Pair of Linear Equations in Two Variables - I

The highest power of the term in the given expression is called degree.
The expression having degree one is a linear expression.
So, 4x – 3y is a linear expression.

The two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 will be parallel if $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.
^_$\because$        $\frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}$, $\frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2}$ and $\frac{c_1}{c_2} = \frac{5}{12} $.
^{_$\therefore$}        $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$
Hence, 2x + 3y = 5 and 4x + 6y = 12 are parallel lines.

Twice the number of mangoes Saurav has is 2y. Adding 13 to this gives the number of mangoes Gautam has, which is x. Therefore, x = 2y + 13.

2x – y + 4 = 2 (0) – (3) + 4 = 0 – 3 + 4 = 1

These two lines N and M do not intersect each other.
$\therefore$ There is no common solution.

The highest power of the term in the given equation is called degree.
An equation having degree one, is called a linear equation.
^_$\because$x = 4 is an equation of degree one, therefore it is a linear equation.

Let Jatin has y number of pens and Lalit has x number of pens.
Jatin has 4 pens more than twice the number of pens Lalit has.
y = 4 + 2x

The highest power of the term in the given equation is called degree.
The equation having a degree one is called a linear equation.
Therefore, 7x + 10 = 0 is the linear equation.

When 2 lines coincide, every point is common, then there are infinite solutions.

x + y – 3
            = 1 + 2 – 3
            = 3 – 3 = 0

The highest power of the term in the given expression is called degree. An expression having degree one is a linear expression. Therefore, 4x + 3y + 2 is the linear expression.

For the pair of equations to represent coincident lines,
$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
In lines x + 2y = 7 and 2x + 4y = 14,
$\frac{a_1}{a_2} = \frac{1}{2}, \frac{b_1}{b_2} =\frac{2}{4} =\frac{1}{2}, \frac{c_1}{c_2} = \frac{7}{14} = \frac{1}{2}$
So, _{$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$}

Let length of the rectangle be y and breadth of the rectangle be x.

As length of the rectangle is 6 units more than twice its breadth, $\therefore$ y = 6 + 2x 

The highest power of the term in the given expression is called degree.
The expression having a degree one is a linear expression.
Therefore, x + z is a linear expression.

Two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are parallel when
$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.
In lines x – y = 7 and 2x – 2y = 15,
$\frac{a_1}{a_2} = \frac{1}{2}, \frac{b_1}{b_2} =\frac{-1}{-2} , \frac{c_1}{c_2} = \frac{7}{15}$
^{_$\therefore$}        $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

Runs scored by Sachin is 25 more than thrice the runs score by Rahul.

Number of red chairs is 5 less than twice the number of yellow ones, 2x = y + 5

x + 2y - 4
= (1) + 2(1) - 4
= 1 + 2 - 4
= - 1

A pair of lines is coincident, if $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.
In lines, x + 2y = 3 and 4x + 8y = 12
$\frac{a_1}{a_2} = \frac{1}{4}, \frac{b_1}{b_2} =\frac{2}{8} = \frac{1}{4} , \frac{c_1}{c_2} = \frac{3}{12} = \frac{1}{4}$
$\therefore$   $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

The highest power of the term in the given equation is called degree.
The equation having a degree one is called a linear equation.

x is the length and y is the breadth. x is equal to 3 more than the twice of y.
Hence x = 2y + 3.

2x - 2y + 4
= 2(4) - 2(0) + 4
= 8 - 0 + 4
= 8 + 4
= 12

When 2 lines intersect at one point, the pair of linear equations has a unique solution.

Denominator is 2 more than thrice its numerator, y = 2 + 3x

The highest power of the term in the given equation is called degree.
The equation having a degree one is called a linear equation.
Therefore, the equation x + y - 3 = 0 is a linear equation.

Denominator is 5 more than twice its numerator, y = 5 + 2x

For two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to be parallel,
$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$.
In lines x – y = 17 and 2x – 2y = 38,
$\frac{a_1}{a_2} = \frac{1}{2}, \frac{b_1}{b_2} =\frac{-1}{-2} = \frac{1}{2} , \frac{c_1}{c_2} = \frac{3}{12} = \frac{17}{38}$
Therefore, ^{_{$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$}}
Hence, the lines are parallel.

The expression having a degree one is called a linear expression. Therefore, 3xy + 7 is not a linear expression because its degree is 2.