Let's go through the questions step by step to find the answers:
A) The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
B) Now we will use this formula to find the slope of the line containing the points \((1, 1)\) and \((3, 5)\):
- Here, \( (x_1, y_1) = (1, 1) \) and \( (x_2, y_2) = (3, 5) \).
- Substituting into the slope formula:
\[ m = \frac{5 - 1}{3 - 1} = \frac{4}{2} = 2 \]
So, the slope \( m = 2 \).
C) The line connecting the points \( (1, 1) \) and \( (3, 5) \) with a slope of 2 would rise 2 units for every 1 unit it moves to the right. When this graph is drawn, it will clearly pass through these points. As for the specific graphs, you would need to refer to the graphs provided, but the one that crosses through \( (1, 1) \) and \( (3, 5) \) is the correct option.
D) Now, to write the equation of the line in slope-intercept form \( y = mx + b \):
- We already found that \( m = 2 \).
- Next, we need to find \( b \), the y-intercept. We can do this by substituting one of the points into the equation.
Using the point \( (1, 1) \):
\[ 1 = 2(1) + b \implies 1 = 2 + b \implies b = 1 - 2 = -1 \]
Thus, the slope-intercept form of the line is:
\[ y = 2x - 1 \]
In terms of the blanks provided:
- Blank 1 would be 2 (for the slope \( m \)).
- Blank 2 would be -1 (for the y-intercept \( b \)).
- Blank 3 would be y = 2x - 1 (the final equation).
So the final answers are:
- Blank 1: 2
- Blank 2: -1
- Blank 3: y = 2x - 1
1 answer