Select The Correct Answer.A Line Passes Through $(10,5)$ And $(13,-4)$. Write The Equation Of The Line In Standard Form.A. 3 X − Y = 5 3x - Y = 5 3 X − Y = 5 B. 3 X − Y = 25 3x - Y = 25 3 X − Y = 25 C. 3 X + Y = 25 3x + Y = 25 3 X + Y = 25 D. 3 X + Y = 35 3x + Y = 35 3 X + Y = 35
Introduction
In mathematics, the equation of a line in standard form is a way to express the relationship between the x and y coordinates of points on the line. The standard form of a line is given by the equation Ax + By = C, where A, B, and C are constants. In this article, we will learn how to find the equation of a line in standard form given two points on the line.
Calculating the Slope of the Line
To find the equation of a line in standard form, we first need to calculate the slope of the line. The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the change in y to the change in x. Given two points (x1, y1) and (x2, y2), the slope of the line is given by:
m = (y2 - y1) / (x2 - x1)
In this case, we are given two points (10, 5) and (13, -4). We can use these points to calculate the slope of the line.
Calculating the Slope
m = (-4 - 5) / (13 - 10) m = -9 / 3 m = -3
Finding the Equation of the Line
Now that we have the slope of the line, we can use it to find the equation of the line in standard form. The equation of a line in standard form is given by:
Ax + By = C
We can use the point-slope form of a line to find the equation of the line. The point-slope form of a line is given by:
y - y1 = m(x - x1)
We can use the point (10, 5) to find the equation of the line.
Finding the Equation
y - 5 = -3(x - 10) y - 5 = -3x + 30 y = -3x + 35
Converting to Standard Form
The equation of the line is in slope-intercept form. We can convert it to standard form by multiplying both sides of the equation by -1.
-1(y) = -1(-3x + 35) -y = 3x - 35 3x + y = 35
Conclusion
In this article, we learned how to find the equation of a line in standard form given two points on the line. We calculated the slope of the line using the two points and then used the point-slope form of a line to find the equation of the line. We then converted the equation to standard form by multiplying both sides of the equation by -1. The final answer is:
The correct answer is D. 3x + y = 35
Q: What is the standard form of a line?
A: The standard form of a line is given by the equation Ax + By = C, where A, B, and C are constants.
Q: How do I calculate the slope of a line?
A: To calculate the slope of a line, you need to know two points on the line. The slope is given by the formula:
m = (y2 - y1) / (x2 - x1)
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)
Q: How do I convert the equation of a line from slope-intercept form to standard form?
A: To convert the equation of a line from slope-intercept form to standard form, you need to multiply both sides of the equation by -1.
Q: What is the equation of a line in standard form given two points (10, 5) and (13, -4)?
A: To find the equation of a line in standard form given two points, you need to calculate the slope of the line and then use the point-slope form of a line to find the equation. The equation of the line is:
3x + y = 35
Q: How do I determine which option is correct when given multiple choices?
A: To determine which option is correct when given multiple choices, you need to plug in the values of x and y from the two points into each option and see which one is true.
Q: What is the importance of finding the equation of a line in standard form?
A: Finding the equation of a line in standard form is important because it allows you to write the equation of a line in a concise and clear way. It is also useful for graphing the line and for solving systems of linear equations.
Q: Can I use the equation of a line in standard form to find the slope of the line?
A: Yes, you can use the equation of a line in standard form to find the slope of the line. The slope is given by the coefficient of x, which is A.
Q: How do I find the equation of a line in standard form given a point and the slope?
A: To find the equation of a line in standard form given a point and the slope, you need to use the point-slope form of a line and then convert it to standard form.
Q: What is the relationship between the equation of a line in standard form and the graph of the line?
A: The equation of a line in standard form is a way to describe the graph of the line. The graph of the line is a set of points that satisfy the equation.
Q: Can I use the equation of a line in standard form to solve systems of linear equations?
A: Yes, you can use the equation of a line in standard form to solve systems of linear equations. You can use the equation to find the intersection point of the two lines.
Q: How do I find the equation of a line in standard form given two parallel lines?
A: To find the equation of a line in standard form given two parallel lines, you need to find the slope of one of the lines and then use it to find the equation of the other line.
Q: What is the difference between the equation of a line in standard form and the equation of a line in slope-intercept form?
A: The equation of a line in standard form is given by the equation Ax + By = C, while the equation of a line in slope-intercept form is given by the equation y = mx + b.