2. Solve and and hence find the value of for which .

Understand the Question We are given a pair of linear equations: and . First, solve this system to determine the values of and using the substitution method. Then, substitute these values of and into the given linear relation to find the value of . Step 1 · Solve for and Given the equations: From equation , express in terms of : Substitute this expression for into equation : Multiply the entire equation by : Substitute into the expression for : Step 2 · Find the Value of Substitute and into : Answer > Common Mistakes Sign Error in Expansion: Incorrectly expanding as instead of . Variable Confusion: Accidentally swapping the coordinates and substituting and into . Division by Negative: Forgetting the negative sign when solving , which incorrectly yields instead of .

Pair of Linear Equations in Two Variables | Exercise 3.2

Question 2

Solve $2x + 3y = 11$ and $2x - 4y = -24$ and hence find the value of $m$ for which $y = mx + 3$ .

Solution

Understand the Question

We are given a pair of linear equations: $2x + 3y = 11$ and $2x - 4y = -24$ .
First, solve this system to determine the values of $x$ and $y$ using the substitution method.
Then, substitute these values of $x$ and $y$ into the given linear relation $y = mx + 3$ to find the value of $m$ .

Step 1 · Solve for $x$ and $y$

Given the equations: $2x + 3y = 11 \quad \dots (1)$ $2x - 4y = -24 \quad \dots (2)$

From equation $(1)$ , express $y$ in terms of $x$ :

\begin{aligned} 3y &= 11 - 2x \\[0.6em] y &= \dfrac{11 - 2x}{3} \end{aligned}

Substitute this expression for $y$ into equation $(2)$ : $2x - 4\left(\dfrac{11 - 2x}{3}\right) = -24$

Multiply the entire equation by $3$ :

\begin{aligned} 3(2x) - 4(11 - 2x) &= 3(-24) \\ 6x - 44 + 8x &= -72 \\ 14x - 44 &= -72 \\ 14x &= -72 + 44 \\ 14x &= -28 \\[0.6em] x &= \dfrac{-28}{14} = -2 \end{aligned}

Substitute $x = -2$ into the expression for $y$ :

\begin{aligned} y &= \dfrac{11 - 2(-2)}{3} \\[0.6em] &= \dfrac{11 + 4}{3} \\[0.6em] &= \dfrac{15}{3} \\[0.6em] &= 5 \end{aligned}

Step 2 · Find the Value of $m$

Substitute $x = -2$ and $y = 5$ into $y = mx + 3$ :

\begin{aligned} 5 &= m(-2) + 3 \\ 5 &= -2m + 3 \\ 5 - 3 &= -2m \\ 2 &= -2m \\[0.6em] m &= \dfrac{2}{-2} = -1 \end{aligned}

Answer

$x = -2, \; y = 5, \; m = -1$

Common Mistakes

Sign Error in Expansion: Incorrectly expanding $-4(11 - 2x)$ as $-44 - 8x$ instead of $-44 + 8x$ .
Variable Confusion: Accidentally swapping the coordinates and substituting $x = 5$ and $y = -2$ into $y = mx + 3$ .
Division by Negative: Forgetting the negative sign when solving $2 = -2m$ , which incorrectly yields $m = 1$ instead of $m = -1$ .

Question 2

More questions in Exercise 3.2