Question

a system of equations has infinitely many solutions. if $2y - 4x = 6$ is one of the equations, which could be the other equation?
\\(\circ\\) $y = 2x + 6$
\\(\circ\\) $y = 4x + 6$
\\(\circ\\) $-y = -2x - 3$
\\(\circ\\) $-y = -4x + 6$

Explanation:

Step1: Recall the condition for infinitely many solutions

A system of linear equations has infinitely many solutions when the two equations are equivalent (i.e., they represent the same line). So we need to simplify the given equation \(2y - 4x=6\) and compare it with the options.

Step2: Simplify the given equation

Start with \(2y - 4x = 6\). Divide the entire equation by 2:
\[
\frac{2y}{2}-\frac{4x}{2}=\frac{6}{2}
\]
Simplify each term: \(y - 2x=3\). Then, we can rewrite it as \(y = 2x + 3\) or multiply both sides by - 1: \(-y=-2x - 3\).

Step3: Analyze each option

• Option 1: \(y = 2x+6\). The slope is 2 (same as our simplified equation \(y = 2x + 3\)), but the y - intercept is 6 vs 3. Not equivalent.
• Option 2: \(y = 4x+6\). Slope is 4, different from 2. Not equivalent.
• Option 3: \(-y=-2x - 3\). Multiply both sides by - 1, we get \(y = 2x+3\), which is equivalent to our simplified equation \(y - 2x = 3\).
• Option 4: \(-y=-4x + 6\). Multiply by - 1: \(y = 4x-6\). Slope is 4, different from 2. Not equivalent.

Answer:

\(-y = - 2x-3\) (the third option)