Solve the system using the linear combination method. 3x-4y=12, 6x-2y=-11

To solve the system of equations using the **linear combination method** (also known as the **elimination method**), we aim to eliminate one of the variables by adding or subtracting the equations.

We are given the system:

\[
3x - 4y = 12 \tag{1}
\]
\[
6x - 2y = -11 \tag{2}
\]

### Step 1: Multiply the equations to align coefficients

We want to eliminate one of the variables, so let’s try to eliminate \(x\). To do this, we’ll need to manipulate the equations so the coefficients of \(x\) are the same.

Notice that if we multiply equation (1) by 2, the coefficient of \(x\) in both equations will match.

Multiplying equation (1) by 2:
\[
2(3x - 4y) = 2(12) \implies 6x - 8y = 24 \tag{3}
\]

Now we have:
\[
6x - 8y = 24 \tag{3}
\]
\[
6x - 2y = -11 \tag{2}
\]

### Step 2: Subtract the equations

Now subtract equation (2) from equation (3) to eliminate \(x\):

\[
(6x - 8y) - (6x - 2y) = 24 - (-11)
\]

Simplifying:
\[
6x - 8y - 6x + 2y = 24 + 11
\]
\[
-6y = 35
\]

### Step 3: Solve for \(y\)

\[
y = \frac{35}{-6} = -\frac{35}{6}
\]

### Step 4: Substitute \(y = -\frac{35}{6}\) into one of the original equations

Let’s substitute \(y = -\frac{35}{6}\) into equation (1):

\[
3x - 4\left(-\frac{35}{6}\right) = 12
\]
\[
3x + \frac{140}{6} = 12
\]
\[
3x + \frac{70}{3} = 12
\]

Multiply through by 3 to eliminate the fraction:
\[
9x + 70 = 36
\]
\[
9x = 36 - 70
\]
\[
9x = -34
\]
\[
x = \frac{-34}{9}
\]

### Final Answer:

The solution to the system of equations is:
\[
x = \frac{-34}{9}, \quad y = \frac{-35}{6}
\]

Would you like more details or have any questions?

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Here are 5 related questions for practice:
1. Solve the system using substitution: \(4x + 5y = 7\) and \(6x - y = 3\).
2. Solve the system using linear combinations: \(2x + 3y = 6\) and \(4x - 3y = -2\).
3. How would the solution method change if the system had no solution?
4. How can you recognize if a system of equations is dependent or independent?
5. What is the graphical interpretation of solving a system of two linear equations?

**Tip**: Always check your solution by substituting the values of \(x\) and \(y\) back into both original equations to verify they satisfy both!

Solve the system using the linear combination method. 3x - 4y = 12, 6x - 2y = -11

Learn how to solve the system of linear equations 3x - 4y = 12 and 6x - 2y = -11 using the linear combination method. This tutorial provides a step-by-step explanation of the elimination method, making it suitable for students in Grades 9-11.

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