To determine which graph matches the equation [tex]\( y + 6 = \frac{3}{4}(x + 4) \)[/tex], we need to rearrange it into the slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Let's break it down step-by-step:
1. Start with the given equation:
[tex]\[
y + 6 = \frac{3}{4}(x + 4)
\][/tex]
2. Distribute [tex]\(\frac{3}{4}\)[/tex] on the right-hand side:
[tex]\[
y + 6 = \frac{3}{4}x + \frac{3}{4} \cdot 4
\][/tex]
3. Simplify the right-hand side:
[tex]\[
y + 6 = \frac{3}{4}x + 3
\][/tex]
4. Subtract 6 from both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = \frac{3}{4}x + 3 - 6
\][/tex]
5. Simplify the equation:
[tex]\[
y = \frac{3}{4}x - 3
\][/tex]
Now we have the equation in slope-intercept form [tex]\( y = mx + b \)[/tex].
- The coefficient of [tex]\( x \)[/tex] ([tex]\( m \)[/tex]) represents the slope of the line. Here, the slope ([tex]\( m \)[/tex]) is [tex]\(\frac{3}{4}\)[/tex] or [tex]\(0.75\)[/tex].
- The constant term ([tex]\( b \)[/tex]) represents the y-intercept of the line. Here, the y-intercept ([tex]\( b \)[/tex]) is [tex]\(-3\)[/tex].
Therefore, the graph of the equation [tex]\( y + 6 = \frac{3}{4}(x + 4) \)[/tex] is a straight line with:
- A slope of [tex]\( 0.75 \)[/tex] (or [tex]\(\frac{3}{4}\)[/tex])
- A y-intercept of [tex]\(-3\)[/tex]
The correct graph will show a line that:
- Crosses the y-axis at [tex]\(-3\)[/tex]
- Rises [tex]\(3\)[/tex] units for every [tex]\(4\)[/tex] units it runs to the right.
