Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Find reliable answers to your questions from a wide community of knowledgeable experts on our user-friendly Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To determine the equation of the line passing through the points [tex]\(A(0, -4)\)[/tex] and [tex]\(B(6, 2)\)[/tex], we need to follow these steps:
1. Calculate the Slope:
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[ y_1 = -4, \quad y_2 = 2, \quad x_1 = 0, \quad x_2 = 6 \][/tex]
[tex]\[ m = \frac{2 - (-4)}{6 - 0} = \frac{2 + 4}{6} = \frac{6}{6} = 1 \][/tex]
2. Determine the Y-Intercept:
The slope-intercept form of a line is:
[tex]\[ y = mx + b \][/tex]
Given that the slope [tex]\(m\)[/tex] is 1 and using the point [tex]\(A(0, -4)\)[/tex]:
[tex]\[ y = 1 \cdot x + b \][/tex]
Since the point [tex]\(A\)[/tex] is on the line, substituting [tex]\(x = 0\)[/tex] and [tex]\(y = -4\)[/tex]:
[tex]\[ -4 = 1 \cdot 0 + b \implies b = -4 \][/tex]
Therefore, the equation of the line is:
[tex]\[ y = x - 4 \][/tex]
3. Verify the Provided Options:
Let's compare this equation with the given options:
- [tex]\( y - 4 = 3x \)[/tex]
[tex]\[ \text{Rewriting: } y = 3x + 4 \quad (\text{This has a slope of 3, not 1}) \][/tex]
- [tex]\( y - 2 = 3(x - 6) \)[/tex]
[tex]\[ \text{Expanding: } y - 2 = 3x - 18 \implies y = 3x - 16 \quad (\text{This has a slope of 3, not 1}) \][/tex]
- [tex]\( y + 4 = x \)[/tex]
[tex]\[ \text{Rewriting: } y = x - 4 \quad (\text{This matches our equation}) \][/tex]
- [tex]\( y + 6 = x - 2 \)[/tex]
[tex]\[ \text{Rewriting: } y = x - 8 \quad (\text{This does not match our equation}) \][/tex]
The equation that correctly represents the line passing through the points [tex]\(A(0, -4)\)[/tex] and [tex]\(B(6, 2)\)[/tex] is:
[tex]\[ \boxed{y + 4 = x} \][/tex]
1. Calculate the Slope:
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the coordinates of points [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[ y_1 = -4, \quad y_2 = 2, \quad x_1 = 0, \quad x_2 = 6 \][/tex]
[tex]\[ m = \frac{2 - (-4)}{6 - 0} = \frac{2 + 4}{6} = \frac{6}{6} = 1 \][/tex]
2. Determine the Y-Intercept:
The slope-intercept form of a line is:
[tex]\[ y = mx + b \][/tex]
Given that the slope [tex]\(m\)[/tex] is 1 and using the point [tex]\(A(0, -4)\)[/tex]:
[tex]\[ y = 1 \cdot x + b \][/tex]
Since the point [tex]\(A\)[/tex] is on the line, substituting [tex]\(x = 0\)[/tex] and [tex]\(y = -4\)[/tex]:
[tex]\[ -4 = 1 \cdot 0 + b \implies b = -4 \][/tex]
Therefore, the equation of the line is:
[tex]\[ y = x - 4 \][/tex]
3. Verify the Provided Options:
Let's compare this equation with the given options:
- [tex]\( y - 4 = 3x \)[/tex]
[tex]\[ \text{Rewriting: } y = 3x + 4 \quad (\text{This has a slope of 3, not 1}) \][/tex]
- [tex]\( y - 2 = 3(x - 6) \)[/tex]
[tex]\[ \text{Expanding: } y - 2 = 3x - 18 \implies y = 3x - 16 \quad (\text{This has a slope of 3, not 1}) \][/tex]
- [tex]\( y + 4 = x \)[/tex]
[tex]\[ \text{Rewriting: } y = x - 4 \quad (\text{This matches our equation}) \][/tex]
- [tex]\( y + 6 = x - 2 \)[/tex]
[tex]\[ \text{Rewriting: } y = x - 8 \quad (\text{This does not match our equation}) \][/tex]
The equation that correctly represents the line passing through the points [tex]\(A(0, -4)\)[/tex] and [tex]\(B(6, 2)\)[/tex] is:
[tex]\[ \boxed{y + 4 = x} \][/tex]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
