Solve the following systems of equations has unique solution and … - Vidyadip

Question

Solve the following systems of equations has unique solution and solve it:
$2x - 3y = 17, 4x + y = 13$

✓

Answer

$2x - 3y = 17, 4x + y = 13$
$2x - 3y - 17 = 0, 4x + y - 13 = 0$
We know that, the system of linear equations $a_1x + b_1y + c_1 = 0, a_2x + b_2y + c_2 = 0$
Has a unique solution if $\frac{\text{a}_1}{\text{a}_2}\neq\frac{\text{b}_1}{\text{b}_2}.$
$\frac{\text{a}_1}{\text{a}_2}=\frac{2}{4}=\frac{1}{2}$
$\frac{\text{b}_1}{\text{b}_2}=\frac{-3}{1}=-3$
Clearly, $\frac{\text{a}_1}{\text{a}_2}\neq\frac{\text{b}_1}{\text{b}_2}.$
So, the system has a unique solution.
$2x - 3y = 17 ...(1)$
$4x + y = 13 ...(2)$
Multiply (2) by 3 and add to (1).
$14x = 56 ? x = 4$
Substituting x = 4 in (1), we get y = -3.
So, the solution is $x = 4, y = -3.$

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