SOLVING SYSTEMS BY ELIMINATION

The elimination method can be used to solve a system of linear equations. In this method, one of the variables is eliminated by adding or subtracting the two equations of the system to obtain a single equation in one variable.

The following steps will be useful to solve system of equations using elimination method. 

Step 1 :

The variable which is eliminated must have the same coefficient in both the equations. If not, make them to be same using least common multiple and multiplication.

Step 2 : 

The variable which is eliminated must have different signs. If not, multiply one of the equations by negative sign. 

Step 3 :

Now add the two equations to eliminate the variable. 

Example 1 :

Solve the system of equations using elimination method. Check the solution by graphing.

2x - 3y  =  12

x + 3y  =  6

Solution :

2x - 3y  =  12 -----(1)

x + 3y  =  6 -----(2)

In the given two equations, the variable y is having the same coefficient. And also, the variable y is having different signs.

So we can eliminate the variable y by adding the two equations. 

Divide both sides by 3.

x = 6

Substitute 6 for x in (2).

(2)----> 6 + 3y = 6

Subtract 6 from each side.

3y = 0

Divide each side by 3.

y = 0

Write the solution as ordered pair.

(x, y) = (6, 0)

Check the solution by graphing. 

To graph the equations, write them in slope-intercept form y = mx + b.

2x - 3y = 12

y = (2/3)x - 4

slope = 2/3

y-intercept = -4

x + 3y = 6

y = -(1/3)x + 2

slope = -1/3

y-intercept = 2

The point of intersection is (6, 0).

Example 2 :

Sum of the cost price of two products is $50. Sum of the selling price of the same two products is $52. If one is sold at 20% profit and other one is sold at 20% loss, find the cost price of each product.

Solution :

Let 'x' and 'y' be the cost prices of two products. 

x + y = 50 ----(1)

Let us assume that x is sold at 20% profit.

Then, the selling price of x :

= 120% of x

= 1.2x

Let us assume that y is sold at 20% loss.

Then, the selling price of y :

= 80% of 'y'

= 0.8y

Given : Selling price of x + selling price of y = 52.

1.2x + 0.8y = 52

Multiply both sides by 10.

12x + 8y = 520

Divide each side by 4.

3x + 2y = 130 ----(2)

Solve (1) and (2).

(2) - (1) ⋅ 2 : 

Substitute 30 for x in (1). 

30 + y = 50

Subtract 30 from each side.

y = 20

So, the cost prices of two products are $30 and $20.

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