines overlap each other, we have the same line so every point is a point of intersection.

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