Question

help me please don't spam​

Answer 1

Step-by-step explanation:

asume x as 2 and y as 1

so 5*2-10*1=0

and

x+y

2+1=3

Answer 2

Question:

Show that the system of equations

5x - 10y = 0 and x + y = 3 has unique solution. Also, find the value of x and y.

Answer:

The given system of equations has unique solution.

The solution of the given simultaneous equations is ( x, y ) = ( 2, 1 ).

Step-by-step-explanation:

We have given two simultaneous equations.

$\sf\:5x\:-\:10y\:=\:0\\\\\\\sf\:x\:-\:2y\:=\:0\:\:\:-\:-\:-\:(\:1\:)\:\:-\:[\:Dividing\:by\:5\:]\\\\\\\sf\:x\:+\:y\:=\:3\:\:\:\:-\:-\:(\:2\:)$

Now,

$\sf\:x\:-\:2y\:=\:0\:\:-\:-\:-\:(\:1\:)\\\\\\\bullet\sf\:a_1\:=\:1\\\\\\\bullet\sf\:b_1\:=\:-\:2\\\\\\\bullet\sf\:c_1\:=\:0\\\\\\\sf\:x\:+\:y\:=\:3\:\:\:-\:-\:(\:2\:)\\\\\\\bullet\sf\:a_2\:=\:1\\\\\\\bullet\sf\:b_2\:=\:1\\\\\\\bullet\sf\:c_2\:=\:3$

Now,

$\sf\:\dfrac{a_1}{a_2}\:=\:\dfrac{1}{1}\\\\\\\implies\pink{\sf\:\dfrac{a_1}{a_2}\:=\:1}\\\\\\\sf\:\dfrac{b_1}{b_2}\:=\:-\:\dfrac{2}{1}\\\\\\\implies\pink{\sf\:\dfrac{b_1}{b_2}\:=\:-\:2}\\\\\\\sf\:\dfrac{c_1}{c_2}\:=\:\dfrac{0}{3}\\\\\\\implies\pink{\sf\:\dfrac{c_1}{c_2}\:=\:0}$

Now, from the above values, we can conclude that,

$\red{\sf\:\dfrac{a_1}{a_2}\:\neq\:\dfrac{b_1}{b_2}\:\neq\:\dfrac{c_1}{c_2}}$

Hence, the given system of linear equations have unique solution.

Now,

$\sf\:x\:-\:2y\:=\:0\:\:\:-\:-\:(\:1\:)\\\\\\\sf\:x\:+\:y\:=\:3\:\:\:-\:-\:(\:2\:)$

Now, by subtracting equation ( 1 ) from equation ( 2 ), we get,

$\sf\:x\:+\:y\:=\:3\:\:\:-\:-\:(\:2\:)\\-\\\underline{\sf\:x\:-\:2y\:=\:0}\sf\:\:\:-\:-\:(\:1\:)\\\\\\\implies\sf\:3y\:=\:3\\\\\\\implies\sf\:y\:=\:\cancel{\frac{3}{3}}\\\\\\\implies\boxed{\red{\sf\:y\:=\:1}}$

Now, by substituting y = 1 in equation ( 2 ), we get,

$\sf\:x\:+\:y\:=\:3\:\:\:-\:-\:(\:2\:)\\\\\\\implies\sf\:x\:+\:1\:=\:3\\\\\\\implies\sf\:x\:=\:3\:-\:1\\\\\\\implies\boxed{\red{\sf\:x\:=\:2}}$

Additional Information:

1. Linear Equations in two variables:

The equation with the highest index (degree) 1 is called as linear equation. If the equation has two different variables, it is called as 'linear equation in two variables'.

The general formula of linear equation in two variables is

ax + by + c = 0

Where, a, b, c are real numbers and

a ≠ 0, b ≠ 0.

2. Solution of a Linear Equation:

The value of the given variable in the given linear equation is called the solution of the linear equation.