The problem involves solving a system of linear equations using elimination and substitution. The equations are:

1. $2x + 3y - z = 9$
2. $3x + 2y + z = 16$
3. $2x + y = 7$

Let’s solve it step by step using elimination and substitution:

Step 1: Simplify Equation (3)

From Equation (3), solve for $y$ in terms of $x$: y = 7 - 2x \tag{4}

Step 2: Eliminate $z$ Using Equations (1) and (2)

Add Equations (1) and (2): $(2x + 3y - z) + (3x + 2y + z) = 9 + 16$ 5x + 5y = 25 \tag{5}

From Equation (5), simplify: x + y = 5 \tag{6}

Step 3: Substitute $y$ From Equation (4) Into Equation (6)

Substitute $y = 7 - 2x$ into $x + y = 5$: $x + (7 - 2x) = 5$ $x + 7 - 2x = 5$ $-x + 7 = 5$ x = 2 \tag{7}

Step 4: Find $y$ Using Equation (4)

Substitute $x = 2$ into $y = 7 - 2x$: $y = 7 - 2(2)$ y = 3 \tag{8}

Step 5: Find $z$ Using Equation (1)

Substitute $x = 2$ and $y = 3$ into Equation (1): $2(2) + 3(3) - z = 9$ $4 + 9 - z = 9$ $13 - z = 9$ z = 4 \tag{9}

Final Solution

The values of $x, y, z$ are: $x = 2, \quad y = 3, \quad z = 4$

Would you like a detailed verification of the solution or further explanation?

Related Questions

1. How does elimination work when there are no directly cancelable terms?
2. How would substitution differ if solving from another equation?
3. What is a geometric interpretation of these three equations?
4. Can we use matrix methods like Gaussian elimination here?
5. How do we check if this system of equations has a unique solution?

Tip

Always verify your final solution by substituting the values of $x, y, z$ into all original equations!