Write The Equation Of The Line That Passes Through The Point { (-6, 1)$}$ And Has A Slope Of { \frac{1}{2}$}$.
Introduction
In mathematics, a linear equation is a type of equation that can be written in the form of y = mx + b, where m is the slope of the line and b is the y-intercept. The slope of a line is a measure of how steep it is, and it can be calculated using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line. In this article, we will learn how to find the equation of a line that passes through a given point and has a known slope.
The Point-Slope Form of a Linear Equation
The point-slope form of a linear equation is given by the formula:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope of the line. This formula is useful when we know the slope of the line and a point on the line, and we want to find the equation of the line.
Example: Finding the Equation of a Line
Let's say we want to find the equation of a line that passes through the point (-6, 1) and has a slope of 1/2. We can use the point-slope form of a linear equation to find the equation of the line.
Step 1: Write the Point-Slope Form of the Equation
The point-slope form of the equation is:
y - 1 = (1/2)(x - (-6))
Step 2: Simplify the Equation
To simplify the equation, we can multiply both sides of the equation by 2 to eliminate the fraction:
2(y - 1) = x + 6
Step 3: Expand and Simplify the Equation
Expanding and simplifying the equation, we get:
2y - 2 = x + 6
Step 4: Write the Equation in Slope-Intercept Form
To write the equation in slope-intercept form, we can add 2 to both sides of the equation:
2y = x + 8
Step 5: Divide Both Sides of the Equation by 2
Dividing both sides of the equation by 2, we get:
y = (1/2)x + 4
Conclusion
In this article, we learned how to find the equation of a line that passes through a given point and has a known slope. We used the point-slope form of a linear equation to find the equation of the line, and we simplified the equation to write it in slope-intercept form. The final equation of the line is y = (1/2)x + 4.
Key Takeaways
- The point-slope form of a linear equation is given by the formula y - y1 = m(x - x1).
- To find the equation of a line that passes through a given point and has a known slope, we can use the point-slope form of a linear equation.
- To simplify the equation, we can multiply both sides of the equation by a constant to eliminate fractions.
- To write the equation in slope-intercept form, we can add or subtract a constant from both sides of the equation.
Practice Problems
- Find the equation of a line that passes through the point (2, 3) and has a slope of 2.
- Find the equation of a line that passes through the point (-3, 4) and has a slope of -1.
- Find the equation of a line that passes through the point (1, 2) and has a slope of 3.
Solutions
- y - 3 = 2(x - 2)
- y - 4 = -1(x + 3)
- y - 2 = 3(x - 1)
Real-World Applications
The equation of a line can be used to model real-world situations, such as the cost of goods, the height of a building, or the distance between two points. For example, a company may use the equation of a line to model the cost of producing a certain product, where the cost is a function of the number of units produced.
Conclusion
Q: What is the point-slope form of a linear equation?
A: The point-slope form of a linear equation is given by the formula 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 that passes through a given point and has a known slope?
A: To find the equation of a line that passes through a given point and has a known slope, you can use the point-slope form of a linear equation. Simply substitute the given point and slope into the formula, and simplify the equation to write it in slope-intercept form.
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is given by the formula 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 linear equation to the slope-intercept form?
A: To convert the point-slope form of a linear equation to the slope-intercept form, you can add or subtract a constant from both sides of the equation. For example, if you have the equation y - 1 = (1/2)(x - (-6)), you can add 1 to both sides to get y = (1/2)x + 4.
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 substitute x = 0 into the equation of the line and solve for y.
Q: What is the slope of a line?
A: The slope of a line is a measure of how steep it is. It can be calculated using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.
Q: How do I find the slope of a line?
A: To find the slope of a line, you can use the formula m = (y2 - y1) / (x2 - x1). Simply substitute the coordinates of two points on the line into the formula and simplify.
Q: What is the equation of a horizontal line?
A: The equation of a horizontal line is given by the formula y = b, where b is the y-intercept of the line.
Q: What is the equation of a vertical line?
A: The equation of a vertical line is given by the formula x = a, where a is the x-intercept of the line.
Q: How do I graph a line?
A: To graph a line, you can use the equation of the line to find the x and y intercepts, and then plot the points on a coordinate plane.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is an equation that can be written in the form of y = mx + b, where m is the slope of the line and b is the y-intercept. A quadratic equation is an equation that can be written in the form of y = ax^2 + bx + c, where a, b, and c are constants.
Conclusion
In conclusion, the equation of a line is a fundamental concept in mathematics that can be used to model real-world situations. By understanding the point-slope form and slope-intercept form of a linear equation, you can find the equation of a line that passes through a given point and has a known slope.