Question

Show that the given system of equations has a unique solution in graph.
3x + 5y = 12
5x+3y=4​

Answer 1

Step-by-step explanation:

hence proved

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Answer 2

★Given:-

• System of equations 3x + 5y = 12 ;5x+3y=4​.

★To prove:-

• The given system of equations has a unique solution in graph.

★Proof:-

Let us have a glance at the concept first:-

If $\sf a_1x+b_1y+c_1 = 0\ ;\ a_2x+b_2y+c_2 = 0$ are two linear equations,then for unique solution we have the condition :

$\mapsto \sf \frac{a_1}{a_2} \neq \frac{b_1}{b_2}$

(The lines are neither parallel nor coincident)

Given ,3x + 5y = 12 ;5x+3y=4​

Here,

=> a1 = 3,a2 = 5

=> b1= 5,  b2 = 3

=> c1 = -12 , c2 = -4

Substituting these values in the above condition,

$:\implies \sf \frac{a_1}{a_2} \neq \frac{b_1}{b_2}\\\\:\implies \sf \frac{3}{5} \neq \frac{5}{3}$

As we can see ,it satisfies the condition.

Hence,the given system of equation has a unique solution.

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★ Know More:-

✦A system of equations is two or more equations containing  same variables.

✦Methods to solve systems of equations are graphing, substitution, elimination and matrices.

✦The system of an equation has infinitely many solutions when the lines are coincident.(Have same y-intercept).

Condition:-

$\mapsto \sf \frac{a_1}{a_2} =\frac{b_1}{b_2} =\frac{c_1}{c_2}$

✦The system of an equation as no solution when the lines are parallel .

Condition:-

$\mapsto \sf \frac{a_1}{a_2} =\frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

✦When there is no solution the equations are called "inconsistent".

✦ For unique or infinitely many solutions,it is called "consistent".

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