Solving Linear Equations: Step-by-Step

We are given a system of three linear equations:

• Equation 1: \(2x + y = 15\)
• Equation 2: \(y + 2z = 17\)
• Equation 3: \(x + 2y = 9\)

The goal is to find the value of the expression \(4x + 3y + z\).

Finding the Value of x

1. From Equation 1, express \(y\) in terms of \(x\):

   \(y = 15 - 2x\)

2. Substitute this expression for \(y\) into Equation 3:

   \(x + 2(15 - 2x) = 9\)

3. Simplify and solve for \(x\):

   \(x + 30 - 4x = 9\)

   \(-3x = 9 - 30\)

   \(-3x = -21\)

   \(x = \frac{-21}{-3}\)

   \(x = 7\)

Finding the Value of y

1. Substitute the value of \(x\) (which is 7) back into the expression for \(y\) derived from Equation 1:

   \(y = 15 - 2(7)\)

2. Calculate \(y\):

   \(y = 15 - 14\)

   \(y = 1\)

Finding the Value of z

1. Substitute the value of \(y\) (which is 1) into Equation 2:

   \(1 + 2z = 17\)

2. Solve for \(z\):

   \(2z = 17 - 1\)

   \(2z = 16\)

   \(z = \frac{16}{2}\)

   \(z = 8\)

Calculating the Final Expression Value

Now substitute the values \(x=7\), \(y=1\), and \(z=8\) into the target expression \(4x + 3y + z\):

\(4x + 3y + z = 4(7) + 3(1) + 8\)

\(= 28 + 3 + 8\)

\(= 31 + 8\)

\(= 39\)

The value of \(4x + 3y + z\) is 39.