Question

which graph matches the equation $y + 3 = 2(x + 3)$?
four graphs with labeled points: first has (0, -3), (-3, -9); second has (0, 3), (-3, -3); third has (0, 6), (-3, 0); fourth has (3, 3), (0, -3)

Explanation:

Step1: Rewrite the equation in slope - intercept form

The given equation is \(y + 3=2(x + 3)\). We can rewrite it in the form \(y=mx + b\) (slope - intercept form, where \(m\) is the slope and \(b\) is the y - intercept) by expanding and simplifying:
First, expand the right - hand side: \(y+3 = 2x+6\).
Then, subtract 3 from both sides: \(y=2x + 6-3=2x + 3\). So the slope \(m = 2\) and the y - intercept \(b = 3\). We can also find the x - intercept by setting \(y = 0\):
\(0=2x + 3\)
\(2x=-3\)
\(x=-\frac{3}{2}=-1.5\) (not necessary for this comparison, but we can also use the point - slope form to find points on the line. The point - slope form \(y - y_1=m(x - x_1)\) has a point \((x_1,y_1)=(-3,-3)\) from the original equation \(y+3 = 2(x + 3)\) (when \(x=-3\), \(y=-3\)).

Step2: Analyze the y - intercept and the point \((-3,-3)\)

The y - intercept of the line \(y = 2x+3\) is \(3\) (when \(x = 0\), \(y=3\)). Also, the line passes through the point \((-3,-3)\) (from the point - slope form \(y+3 = 2(x + 3)\), when \(x=-3\), \(y=-3\)).

• For the first graph: The y - intercept is \(-3\) (when \(x = 0\), \(y=-3\)) and the point \((-3,-9)\) is on it. This does not match our line since our y - intercept is \(3\) and the point \((-3,-3)\) should be on the line, not \((-3,-9)\).
• For the second graph: The y - intercept is \(3\) (when \(x = 0\), \(y = 3\)) and the point \((-3,-3)\) is on it. This matches the line \(y=2x + 3\) (since when \(x=-3\), \(y=2\times(-3)+3=-6 + 3=-3\) and when \(x = 0\), \(y=3\)).
• For the third graph: The y - intercept is \(6\) (when \(x = 0\), \(y = 6\)) and the point \((-3,0)\) is on it. This does not match our line.
• For the fourth graph: The y - intercept is \(-3\) (when \(x = 0\), \(y=-3\)) and the point \((3,3)\) is on it. Let's check if \((3,3)\) is on \(y = 2x+3\): \(y=2\times3+3=6 + 3=9

eq3\). So this does not match.

Answer:

The second graph (with points \((0,3)\) and \((-3,-3)\) marked on it)