Linear Inequalities

Concept:
Linear Programming - Graphical Solution of Linear Inequalities.

Step 1: Identify the system of inequalities.
The given constraints are: 1) , 2) , 3) , and the non-negativity constraints , .

Step 2: Analyze the boundary conditions for the specific regions.
Looking at a standard graph for these equations, the region is typically bounded between the lines , , and . To verify, we select a test point that lies comfortably inside the expected region.

Step 3: Select and test a coordinate point.
Let's select the test point . We will substitute and into each inequality to see if they all hold true.

Step 4: Evaluate the first two inequalities.
Substitute into : . Since is true, the point satisfies the first condition. Substitute into : . This is also true.

Step 5: Evaluate the remaining constraints to finalize the region.
Check : Since , this is true. Check non-negativity: and are both true. Because the test point satisfies all given inequalities simultaneously, the region containing this point is the correct solution set. $ $

For (the identity matrix), we verify the values of and such that:
Step 1: Simplify for
Consider the third row of and the first column of :
Simplify:

Step 2: Simplify for
Consider the second row of and the third column of :
Simplify:

Final Answer:

Step 1: Use the substitution or elimination method We are given the system of equations: 1. 2. We will use the elimination method. First, add both equations to eliminate . Step 2: Add the two equations Add equation 1 and equation 2: Simplify: Step 3: Substitute back into the first equation Substitute into the first equation : Answer: Therefore, the solution to the system of equations is and . So, the correct answer is option (2).

Step 1: Use the substitution or elimination method We are given the system of equations: 1. 2. We will use the substitution method. Step 2: Solve one equation for one variable From the first equation , solve for : Step 3: Substitute into the second equation Substitute into the second equation : Step 4: Solve for Now substitute back into : Answer: Therefore, the solution to the system of equations is and . So, the correct answer is option (1).

Let the age of the son be years. The father's age is years (since the father is three times as old as the son). The sum of their ages is 60 years, so: Thus, the son's age is 15 years. Therefore, the correct answer is option (1).