Question

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Explanation:

Step1: Rewrite first equation in slope - intercept form

The slope - intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.
For the equation \(2x+3y = 6\), we solve for \(y\):
Subtract \(2x\) from both sides: \(3y=-2x + 6\)
Divide both sides by 3: \(y=-\frac{2}{3}x + 2\)

Step2: Rewrite second equation in slope - intercept form

For the equation \(6x=-9y + 18\), we solve for \(y\):
First, add \(9y\) to both sides: \(9y+6x = 18\)
Subtract \(6x\) from both sides: \(9y=-6x + 18\)
Divide both sides by 9: \(y=-\frac{6}{9}x + 2=-\frac{2}{3}x + 2\)

Step3: Analyze the two lines

Since both equations, when rewritten in slope - intercept form, are \(y =-\frac{2}{3}x+2\), this means that the two lines are identical (they have the same slope and the same y - intercept). When two lines in a system of linear equations are identical, the system has infinitely many solutions, and the solution set is all the points that lie on the line \(y =-\frac{2}{3}x + 2\) (or equivalently, on the line represented by either of the original equations).

Answer:

The system of equations has infinitely many solutions, and the solution set is all ordered pairs \((x,y)\) such that \(2x + 3y=6\) (or \(6x=-9y + 18\)), which can be written as \(\{(x,y)\mid y =-\frac{2}{3}x + 2,x\in\mathbb{R}\}\)