If 
[x
y
z]
​​ is a solution of the system of equations
7x+2y+z=1 3y−z=5 −3x+4y−2z=8​, then the value of x+y+z is

We are given the system of linear equations:

\[
\begin{aligned}
7x + 2y + z &= 1 \quad \text{(Equation 1)} \\
3y - z &= 5 \quad \text{(Equation 2)} \\
-3x + 4y - 2z &= 8 \quad \text{(Equation 3)}
\end{aligned}
\]

Our goal is to find the value of \( x + y + z \).

### Step 1: Solve for \( z \) using Equation 2

From Equation 2:

\[
3y - z = 5
\]

Solve for \( z \):

\[
z = 3y - 5
\]

### Step 2: Substitute \( z = 3y - 5 \) into the other two equations

Now substitute \( z = 3y - 5 \) into Equation 1 and Equation 3.

**Substitute into Equation 1:**

\[
7x + 2y + (3y - 5) = 1
\]

Simplify:

\[
7x + 5y - 5 = 1
\]

\[
7x + 5y = 6 \quad \text{(Equation 4)}
\]

**Substitute into Equation 3:**

\[
-3x + 4y - 2(3y - 5) = 8
\]

Simplify:

\[
-3x + 4y - 6y + 10 = 8
\]

\[
-3x - 2y = -2
\]

\[
3x + 2y = 2 \quad \text{(Equation 5)}
\]

### Step 3: Solve the system of two equations

We now solve the system of two equations:

1. \( 7x + 5y = 6 \)
2. \( 3x + 2y = 2 \)

Multiply Equation 5 by 5 and Equation 4 by 2 to eliminate \( y \):

\[
5(3x + 2y) = 5(2) \quad \Rightarrow \quad 15x + 10y = 10
\]

\[
2(7x + 5y) = 2(6) \quad \Rightarrow \quad 14x + 10y = 12
\]

Now subtract the second equation from the first:

\[
(15x + 10y) - (14x + 10y) = 10 - 12
\]

\[
x = -2
\]

### Step 4: Substitute \( x = -2 \) back into one of the equations

Substitute \( x = -2 \) into Equation 5:

\[
3(-2) + 2y = 2
\]

\[
-6 + 2y = 2
\]

\[
2y = 8
\]

\[
y = 4
\]

### Step 5: Find \( z \)

Substitute \( y = 4 \) into \( z = 3y - 5 \):

\[
z = 3(4) - 5 = 12 - 5 = 7
\]

### Step 6: Calculate \( x + y + z \)

Now that we know \( x = -2 \), \( y = 4 \), and \( z = 7 \), we can calculate:

\[
x + y + z = -2 + 4 + 7 = 9
\]

### Final Answer:
The value of \( x + y + z \) is \( \boxed{9} \).

---

Would you like more details on any part of the solution? Here are some related questions:

1. How do you solve systems of linear equations using substitution?
2. What is the method of elimination for solving systems of equations?
3. How do you solve a system of equations with three variables?
4. Can the value of \( x + y + z \) change if the system of equations has no solution?
5. What would happen if we used matrices to solve this system?

**Tip:** When solving systems of equations, always look for substitutions that can simplify the equations, like solving for one variable in terms of others!

If [x, y, z] is a solution of the system of equations 7x+2y+z=1, 3y−z=5, −3x+4y−2z=8, then the value of x+y+z is?

Learn how to solve a system of linear equations using substitution and elimination methods. This problem involves solving for x, y, and z in a three-variable system and determining the sum x+y+z. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation