System of Two Equations: Addition/Elimination Method

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Equation (1):    X + Y =

Equation (2):    X + Y =

Equation (1): 3X+4Y = 12

Equation (2): 2X - 5Y = 10

Now we will eliminate the y terms by making the coefficients of the y terms become opposite.

Note that the coefficients of the y terms are 4 and 5.
Since the Least Common Multiple of 4 and 5 is 20, we want the y coefficient in Equation (1) to be 20
and the y coefficient in Equation (2) to be -20.

Equation (1): 15X+20Y = 60 Multiply each term in Equation (1) by 5

Equation (2): 8X - 20Y = 40 Multiply each term in Equation (2) by 4

Add Equation (1) and Equation (2) together and then solve for x:

15X+20Y = 60

8X - 20Y = 40

23X = 100

X = 100/23 divide both sides by 23

Use Equation (1) to find y:

15X+20Y = 60

15(100/23)+20Y = 60

1500/23 + (20)Y = 60

20Y = 60 + (-1500/23)

20Y = -120/23

Y = -6/23

Solution is (100/23, -6/23)

The two lines intersect at one point.
There is one solution.
The system of equations is said to be 'consistent'.

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System of Two Equations: Addition/Elimination Method

Equation (1): 3X+4Y = 12

Equation (2): 2X - 5Y = 10

Now we will eliminate the y terms by making the coefficients of the y terms become opposite.

Note that the coefficients of the y terms are 4 and 5. Since the Least Common Multiple of 4 and 5 is 20, we want the y coefficient in Equation (1) to be 20 and the y coefficient in Equation (2) to be -20.

Equation (1): 15X+20Y = 60 Multiply each term in Equation (1) by 5

Equation (2): 8X - 20Y = 40 Multiply each term in Equation (2) by 4

Add Equation (1) and Equation (2) together and then solve for x:

15X+20Y = 60

8X - 20Y = 40

23X = 100

X = 100/23 divide both sides by 23

Use Equation (1) to find y:

15X+20Y = 60

15(100/23)+20Y = 60

1500/23 + (20)Y = 60

20Y = 60 + (-1500/23)

20Y = -120/23

Y = -6/23

Solution is (100/23, -6/23)

The two lines intersect at one point. There is one solution. The system of equations is said to be 'consistent'.

Absolute Value and Vertical Bars:

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Note that the coefficients of the y terms are 4 and 5.
Since the Least Common Multiple of 4 and 5 is 20, we want the y coefficient in Equation (1) to be 20
and the y coefficient in Equation (2) to be -20.

The two lines intersect at one point.
There is one solution.
The system of equations is said to be 'consistent'.