Solve `2x+ 3y= 11 and 2x-4y= -24` and hence find the value of 'm' for which y= mx+ 3.

Answer

To solve the system of equations \(2x + 3y = 11\) and \(2x - 4y = -24\), we will use the elimination method.

### Step 1: Write down the equations
We have:
1. \(2x + 3y = 11\) (Equation 1)
2. \(2x - 4y = -24\) (Equation 2)

### Step 2: Eliminate \(x\)
We can eliminate \(x\) by subtracting Equation 2 from Equation 1.

\[ (2x + 3y) - (2x - 4y) = 11 - (-24) \]

This simplifies to:

\[ 3y + 4y = 11 + 24 \]

\[ 7y = 35 \]

### Step 3: Solve for \(y\)
Now, divide both sides by 7:

\[ y = \frac{35}{7} = 5 \]

### Step 4: Substitute \(y\) back to find \(x\)
Now that we have \(y = 5\), we can substitute this value back into either Equation 1 or Equation 2 to find \(x\). We will use Equation 1:

\[ 2x + 3(5) = 11 \]

This simplifies to:

\[ 2x + 15 = 11 \]

Subtract 15 from both sides:

\[ 2x = 11 - 15 \]

\[ 2x = -4 \]

Now, divide both sides by 2:

\[ x = \frac{-4}{2} = -2 \]

### Step 5: Write the solution
The solution to the system of equations is \(x = -2\) and \(y = 5\).

### Step 6: Find the value of \(m\)
We need to find the value of \(m\) such that \(y = mx + 3\). We already have \(y = 5\) and \(x = -2\). Substitute these values into the equation:

\[ 5 = m(-2) + 3 \]

Rearranging gives:

\[ 5 - 3 = -2m \]

\[ 2 = -2m \]

Now, divide both sides by -2:

\[ m = \frac{2}{-2} = -1 \]

### Final Answer
The value of \(m\) is \(-1\).

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