# Pair of Linear Equations in Two Variables - Class 10 - Mathematics

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## Extra Questions - Pair of Linear Equations in Two Variables | NCERT | Mathematics | Class 10

Form the pair of linear equations for the following problem and find their solution by substitution method: The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.

Let the **larger angle** be denoted by $x$ and the **smaller angle** by $y$. Since both angles are supplementary (they add up to $180^\circ$), we can write their relationship as:

$$ x + y = 180 \quad \text{(i)} $$

It is given that the difference between the larger angle and the smaller angle is $18^\circ$. Thus, we have:

$$ x - y = 18 \quad \text{(ii)} $$

### Using the Substitution Method:

From equation (i), solve for $x$:

$$ x = 180 - y \quad \text{(iii)} $$

Substitute the expression from equation (iii) into equation (ii):

$$ (180 - y) - y = 18 $$

Combine like terms:

$$ 180 - 2y = 18 $$

To isolate $y$, subtract $180$ from both sides:

$$ -2y = 18 - 180 \ -2y = -162 $$

Divide by -2:

$$ y = \frac{-162}{-2} = 81 $$

Hence, the **smaller angle** $y$ is $81^\circ$.

Now, substitute $y = 81$ back into equation (iii) to find $x$:

$$ x = 180 - 81 = 99 $$

Thus, the **larger angle** $x$ is $99^\circ$.

### Conclusion:

The two supplementary angles are **$99^\circ$** and **$81^\circ$**, where the larger angle exceeds the smaller by $18^\circ$.

Given graph represents a pair of linear equations having a solution(s).

A) Consistent, unique B) Inconsistent, zero C) Dependent, infinite D) None of these

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1 (vi)

Solve the following pair of linear equations by the substitution method.

$$ \frac{3x}{2} - \frac{5y}{3} = -2 ; \frac{x}{3} + \frac{y}{2} = \frac{13}{6} $$

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Find the equation of the line passing through $(2,1)$ and parallel to the line $2x-y=4$.

(A) $y=\frac{2}{5}x-1$

(B) $y=5x-2$ (C) $y=2x-3$

(D) None of these

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Put the equation $\frac{x}{a} + \frac{y}{b} = 1$ into slope-intercept form and find its slope and $y$-intercept.

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Equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$ and $\frac{x}{4}=\frac{y}{2}=\frac{z}{3}$ is

A) $x+2y-2z=0$

B) $3x+2y-2z=0$

C) $x-2y+z=0$

D) $5x+2y-4z=0$

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Which of the following is true if $A$ and $C$ are coefficient and augmented matrices, respectively, for a system of linear equations? Here $n =$ number of unknowns.

A. Rank $A = \text{Rank of } C = n$, Infinite solutions

B. Rank $A = \text{Rank of } C = r, r < n$, Infinite solutions

C. Rank $A = \text{Rank of } C = n$, unique solution

D. Rank $A = \text{Rank of } C = r, r < n$, unique solution

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Solve the pair of equations using the method of substitution: $3x - y = 8$, $5x + y = 0$ [2 MARKS].