If the straight line y = `mx +c` passes through the points (2,4) and `(-3,6)` , find the values of m and c .

Answer

To find the values of \( m \) and \( c \) for the line \( y = mx + c \) that passes through the points \( (2, 4) \) and \( (-3, 6) \), we can follow these steps:

### Step 1: Substitute the first point into the line equation
We start with the equation of the line:
\[ y = mx + c \]
Substituting the first point \( (2, 4) \):
\[ 4 = m(2) + c \]
This simplifies to:
\[ 2m + c = 4 \quad \text{(Equation 1)} \]

### Step 2: Substitute the second point into the line equation
Next, we substitute the second point \( (-3, 6) \):
\[ 6 = m(-3) + c \]
This simplifies to:
\[ -3m + c = 6 \quad \text{(Equation 2)} \]

### Step 3: Set up the system of equations
Now we have a system of two equations:
1. \( 2m + c = 4 \) (Equation 1)
2. \( -3m + c = 6 \) (Equation 2)

### Step 4: Eliminate \( c \)
To eliminate \( c \), we can subtract Equation 1 from Equation 2:
\[ (-3m + c) - (2m + c) = 6 - 4 \]
This simplifies to:
\[ -3m - 2m + c - c = 2 \]
\[ -5m = 2 \]
Thus, we find:
\[ m = -\frac{2}{5} \]

### Step 5: Substitute \( m \) back to find \( c \)
Now that we have \( m \), we can substitute it back into Equation 1 to find \( c \):
\[ 2m + c = 4 \]
Substituting \( m = -\frac{2}{5} \):
\[ 2\left(-\frac{2}{5}\right) + c = 4 \]
This simplifies to:
\[ -\frac{4}{5} + c = 4 \]
Adding \( \frac{4}{5} \) to both sides:
\[ c = 4 + \frac{4}{5} \]
Converting 4 to a fraction:
\[ c = \frac{20}{5} + \frac{4}{5} = \frac{24}{5} \]

### Final Result
Thus, the values of \( m \) and \( c \) are:
\[ m = -\frac{2}{5}, \quad c = \frac{24}{5} \]

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