To determine which graph matches the equation [tex]\( y + 6 = \frac{3}{4}(x + 4) \)[/tex], we should rewrite the equation in slope-intercept form, which is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Here is a step-by-step solution:
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. Isolate [tex]\( y \)[/tex] by subtracting 6 from both sides:
[tex]\[ y = \frac{3}{4}x + 3 - 6 \][/tex]
5. Simplify further:
[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]
From this equation, we can see:
- The slope (m) is [tex]\(\frac{3}{4}\)[/tex] or 0.75.
- The y-intercept (b) is -3.
To match the graph with this equation:
- Look for a line that crosses the y-axis at [tex]\( y = -3 \)[/tex].
- Ensure that the line has a slope of [tex]\(\frac{3}{4}\)[/tex], which means for every 4 units you move to the right on the x-axis, the line moves up 3 units.
By verifying these characteristics, you can identify the correct graph.
