To determine which graph matches the given equation [tex]\( y + 3 = 2(x + 3) \)[/tex], we need to rewrite this equation in the slope-intercept form, which is [tex]\( y = mx + b \)[/tex] where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
Here are the steps:
1. Expand the Equation:
Start with the given equation:
[tex]\[
y + 3 = 2(x + 3)
\][/tex]
Distribute the [tex]\( 2 \)[/tex] on the right-hand side:
[tex]\[
y + 3 = 2x + 6
\][/tex]
2. Isolate [tex]\( y \)[/tex]:
To get the equation into the form [tex]\( y = mx + b \)[/tex], subtract [tex]\( 3 \)[/tex] from both sides:
[tex]\[
y + 3 - 3 = 2x + 6 - 3
\][/tex]
Simplify the equation:
[tex]\[
y = 2x + 3
\][/tex]
3. Identify the Slope and [tex]\( y \)[/tex]-Intercept:
Now the equation is in the slope-intercept form [tex]\( y = 2x + 3 \)[/tex]. From this equation, we can identify:
- The slope ([tex]\( m \)[/tex]) is [tex]\( 2 \)[/tex]
- The [tex]\( y \)[/tex]-intercept ([tex]\( b \)[/tex]) is [tex]\( 3 \)[/tex]
These values indicate how the line will appear on the graph:
- The slope of [tex]\( 2 \)[/tex] means that for each unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by [tex]\( 2 \)[/tex] units.
- The [tex]\( y \)[/tex]-intercept of [tex]\( 3 \)[/tex] means that the line crosses the [tex]\( y \)[/tex]-axis at the point [tex]\( (0, 3) \)[/tex].
4. Graphing:
To graph the line [tex]\( y = 2x + 3 \)[/tex]:
- Start by plotting the [tex]\( y \)[/tex]-intercept, which is the point [tex]\( (0, 3) \)[/tex].
- From this point, use the slope to find another point on the line. Since the slope is [tex]\( 2 \)[/tex], move up [tex]\( 2 \)[/tex] units and right [tex]\( 1 \)[/tex] unit from the [tex]\( y \)[/tex]-intercept to find the next point, which would be [tex]\( (1, 5) \)[/tex].
- Draw a straight line through the points.
Thus, the graph that matches the equation [tex]\( y + 3 = 2(x + 3) \)[/tex] is a line with a slope of [tex]\( 2 \)[/tex] and a [tex]\( y \)[/tex]-intercept of [tex]\( 3 \)[/tex], represented graphically as [tex]\( y = 2x + 3 \)[/tex]. This line goes through the points [tex]\( (0, 3) \)[/tex] and [tex]\( (1, 5) \)[/tex] on the Cartesian plane.
