Question

muriel says she has written a system of two linear equations that has an infinite number of solutions. one of the equations of the system is $3y = 2x - 9$. which could be the other equation?\
\\(\boldsymbol{\circ}\\) $2y = x - 4.5$\
\\(\boldsymbol{\circ}\\) $y = \frac{2}{3}x - 3$\
\\(\boldsymbol{\circ}\\) $6y = 6x - 27$\
\\(\boldsymbol{\circ}\\) $y = \frac{3}{2}x - 4.5$

Explanation:

Step1: Recall the condition for infinite solutions

A system of linear equations \(a_1x + b_1y = c_1\) and \(a_2x + b_2y = c_2\) has infinite solutions if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\) (they are the same line, i.e., one equation is a multiple of the other). First, rewrite the given equation \(3y = 2x - 9\) in slope - intercept form \(y=mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept). Divide both sides of \(3y = 2x-9\) by 3: \(y=\frac{2}{3}x - 3\).

Step2: Analyze each option

• Option 1: \(2y=x - 4.5\). Rewrite in slope - intercept form by dividing by 2: \(y=\frac{1}{2}x-2.25\). The slope of this line is \(\frac{1}{2}\), and the slope of the given line is \(\frac{2}{3}\). Since \(\frac{1}{2}

eq\frac{2}{3}\), this line is not the same as the given line.

• Option 2: \(y = \frac{2}{3}x-3\). This is the same as the line we obtained from the given equation \(3y = 2x - 9\) (after dividing by 3). So, the two equations represent the same line, and the system will have an infinite number of solutions.
• Option 3: \(6y=6x - 27\). Rewrite in slope - intercept form by dividing by 6: \(y=x - 4.5\). The slope of this line is \(1\), and the slope of the given line is \(\frac{2}{3}\). Since \(1

eq\frac{2}{3}\), this line is not the same as the given line.

• Option 4: \(y=\frac{3}{2}x - 4.5\). The slope of this line is \(\frac{3}{2}\), and the slope of the given line is \(\frac{2}{3}\). Since \(\frac{3}{2}

eq\frac{2}{3}\), this line is not the same as the given line.

Answer:

B. \(y=\frac{2}{3}x - 3\) (where the option is the second one in the list of options: \(y=\frac{2}{3}x - 3\))