What ordered pair is the solution to the following system of linear inequalities?
[tex]4 x-5 y \geq 11[/tex]
[tex]3 x+6 y\ \textless \ 1[/tex]
A. (-1,1)
B. (2,-1)
C. (0,0)
D. (4,2)
Answer (2)
Substitute each ordered pair into the inequalities.
Check if both inequalities are true for each ordered pair.
The ordered pair ( 2 , − 1 ) satisfies both inequalities: 4 ( 2 ) − 5 ( − 1 ) ≥ 11 and 3 ( 2 ) + 6 ( − 1 ) < 1 .
Therefore, the solution is ( 2 , − 1 ) .
Explanation
Analyze the problem and data We are given the following system of linear inequalities:
= 11"> 4 x − 5 y " >= 11 3 x + 6 y < 1
We need to find which of the given ordered pairs satisfies both inequalities. We will test each ordered pair to see if it is a solution.
Check each ordered pair
Checking the ordered pair ( − 1 , 1 ) :
Substitute x = − 1 and y = 1 into the first inequality: = 11"> 4 ( − 1 ) − 5 ( 1 ) " >= 11 = 11"> − 4 − 5" >= 11 = 11"> − 9" >= 11 This is false.
Since the first inequality is not satisfied, we don't need to check the second inequality. The ordered pair ( − 1 , 1 ) is not a solution.
Checking the ordered pair ( 2 , − 1 ) :
Substitute x = 2 and y = − 1 into the first inequality: = 11"> 4 ( 2 ) − 5 ( − 1 ) " >= 11 = 11"> 8 + 5" >= 11 = 11"> 13" >= 11 This is true.
Substitute x = 2 and y = − 1 into the second inequality: 3 ( 2 ) + 6 ( − 1 ) < 1 6 − 6 < 1 0 < 1 This is true.
Since both inequalities are satisfied, the ordered pair ( 2 , − 1 ) is a solution.
Checking the ordered pair ( 0 , 0 ) :
Substitute x = 0 and y = 0 into the first inequality: = 11"> 4 ( 0 ) − 5 ( 0 ) " >= 11 = 11"> 0" >= 11 This is false.
Since the first inequality is not satisfied, we don't need to check the second inequality. The ordered pair ( 0 , 0 ) is not a solution.
Checking the ordered pair ( 4 , 2 ) :
Substitute x = 4 and y = 2 into the first inequality: = 11"> 4 ( 4 ) − 5 ( 2 ) " >= 11 = 11"> 16 − 10" >= 11 = 11"> 6" >= 11 This is false.
Since the first inequality is not satisfied, we don't need to check the second inequality. The ordered pair ( 4 , 2 ) is not a solution.
Final Answer Only the ordered pair ( 2 , − 1 ) satisfies both inequalities. Therefore, the solution to the system of linear inequalities is ( 2 , − 1 ) .
Examples
Systems of inequalities are used in various real-world applications, such as linear programming, where the goal is to optimize a certain objective function subject to constraints. For example, a company might want to maximize its profit while adhering to limitations on resources like labor and materials. These constraints can be expressed as linear inequalities, and the solution to the system of inequalities represents the feasible region within which the optimal solution lies. Another example is in diet planning, where one might want to minimize the cost of a diet while meeting certain nutritional requirements, which can be formulated as a system of linear inequalities.
The ordered pair that satisfies the system of linear inequalities is ( 2 , − 1 ) , as it meets the criteria for both inequalities. Other pairs do not satisfy one or both inequalities. Thus, the solution is ( 2 , − 1 ) .
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