Linear System

Algebra Level 3

Check the option that identifies the locus of all ordered pairs (a, b) ∈ R² that make the linear system impossible

$S :\left\{\begin{array}{l}{-x+5 y=10} \\ {\left(\frac{a^{2}}{5}+5 b^{2}\right) x+10 a b y=1}\end{array}\right.$

A hyperbole A parable An ellipse A line

2 solutions

Δrchish Ray
Jul 21, 2019

In order for the system of the equations to be unsolvable, the ratios of the coefficients of $x$ and $y$ must be the same in each equation. For example:

$\begin{cases} 3x + 5y = 10 \\ 6x + 10y = 14 \end{cases}$

Therefore, $\displaystyle \frac{10ab}{ \frac{ a^{2} }{5} + 5b^{2}} = \frac{5}{-1}$

$\displaystyle \Longrightarrow 10ab = (-5)(\frac{ a^{2} }{5} + 5b^{2})$

$\displaystyle \Longrightarrow 10ab = -a^{2} - 25b^{2}$

$\displaystyle \Longrightarrow 10ab + a^{2} + 25b^{2} = 0$

$\displaystyle \Longrightarrow (a+5b)^{2} = 0$

$\displaystyle \Longrightarrow a+5b = 0$

$\displaystyle \Longrightarrow a = -5b$

The final equation gives us that the relationship between $a$ and $b$ is linear, and thus the graph will form a $\fbox{line}$ .

A Former Brilliant Member
Jul 21, 2019

There are lots of spelling mistakes. (I) parabola (ii) hyperbola. Also, simply a line is not enough. Lines can be straight or curved. The correct option should be a straight line.

Euclid defined the word line to be a straight line. The person may not speak English well and/or autocorrect may be to blame for the misspellings, though they do not take away anything from the problem.

But just report the problem instead of posting a solution.

Δrchish Ray - 1 year, 10 months ago

Linear System

2 solutions

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