Solving Simultaneous Linear Inequalities Through Graphing

Simultaneous linear inequalities involve solving a set of linear inequalities where the solutions must satisfy all the given inequalities simultaneously. This process can be effectively managed using both algebraic and graphical methods. This article focuses on the graphical approach, which involves determining the region where the shaded areas of the inequalities overlap.

Understanding Simultaneous Linear Inequalities

A system of simultaneous linear inequalities can be written in the form:

a_1x b_1y a_2x b_2y

where a_i, b_i, c_i are constants, and x and y are the variables. The goal is to determine the region on the coordinate plane where all these inequalities hold true.

Graphical Solution

To solve the system graphically, follow these steps:

Graph each inequality. Each inequality represents a half-plane. For instance, the inequality 4x y can be written as y . This equation can be graphed by plotting the line 4x y 13. The line will intersect the x-axis at (13/4, 0) and the y-axis at (0, 13). Shade the region below this line to represent the inequality 4x y . Repeat for the second inequality. Similarly, the inequality 3x - 2y can be written as y > (3/2)x - 8. This equation can be graphed by plotting the line 3x - 2y 16. The line will intersect the x-axis at (16/3, 0) and the y-axis at (0, -8). Shade the region above this line to represent the inequality 3x - 2y . Identify the overlapping region. The solution to the system of inequalities is the region where the shaded areas of both inequalities overlap. This region represents the set of all points that satisfy both inequalities simultaneously.

Intersection Points

The lines 4x y 13 and 3x - 2y 16 will intersect at a point where both inequalities are satisfied. To find this point, solve the system of equations:

Step-by-step solution:

Express one variable in terms of the other. From 4x y 13, we get y 13 - 4x. Substitute this expression into the second equation. 3x - 2(13 - 4x) 16 3x - 26 8x 16 11x 42 x 42/11 ≈ 3.818 Solve for the other variable. Substitute x 42/11 back into y 13 - 4x to get: y 13 - 4(42/11) 13 - 168/11 ≈ 2.273

The intersection point is approximately (3.818, 2.273). Checking this point in both inequalities confirms it is a valid solution.

Conclusion and Importance of Verification

While the graphical method provides a visual understanding of the solution, algebraic methods ensure accuracy. Both approaches are crucial in understanding and solving simultaneous linear inequalities. Verification through substitution ensures the correctness of the solution, highlighting the importance of double-checking in mathematical problem-solving.