CBSE · Class 10 · Mathematics · Pair of Linear Equations in Two VariablesSolve the following pair of linear equations by the elimination method: $$2x + 3y = 11$$ $$2x - 4y = -24$$\nHence, find the value of '$m$' for which $y = mx + 3$.

Given equations:

1. $2x + 3y = 11$ --- (Equation 1)
2. $2x - 4y = -24$ --- (Equation 2)

Step 1: Eliminate $x$ by subtracting Equation (2) from Equation (1) $$(2x + 3y) - (2x - 4y) = 11 - (-24)$$ $$2x - 2x + 3y + 4y = 11 + 24$$ $$7y = 35$$ $$y = \frac{35}{7} = 5$$

Step 2: Substitute $y = 5$ in Equation (1) to find $x$ $$2x + 3(5) = 11$$ $$2x + 15 = 11$$ $$2x = 11 - 15$$ $$2x = -4$$ $$x = \frac{-4}{2} = -2$$ \nSo, $x = -2$ and $y = 5$.

Step 3: Find the value of '$m$' using $y = mx + 3$\nSubstitute $x = -2$ and $y = 5$ into the equation: $$5 = m(-2) + 3$$ $$5 - 3 = -2m$$ $$2 = -2m$$ $$m = \frac{2}{-2} = -1$$

Final Answer: $x = -2$, $y = 5$, and $m = -1$.