To determine which graph matches the equation [tex]\( y + 6 = \frac{3}{4}(x + 4) \)[/tex], we need to transform this equation 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. Here’s the step-by-step solution:
1. Start with the given equation:
[tex]\[
y + 6 = \frac{3}{4}(x + 4)
\][/tex]
2. Distribute the [tex]\(\frac{3}{4}\)[/tex] on the right side:
[tex]\[
y + 6 = \frac{3}{4} \cdot x + \frac{3}{4} \cdot 4
\][/tex]
3. Simplify the right side:
[tex]\[
y + 6 = \frac{3}{4} x + 3
\][/tex]
4. Isolate [tex]\( y \)[/tex] by subtracting 6 from both sides:
[tex]\[
y = \frac{3}{4} x + 3 - 6
\][/tex]
5. Simplify the right side:
[tex]\[
y = \frac{3}{4} x - 3
\][/tex]
Now we have the equation in slope-intercept form [tex]\( y = \frac{3}{4}x - 3 \)[/tex].
- Slope ([tex]\(m\)[/tex]): [tex]\( \frac{3}{4} \)[/tex] or [tex]\( 0.75 \)[/tex]
- Y-intercept ([tex]\(b\)[/tex]): [tex]\( -3 \)[/tex]
To identify the correct graph:
- Look for the line that crosses the y-axis at [tex]\( (0, -3) \)[/tex].
- Ensure the line rises [tex]\( \frac{3}{4} \)[/tex] units up for every 1 unit it moves to the right, which indicates a slope of [tex]\( 0.75 \)[/tex].
Verify the graph by checking these characteristics. The graph should accurately reflect these parameters to match the given equation.
