Terri Wrote The Equation Using Slope-intercept Form For The Line That Passes Through The Points $(4,6$\] And $(-2,3$\].Step 1: $m=\frac{4-(-2)}{6-3}=\frac{6}{3}=2$Step 2: $3=2(-2)+b$Step 3: $3-4+b$
Introduction
In mathematics, linear equations are a fundamental concept that plays a crucial role in various fields, including algebra, geometry, and calculus. One of the most common forms of linear equations is the slope-intercept form, which is represented as y = mx + b, where m is the slope and b is the y-intercept. In this article, we will explore how to solve linear equations using the slope-intercept form, with a focus on finding the equation of a line that passes through two given points.
Step 1: Finding the Slope
The first step in solving a linear equation using the slope-intercept form is to find the slope (m) of the line. The slope is a measure of how steep the line is and can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two given points.
Let's consider the example given in the problem statement, where the two points are (4, 6) and (-2, 3). To find the slope, we can use the formula:
m = (6 - 3) / (4 - (-2)) m = 3 / 6 m = 1/2
However, in the problem statement, the slope is given as 2. This is an error, and we will use the correct slope of 1/2 in our calculations.
Step 2: Finding the Y-Intercept
Once we have the slope, we can use it to find the y-intercept (b) of the line. The y-intercept is the point where the line intersects the y-axis, and it can be found using the formula:
b = y - mx
where (x, y) is a point on the line.
Using the point (4, 6), we can substitute the values into the formula:
b = 6 - (1/2)(4) b = 6 - 2 b = 4
However, in the problem statement, the equation is given as 3 = 2(-2) + b. This is an error, and we will use the correct equation 6 = (1/2)(4) + b in our calculations.
Step 3: Solving for the Y-Intercept
Now that we have the correct equation, we can solve for the y-intercept (b). To do this, we can isolate the variable b by subtracting the product of the slope and the x-coordinate from both sides of the equation:
b = 6 - (1/2)(4) b = 6 - 2 b = 4
However, in the problem statement, the equation is given as 3 - 4 + b. This is an error, and we will use the correct equation 6 - 2 + b in our calculations.
Conclusion
In conclusion, solving linear equations using the slope-intercept form requires finding the slope and y-intercept of the line. The slope can be calculated using the formula m = (y2 - y1) / (x2 - x1), and the y-intercept can be found using the formula b = y - mx. By following these steps, we can find the equation of a line that passes through two given points.
Example Solution
Let's consider the example given in the problem statement, where the two points are (4, 6) and (-2, 3). To find the equation of the line, we can use the slope-intercept form:
y = mx + b
where m is the slope and b is the y-intercept.
Using the correct slope of 1/2, we can substitute the values into the equation:
y = (1/2)x + b
To find the y-intercept, we can use the point (4, 6):
6 = (1/2)(4) + b 6 = 2 + b b = 4
Now that we have the correct equation, we can write the final answer:
y = (1/2)x + 4
Discussion
The problem statement given in the example is incorrect, and we have used the correct slope and y-intercept to find the equation of the line. This highlights the importance of double-checking the given information and using the correct formulas to solve linear equations.
Common Mistakes
When solving linear equations using the slope-intercept form, there are several common mistakes to watch out for:
- Using the incorrect slope or y-intercept
- Not following the correct order of operations
- Not isolating the variable b
- Not checking the given information for errors
By being aware of these common mistakes, we can avoid making errors and ensure that our solutions are accurate.
Real-World Applications
Linear equations have numerous real-world applications, including:
- Modeling population growth and decline
- Describing the motion of objects
- Calculating the cost of goods and services
- Determining the area and perimeter of shapes
By understanding how to solve linear equations using the slope-intercept form, we can apply these concepts to real-world problems and make informed decisions.
Conclusion
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
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), where (x1, y1) and (x2, y2) are the coordinates of two points on the line.
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 can be found using the formula b = y - mx, where (x, y) is a point on the line.
Q: How do I solve a linear equation using the slope-intercept form?
A: To solve a linear equation using the slope-intercept form, you need to find the slope and y-intercept of the line. Once you have these values, you can substitute them into the equation y = mx + b to find the equation of the line.
Q: What are some common mistakes to watch out for when solving linear equations?
A: Some common mistakes to watch out for when solving linear equations include:
- Using the incorrect slope or y-intercept
- Not following the correct order of operations
- Not isolating the variable b
- Not checking the given information for errors
Q: What are some real-world applications of linear equations?
A: Linear equations have numerous real-world applications, including:
- Modeling population growth and decline
- Describing the motion of objects
- Calculating the cost of goods and services
- Determining the area and perimeter of shapes
Q: How do I determine if a line is parallel or perpendicular to another line?
A: To determine if a line is parallel or perpendicular to another line, you can use the slope formula. If the slopes are equal, the lines are parallel. If the product of the slopes is -1, the lines are perpendicular.
Q: Can I use the slope-intercept form to find the equation of a line that passes through three points?
A: Yes, you can use the slope-intercept form to find the equation of a line that passes through three points. However, you will need to find the slope of the line using the formula m = (y2 - y1) / (x2 - x1), and then use the point-slope form to find the equation of the line.
Q: How do I graph a line using the slope-intercept form?
A: To graph a line using the slope-intercept form, you can use the following steps:
- Find the y-intercept of the line by substituting x = 0 into the equation.
- Find the slope of the line by substituting the values of x and y into the equation.
- Plot the y-intercept on the graph.
- Use the slope to determine the direction of the line.
- Plot additional points on the graph to complete the line.
Q: Can I use the slope-intercept form to find the equation of a line that passes through a point and has a given slope?
A: Yes, you can use the slope-intercept form to find the equation of a line that passes through a point and has a given slope. To do this, you can substitute the values of x and y into the equation y = mx + b, and then solve for b.
Conclusion
In conclusion, solving linear equations using the slope-intercept form requires finding the slope and y-intercept of the line. By following the correct steps and avoiding common mistakes, you can find the equation of a line that passes through two or more points. The real-world applications of linear equations are numerous, and understanding how to solve them is essential for making informed decisions in various fields.