Question

Decide whether the graphs of the two equations are $$ 3 x+9 y+2=0 ; 2 y=-6 x+3 $$

Short answer

No, the graphs of the two equations are not the same.

Step-by-step solution

Simplify the first equation

The first equation is \(3x + 9y + 2 = 0\). To simplify this equation into slope-intercept form \(y=mx+b\), first isolate y. The first step is to subtract \(3x + 2\) from both sides, which results in \(9y = -3x - 2\). Divide every term by 9 to get \(y = -\frac{1}{3}x - \frac{2}{9}\).

Simplify the second equation

The second equation is \(2y = -6x + 3\). To simplify this equation into slope-intercept form \(y=mx+b\), divide every term by 2 to get \(y = -3x + \frac{3}{2}\).

Compare the two equations

From the above simplifications, first equation is \(y = -\frac{1}{3}x - \frac{2}{9}\) and the second equation is \(y = -3x + \frac{3}{2}\). Compare the slopes and the y-intercepts of the two equations. The slope of the first equation is -\frac{1}{3} and the slope of the second equation is -3. Similarly, the y-intercept of the first equation is -\frac{2}{9} and the y-intercept of the second equation is \frac{3}{2}. Since both the slopes and the y-intercepts are not equal, the given equations are not the same.