Math, asked by karishamaisrani, 10 months ago

Show algebraically that the
system
2x +3y =8 & 6x+9y =24
is consistent & have infinite solutions ​

Answers

Answered by TheValkyrie
34

Answer:

\Large{\underline{\underline{\it{Given:}}}}

  • 2x+3y = 8
  • 6x+9y = 24

\Large{\underline{\underline{\it{To\:Show:}}}}

  • The pair of equations have consistent and infinite solutions

\Large{\underline{\underline{\it{Solution:}}}}

\dag A pair of linear equations have infinite and consistent solution if

 \dfrac{a_1}{a_2} =\dfrac{b_1}{b_2} =\dfrac{c_1}{c_2}

 where a₁ = 2, a₂ = 6, b₁=3, b₂=9, c₁=8, c₂=24

  \dfrac{a_1}{a_2} =\dfrac{2}{6}

 \dfrac{b_1}{b_2} =\dfrac{3}{9}

 \dfrac{c_1}{c_2} =\dfrac{8}{24}

\dfrac{2}{6} =\dfrac{3}{9} =\dfrac{8}{24}

\dfrac{1}{3} =\dfrac{1}{3} =\dfrac{1}{3}

\dag Here

\dfrac{a_1}{a_2} =\dfrac{b_1}{b_2} =\dfrac{c_1}{c_2}

\dag Hence the pair of equation has infinite umber of solution and is consistent.

\Large{\underline{\underline{\it{Notes:}}}}

\dag If a pair of equations has a unique solution and is consistent,

 \dfrac{a_1}{a_2}\neq  \dfrac{b_1}{b_2}

\dag If a pair of equations has infinte solutions and is consistent

 \dfrac{a_1}{a_2} =\dfrac{b_1}{b_2} =\dfrac{c_1}{c_2}

\dag If a pair of equations has no solution and is inconsistent,

 \dfrac{a_1}{a_2} =\dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}

Answered by varadad25
9

Answer:

The given pair of linear equations has consistent and infinite solutions.

Step-by-step-explanation:

The given simultaneous equations are

\sf\:2x\:+\:3y\:=\:8\:\:-\:-\:(\:1\:)\:\&\\\\\sf\:6x\:+\:9y\:=\:24\:\:\:-\:-\:(\:2\:)

For equation ( 1 ),

\sf\:2x\:+\:3y\:=\:8\:\:\:-\:-\:(\:1\:)\\\\\\\bullet\sf\:a_1\:=\:2\\\\\\\bullet\sf\:b_1\:=\:3\\\\\\\bullet\sf\:c_1\:=\:8

For equation ( 2 ),

\sf\:6x\:+\:9y\:=\:24\:\:\:-\:-\:(\:2\:)\\\\\\\bullet\sf\:a_2\:=\:6\\\\\\\bullet\sf\:b_2\:=\:9\\\\\\\bullet\sf\:c_2\:=\:24

Now,

\sf\:\dfrac{a_1}{a_2}\:=\:\cancel{\dfrac{2}{6}}\\\\\\\implies\boxed{\red{\sf\:\dfrac{a_1}{a_2}\:=\:\dfrac{1}{3}}}\sf\:\:\:-\:-\:(\:3\:)

Now,

\sf\:\dfrac{b_1}{b_2}\:=\:\cancel{\dfrac{3}{9}}\\\\\\\implies\boxed{\red{\sf\:\dfrac{b_1}{b_2}\:=\:\dfrac{1}{3}}}\sf\:\:\:-\:-\:(\:4\:)

Now,

\sf\:\dfrac{c_1}{c_2}\:=\:\cancel{\dfrac{8}{24}}\\\\\\\implies\boxed{\red{\sf\:\dfrac{c_1}{c_2}\:=\:\dfrac{1}{3}}}\sf\:\:\:-\:-\:(\:5\:)

From equations ( 3 ), ( 4 ) & ( 5 ),

\pink{\sf\:\dfrac{a_1}{a_2}\:=\:\dfrac{b_1}{b_2}\:=\:\dfrac{c_1}{c_2}}

Hence, the given linear equations have consistent and infinite solutions.

\\

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.

3. Conditions and the nature of solutions:

\begin{array}{|c|c|} \cline{1-2}\sf\:Relation\:between\:coefficients & \sf\:Nature\:of\:solutions\\\cline{1-2}\sf\:\dfrac{a_1}{a_2}\:=\:\dfrac{b_1}{b_2}\:\neq\:\dfrac{c_1}{c_2} & \sf\:No\:solution\\\cline{1-2}\sf\:\dfrac{a_1}{a_2}\:=\:\dfrac{b_1}{b_2}\:=\:\dfrac{c_1}{c_2} & \sf\:Infinite\:solutions\\\cline{1-2}\sf\:\dfrac{a_1}{a_2}\:\neq\:\dfrac{b_1}{b_2} & \sf\:Unique\:solution\\\cline{1-2}\end{array}

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