Question

determining the number of solutions to a system of equations
alvins first step in solving the given system of equations is to multiply the first equation by 2 and the second equation by -3. which linear combination of alvins system of equations reveals the number of solutions to the system?
$9x + 4y = 36$
$6x + 2.5y = 24$

• infinite solutions: $0x = 0y = 0$
• no solutions: $0x + 15.5y = 144$
• one solution: $0x + 0.5y = 0$
• two solutions: $0x - 0.5y = 60$

Explanation:

Step1: Multiply first equation by 2

$2\times(9x + 4y) = 2\times36 \implies 18x + 8y = 72$

Step2: Multiply second equation by -3

$-3\times(6x + 2.5y) = -3\times24 \implies -18x - 7.5y = -72$

Step3: Add the two new equations

$(18x + 8y) + (-18x - 7.5y) = 72 + (-72)$
$\implies 0x + 0.5y = 0$

Step4: Analyze the result

A non-trivial $0x + ay = 0$ (with $a
eq0$) means the system has one unique solution.

Answer:

One solution: $0x + 0.5y = 0$