To solve this system of equations using the substitution method, let's go through it step-by-step.

Given: $y = 3x - 6$ $x - 4y = 11$

Step 1: Substitute $y$ from the first equation into the second equation.

Since $y = 3x - 6$, we substitute $3x - 6$ for $y$ in the second equation: $x - 4(3x - 6) = 11$

Step 2: Distribute and simplify.

Expanding $-4(3x - 6)$: $x - 12x + 24 = 11$

Combine like terms: $-11x + 24 = 11$

Step 3: Solve for $x$.

Isolate $x$ by moving constants to the other side: $-11x = 11 - 24$ $-11x = -13$

Now, divide by $-11$: $x = \frac{13}{11}$

Step 4: Substitute $x = \frac{13}{11}$ back into the first equation to solve for $y$.

Using the first equation $y = 3x - 6$: $y = 3\left(\frac{13}{11}\right) - 6$

Simplify: $y = \frac{39}{11} - \frac{66}{11} = \frac{-27}{11}$

Solution

The solution to the system is: $x = \frac{13}{11}, \quad y = -\frac{27}{11}$

Would you like further details or have any questions?

Here are five related questions for further practice:

1. Solve using substitution: $y = 2x + 3$ and $3x - y = 4$.
2. Use substitution for $y = 5x - 7$ and $2x + 3y = 11$.
3. Solve $y = 4x + 2$ and $x + 3y = 10$ by substitution.
4. Apply substitution to $y = \frac{1}{2}x + 1$ and $3x - y = 7$.
5. Try solving $y = -2x + 6$ and $x + 4y = -3$ by substitution.

Tip: Substitution is efficient when one of the equations is already solved for one variable, making it easier to plug into the other.