The number of solutions of the system of in equations x + 2y ≤ 3, 3x + 4y ≥ 12, x ≥ 0, y ≥ 1 is
Equations are mathematical statements that assert the equality of two expressions. They consist of variables constants and operators. Equations are used to represent relationships between different quantities. Solving an equation involves finding the value(s) of the variable(s) that make the equation true. Common types include linear quadratic and polynomial equations.
Linear Programming is a mathematical technique used to optimize a linear objective function subject to linear constraints. It involves finding the feasible region that satisfies all constraints. The solution lies at the vertex of the feasible region. Linear Programming is widely used in economics business and industrial applications to make optimal decisions.
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To graph the system of inequalities:
1. x + 2y ≤ 3
2. 3x + 4y ≥ 12
3. x ≥ 0
4. y ≥ 1
We draw a graph of the inequalities.
For x + 2y ≤ 3, the line x + 2y = 3 crosses the x-axis at (3, 0) and the y-axis at (0, 1.5).
– For 3x + 4y ≥ 12, the line 3x + 4y = 12 cuts the x-axis at the point (4, 0) and y-axis at (0, 3).
-x ≥ 0 and y ≥ 1 limits the feasible region to the first quadrant above the line y = 1.
When we plot these constraints, we see that the feasible region is empty because the two lines do not intersect within the given constraints.
Thus, the number of solutions is zero.
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