A Line Has A Slope Of -2 And Passes Through The Point { (3,5)$}$. Which Of The Following Is The Equation Of The Line?A. { Y = -2x + 3$}$ B. { Y = -2x + 11$}$ C. { Y = -2x + 5$}$ D. { Y = -2x - 1$}$
Introduction
In mathematics, a line is defined by its slope and a point it passes through. The slope of a line is a measure of how steep it is, and it can be calculated using the coordinates of two points on the line. In this article, we will explore how to find the equation of a line given its slope and a point it passes through. We will use the point-slope form of a line, which is a powerful tool for finding the equation of a line.
The Point-Slope Form of a Line
The point-slope form of a line is given by the equation:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope of the line. This equation is useful because it allows us to find the equation of a line given its slope and a point it passes through.
Finding the Equation of a Line with a Slope of -2
We are given that the line has a slope of -2 and passes through the point (3, 5). We can use the point-slope form of a line to find the equation of the line. Plugging in the values of the slope and the point, we get:
y - 5 = -2(x - 3)
Simplifying the Equation
To simplify the equation, we can expand the right-hand side and combine like terms:
y - 5 = -2x + 6
Adding 5 to Both Sides
To isolate y, we can add 5 to both sides of the equation:
y = -2x + 11
Conclusion
We have found the equation of the line with a slope of -2 and passing through the point (3, 5). The equation is y = -2x + 11. This equation is in the slope-intercept form, which is a common way to write the equation of a line.
Comparing the Options
Let's compare the equation we found with the options given:
A. y = -2x + 3 B. y = -2x + 11 C. y = -2x + 5 D. y = -2x - 1
Only option B matches the equation we found.
Why the Other Options are Incorrect
The other options are incorrect because they do not match the equation we found. Option A has a different constant term, option C has a different constant term, and option D has a negative slope.
Conclusion
In conclusion, the equation of the line with a slope of -2 and passing through the point (3, 5) is y = -2x + 11. This equation is in the slope-intercept form, which is a common way to write the equation of a line.
Final Answer
Introduction
In our previous article, we explored how to find the equation of a line given its slope and a point it passes through. We used the point-slope form of a line to find the equation of a line with a slope of -2 and passing through the point (3, 5). In this article, we will answer some common questions related to finding the equation of a line.
Q: What is the point-slope form of a line?
A: The point-slope form of a line is given by the equation:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope of the line.
Q: How do I find the equation of a line given its slope and a point it passes through?
A: To find the equation of a line given its slope and a point it passes through, you can use the point-slope form of a line. Plug in the values of the slope and the point into the equation, and simplify to find the equation of the line.
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is given by the equation:
y = mx + b
where m is the slope of the line, and b is the y-intercept.
Q: How do I convert the point-slope form of a line to the slope-intercept form?
A: To convert the point-slope form of a line to the slope-intercept form, you can add the constant term to both sides of the equation and simplify.
Q: What is the y-intercept of a line?
A: The y-intercept of a line is the point where the line intersects the y-axis. It is the value of y when x is equal to 0.
Q: How do I find the y-intercept of a line?
A: To find the y-intercept of a line, you can plug in x = 0 into the equation of the line and solve for y.
Q: What is the equation of a line with a slope of 0?
A: The equation of a line with a slope of 0 is a horizontal line. It can be written in the form y = b, where b is the y-intercept.
Q: What is the equation of a line with a slope of infinity?
A: The equation of a line with a slope of infinity is a vertical line. It can be written in the form x = a, where a is the x-intercept.
Q: How do I find the equation of a line given two points it passes through?
A: To find the equation of a line given two points it passes through, you can use the two-point form of a line. Plug in the values of the two points into the equation, and simplify to find the equation of the line.
Conclusion
In conclusion, finding the equation of a line is a fundamental concept in mathematics. By using the point-slope form of a line, we can find the equation of a line given its slope and a point it passes through. We can also convert the point-slope form to the slope-intercept form and find the y-intercept of a line.
Final Answer
The final answer is that the equation of a line with a slope of -2 and passing through the point (3, 5) is y = -2x + 11.