To determine which graph represents the equation [tex]\( y + 2 = 3(x - 1) \)[/tex], we need to convert this equation into slope-intercept form, which is [tex]\( y = mx + b \)[/tex]. Here are the steps to convert and analyze the equation:
1. Start with the given equation:
[tex]\[ y + 2 = 3(x - 1) \][/tex]
2. Distribute the 3 on the right side:
[tex]\[ y + 2 = 3x - 3 \][/tex]
3. Isolate [tex]\( y \)[/tex] by subtracting 2 from both sides:
[tex]\[ y = 3x - 3 - 2 \][/tex]
4. Simplify the right-hand side:
[tex]\[ y = 3x - 5 \][/tex]
So, the equation in slope-intercept form is [tex]\( y = 3x - 5 \)[/tex].
Now, we can interpret this equation:
- The slope ([tex]\(m\)[/tex]) is 3, which means the line rises 3 units vertically for every 1 unit it moves horizontally.
- The y-intercept ([tex]\(b\)[/tex]) is -5, which means the line crosses the y-axis at [tex]\( (0, -5) \)[/tex].
The correct graph will be the one that has a slope of 3 and crosses the y-axis at -5.
So, the graph that represents the equation [tex]\( y = 3x - 5 \)[/tex] is the correct answer.
