Question

elena is solving this system of equations:
\\( \left\\{ \

$$\begin{array} { l } { 10 x - 6 y = 16 } \\\\ { 5 x - 3 y = 8 } \\end{array}$$

\
ight. \\)
she multiplies the second equation by 2, then subtracts the resulting equation from the first. to her surprise, she gets the equation \\( 0 = 0 \\).
what is special about this system of equations? why does she get this result and what does it mean about the solutions? (if you are not sure, try graphing them.)
\\( \bigcirc \\) the two equations have the same slope. the graphs show parallel lines which indicates there is no solution
\\( \bigcirc \\) the two equations are multiples of each other and are equivalent. equivalent equations are represented by the same graph and have the same solutions as one another. this system has infinitely many solutions.

Explanation:

Step1: Analyze the second equation multiplied by 2

Multiply the second equation \(10x - 6y=16\) by \(2\). We get \(2\times(10x - 6y)=2\times16\), which simplifies to \(20x-12y = 32\). But when Elena subtracts the first equation \(5x - 3y=8\) (after multiplying by 4: \(20x-12y = 32\)) from the doubled - second equation (which is also \(20x - 12y=32\)), we have \((20x-12y)-(20x - 12y)=32 - 32\).

Step2: Simplify the subtraction

\(20x-12y-20x + 12y=0\), which gives \(0 = 0\). This means the two equations are equivalent.

Answer:

The two equations are multiples of each other and are equivalent. Equivalent equations are represented by the same graph and have the same solutions as one another. This system has infinitely many solutions.