Solve 2x + 3y = 11 and 2x – 4y = – 24 and hence find the value of ‘m’ for which y = mx + 3.
Step-by-Step Solution
Key Concept: Use the method of elimination (or substitution) to solve the pair of linear equations. Once the ordered pair (x, y) is obtained, substitute it in the given linear equation y = mx + 3 to determine the required value of m.
1. Write the two equations\
\[\begin{aligned}
& (1)\; 2x + 3y = 11 \\
& (2)\; 2x - 4y = -24
\end{aligned}\]
2. Eliminate one variable – subtract equation (2) from equation (1):\
\[ (2x+3y) - (2x-4y) = 11 - (-24) \]
\[ 2x + 3y - 2x + 4y = 35 \]
\[ 7y = 35 \]
\[ y = 5 \]
3. Find x by substituting y = 5 in either equation (use (1)):\
\[ 2x + 3(5) = 11 \]
\[ 2x + 15 = 11 \]
\[ 2x = -4 \]
\[ x = -2 \]
Hence the solution of the simultaneous equations is \[ (x, y) = (-2,\;5). \]
4. Find m such that the line \( y = mx + 3 \) passes through the point \((-2,5)\). Substitute x = -2, y = 5:\
\[ 5 = m(-2) + 3 \]
\[ 5 = -2m + 3 \]
\[ -2m = 2 \]
\[ m = -1 \]
5. Result: The required value of \( m \) is \( -1 \).
Correct Answer: m = -1