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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
- Q1(i): Solve the following pair of linear equations by the substitution method. (i) x + y = 14; x – y = 4
- Q1(ii): Solve the following pair of linear equations by the substitution method. (ii) s – t = 3; $\frac{s}{3} + \frac{t}{2} = 6$
- Q1(iii): Solve the following pair of linear equations by the substitution method. (iii) 3x – y = 3; 9x – 3y = 9
- Q1(iv): Solve the following pair of linear equations by the substitution method. (iv) 0.2x + 0.3y = 1.3; 0.4x + 0.5y = 2.3
- Q1(v): Solve the following pair of linear equations by the substitution method. (v) $\sqrt{2}x + \sqrt{3}y = 0$; $\sqrt{3}x - \sqrt{8}y = 0$
- Q1(vi): Solve the following pair of linear equations by the substitution method. (vi) $\frac{3x}{2} - \frac{5y}{3} = -2$; $\frac{x}{3} + \frac{y}{2} = \frac{13}{6}$
- Q3(i): Form the pair of linear equations for the following problems and find their solution by substitution method. (i) The difference between two numbers is 26 and one number is three times the other. Find them.
- Q3(ii): Form the pair of linear equations for the following problems and find their solution by substitution method. (ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.
- Q3(iii): Form the pair of linear equations for the following problems and find their solution by substitution method. (iii) The coach of a cricket team buys 7 bats and 6 balls for ` 3800. Later, she buys 3 bats and 5 balls for ` 1750. Find the cost of each bat and each ball.
- Q3(iv): Form the pair of linear equations for the following problems and find their solution by substitution method. (iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is ` 105 and for a journey of 15 km, the charge paid is ` 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?
- Q3(v): Form the pair of linear equations for the following problems and find their solution by substitution method. (v) A fraction becomes $\frac{9}{11}$, if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes $\frac{5}{6}$. Find the fraction.
- Q3(vi): Form the pair of linear equations for the following problems and find their solution by substitution method. (vi) Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?
CBSE Solutions for Class 10 Mathematics Pair of linear equations in two variable
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