What will be the value of m and c if the straight line `y=mx+c` passes through the points (3, -4) and (-1, 2) ?

Answer

To find the values of \( m \) and \( c \) in the equation of the straight line \( y = mx + c \) that passes through the points \( (3, -4) \) and \( (-1, 2) \), we can follow these steps:

### Step 1: Set up the equations using the given points

Using the first point \( (3, -4) \):
\[ y = mx + c \implies -4 = m(3) + c \]
This can be rewritten as:
\[ 3m + c = -4 \quad \text{(Equation 1)} \]

Using the second point \( (-1, 2) \):
\[ y = mx + c \implies 2 = m(-1) + c \]
This can be rewritten as:
\[ -m + c = 2 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations

Now, we have the following system of equations:
1. \( 3m + c = -4 \) (Equation 1)
2. \( -m + c = 2 \) (Equation 2)

To eliminate \( c \), we can subtract Equation 2 from Equation 1:
\[ (3m + c) - (-m + c) = -4 - 2 \]
This simplifies to:
\[ 3m + c + m - c = -6 \]
\[ 4m = -6 \]
Now, solve for \( m \):
\[ m = \frac{-6}{4} = -\frac{3}{2} \]

### Step 3: Substitute \( m \) back to find \( c \)

Now that we have \( m \), we can substitute it back into either Equation 1 or Equation 2 to find \( c \). Let's use Equation 1:
\[ 3m + c = -4 \]
Substituting \( m = -\frac{3}{2} \):
\[ 3\left(-\frac{3}{2}\right) + c = -4 \]
\[ -\frac{9}{2} + c = -4 \]
To isolate \( c \), we add \( \frac{9}{2} \) to both sides:
\[ c = -4 + \frac{9}{2} \]
Convert -4 to a fraction:
\[ c = -\frac{8}{2} + \frac{9}{2} = \frac{1}{2} \]

### Final Result

Thus, the values are:
\[ m = -\frac{3}{2}, \quad c = \frac{1}{2} \]

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