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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