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Q2:
Solve 2x + 3y = 11 and 2x – 4y = – 24 and hence find the value of ‘$m$’ for which y = mx + 3.

Solution :

Given: A system of two linear equations in two variables:

(i) $2x + 3y = 11$

(ii) $2x - 4y = -24$

Additionally, a linear relation: $y = mx + 3$.

To Find: The values of $x$ and $y$ that satisfy the system of equations, and subsequently, the value of the constant $m$.

Step 1: Solving the system of linear equations using the Elimination Method.

We have the system:

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

$2x - 4y = -24$ --- (Equation 2)

To eliminate the variable $x$, we subtract Equation 2 from Equation 1:

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

$2x + 3y - 2x + 4y = 11 + 24$ [Distributing the negative sign]

$7y = 35$ [Combining like terms: $2x - 2x = 0$ and $3y + 4y = 7y$]

$y = \frac{35}{7}$ [Dividing both sides by 7]

$y = 5$

Step 2: Finding the value of $x$.

Substitute the value $y = 5$ into Equation 1:

$2x + 3(5) = 11$

$2x + 15 = 11$ [Performing multiplication]

$2x = 11 - 15$ [Transposing 15 to the right side]

$2x = -4$

$x = \frac{-4}{2}$ [Dividing both sides by 2]

$x = -2$

Step 3: Finding the value of $m$.

We are given the equation $y = mx + 3$. Substitute $x = -2$ and $y = 5$ into this equation:

$5 = m(-2) + 3$

$5 = -2m + 3$

$5 - 3 = -2m$ [Transposing 3 to the left side]

$2 = -2m$

$m = \frac{2}{-2}$ [Dividing both sides by -2]

$m = -1$

Final Answer: The solution to the system is $x = -2$ and $y = 5$. The value of $m$ for which $y = mx + 3$ is $m = -1$.


More Questions from Class 10 Mathematics Pair of linear equations in two variable EXERCISE 3.2


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