We are given the system of three linear equations:

1. $-5x + 3y + 6z = 4$
2. $-3x + y + 5z = -5$
3. $-4x + 2y + z = 13$

Let's solve this system step by step using the elimination or substitution method.

Step 1: Simplify the equations

The equations are already in standard form:
$Ax + By + Cz = D$
We proceed without any modifications.

Step 2: Eliminate one variable

To start, we can eliminate $z$ by combining pairs of equations.

1. From equations (1) and (2):
   $-5x + 3y + 6z = 4 \quad \text{and} \quad -3x + y + 5z = -5$
   Multiply the second equation by $-6$ to align the coefficients of $z$:
   $-6(-3x + y + 5z) = -6(-5)$
   $18x - 6y - 30z = 30$

   Now add this to the first equation:
   $(-5x + 3y + 6z) + (18x - 6y - 30z) = 4 + 30$
   Simplify:
   $13x - 3y - 24z = 34 \quad \text{(Equation 4)}$

2. From equations (2) and (3):
   $-3x + y + 5z = -5 \quad \text{and} \quad -4x + 2y + z = 13$
   Multiply the second equation by $-5$ to align the coefficients of $z$:
   $-5(-4x + 2y + z) = -5(13)$
   $20x - 10y - 5z = -65$

   Now add this to the second equation:
   $(-3x + y + 5z) + (20x - 10y - 5z) = -5 + (-65)$
   Simplify:
   $17x - 9y = -70 \quad \text{(Equation 5)}$

Step 3: Solve for $x$ and $y$

Now we solve the system of two equations in $x$ and $y$:

1. Equation 4: $13x - 3y - 24z = 34$
2. Equation 5: $17x - 9y = -70$

Let’s isolate $x$ and $y$, and substitute into the equations to solve.

I'll now compute these values step-by-step. Give me a moment. sympy import symbols, Eq, solve

Define variables

x, y, z = symbols('x y z')

Define the equations

eq1 = Eq(-5x + 3y + 6z, 4) eq2 = Eq(-3x + y + 5z, -5) eq3 = Eq(-4x + 2*y + z, 13)

Solve the system of equations

solution = solve((eq1, eq2, eq3), (x, y, z)) solution### Solution: The solution to the system of equations is:
$x = -2, \, y = 4, \, z = -3$

Verification:

1. Substitute $x = -2$, $y = 4$, and $z = -3$ into the original equations:
   • For $-5x + 3y + 6z = 4$:
     $-5(-2) + 3(4) + 6(-3) = 4$ → $10 + 12 - 18 = 4$ ✔
   • For $-3x + y + 5z = -5$:
     $-3(-2) + 4 + 5(-3) = -5$ → $6 + 4 - 15 = -5$ ✔
   • For $-4x + 2y + z = 13$:
     $-4(-2) + 2(4) + (-3) = 13$ → $8 + 8 - 3 = 13$ ✔

Thus, the solution is correct.

Do you want me to explain any step further or solve a related question?

Related Questions:

1. Solve another system of linear equations with three variables.
2. How to solve a system of equations using substitution?
3. Explain the elimination method for solving linear equations.
4. What is Gaussian elimination for systems of linear equations?
5. How can matrices be used to solve systems of equations?

Tip:

For systems with three variables, eliminate one variable first to reduce it to two equations with two variables, then solve systematically.