My problem is to graph a system of inequalities (which ive done) and introduce slack variables to convert the system of inequalities to a system of equations. So I am building my table and all is easy until I get to the point where S1 and S2 are both zeros or S1 and S3 are zeroes or S2 and S3 are both zeros???? Below is the equation but how am i supposed to solve besides continuaslly guessing and checking (which is how I did 2 of them)

3* x1 + x2 + S1 =105
x1 + x2 + S2 =37
x1 + 3* x2 + S3 =48

So when S1 & S2 were 0's i figured it out x1=34, x2=3 & S3=5 Then when S1 & S3 were both 0's I figured that out which took forever at 33.375,4.875 & -1.2. But i was just guessing and checking. Can someone please help me understand the correct way to solve these?

Thanks in advance.

An athlete who is three standard deviations above the mean would weigh_pounds?

To solve a system of inequalities and introduce slack variables, you can follow these steps:

Step 1: Graph the system of inequalities on a coordinate plane. This will help you visualize the feasible region.

Step 2: Identify the vertices of the feasible region. These are the points where two or more boundary lines intersect.

Step 3: For each vertex, assign values of zero to the slack variables to create a system of equations. Each equation corresponds to one inequality in the system.

Step 4: Solve the resulting system of equations to find the values of the original variables.

Let's apply these steps to your example:

Step 1: Graph the system of inequalities:

3x1 + x2 + S1 ≤ 105
x1 + x2 + S2 ≤ 37
x1 + 3x2 + S3 ≤ 48

Step 2: Identify the vertices of the feasible region. From your description, it seems like you have already done this and found three vertices.

Step 3: Assign zero values to the slack variables at each vertex:

At vertex 1 (S1 and S2 are zero):
3x1 + x2 + 0 = 105
x1 + x2 + 0 = 37
x1 + 3x2 + S3 = 48

At vertex 2 (S1 and S3 are zero):
3x1 + x2 + S1 = 105
x1 + x2 + 0 = 37
x1 + 3x2 + 0 = 48

At vertex 3 (S2 and S3 are zero):
3x1 + x2 + S1 = 105
x1 + x2 + S2 = 37
x1 + 3x2 + 0 = 48

Step 4: Solve the resulting system of equations at each vertex.

You can solve the system of equations using different methods, such as substitution or elimination. By solving the equations at each vertex, you can find the values of the original variables (x1 and x2) and the slack variables.

If you are facing difficulty in solving these systems, you can use tools like graphing calculators or software to find the solutions efficiently. These tools can provide accurate solutions without the need for manual guesswork and checking.

Remember, graphing the system of inequalities and using the vertices to form the equations will help you in finding the solutions more systematically and avoid relying purely on guesswork.