Graph A Line That Contains The Point \[$(2,0)\$\] And Has A Slope Of \[$-5\$\].

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Introduction

Graphing a line with a given point and slope is a fundamental concept in mathematics, particularly in algebra and geometry. In this article, we will explore how to graph a line that contains the point {(2,0)$}$ and has a slope of {-5$}$. We will use the point-slope form of a linear equation, which is a powerful tool for graphing lines.

Understanding the Point-Slope Form

The point-slope form of a linear equation is given by:

y−y1=m(x−x1)y - y_1 = m(x - x_1)

where {(x_1, y_1)$}$ is a point on the line, and {m$}$ is the slope of the line. This form is useful because it allows us to easily graph a line with a given point and slope.

Using the Point-Slope Form to Graph a Line

To graph a line with a given point and slope, we can use the point-slope form of a linear equation. Let's use the point {(2,0)$}$ and the slope {-5$}$ as an example.

Step 1: Write the Point-Slope Form of the Equation

Using the point {(2,0)$}$ and the slope {-5$}$, we can write the point-slope form of the equation as:

y−0=−5(x−2)y - 0 = -5(x - 2)

Step 2: Simplify the Equation

Simplifying the equation, we get:

y=−5x+10y = -5x + 10

Step 3: Graph the Line

To graph the line, we can use the equation {y = -5x + 10$}$. We can start by plotting the point {(2,0)$}$ on the coordinate plane. Then, we can use the slope {-5$}$ to find another point on the line.

Step 4: Find Another Point on the Line

Using the slope {-5$}$, we can find another point on the line by moving 1 unit to the right and 5 units down from the point {(2,0)$}$. This gives us the point {(3,-5)$}$.

Step 5: Draw the Line

Now that we have two points on the line, we can draw the line by connecting the points with a straight line.

Conclusion

Graphing a line with a given point and slope is a straightforward process that involves using the point-slope form of a linear equation. By following the steps outlined in this article, we can easily graph a line with a given point and slope.

Example Use Cases

Graphing a line with a given point and slope has many practical applications in mathematics and science. Some example use cases include:

  • Linear Regression: Graphing a line with a given point and slope is a key step in linear regression, which is a statistical technique used to model the relationship between two variables.
  • Physics: Graphing a line with a given point and slope is used to model the motion of objects in physics, such as the trajectory of a projectile.
  • Engineering: Graphing a line with a given point and slope is used to design and optimize systems in engineering, such as the design of a bridge.

Tips and Tricks

Here are some tips and tricks for graphing a line with a given point and slope:

  • Use the Point-Slope Form: The point-slope form of a linear equation is a powerful tool for graphing lines. Make sure to use it when graphing a line with a given point and slope.
  • Simplify the Equation: Simplifying the equation can make it easier to graph the line.
  • Use a Graphing Calculator: A graphing calculator can be a useful tool for graphing a line with a given point and slope.

Common Mistakes

Here are some common mistakes to avoid when graphing a line with a given point and slope:

  • Not Using the Point-Slope Form: Failing to use the point-slope form of a linear equation can make it difficult to graph the line.
  • Not Simplifying the Equation: Failing to simplify the equation can make it difficult to graph the line.
  • Not Using a Graphing Calculator: Failing to use a graphing calculator can make it difficult to graph the line.

Conclusion

Graphing a line with a given point and slope is a fundamental concept in mathematics, particularly in algebra and geometry. By using the point-slope form of a linear equation, we can easily graph a line with a given point and slope. This article has provided a step-by-step guide on how to graph a line with a given point and slope, as well as some tips and tricks for graphing lines.

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Q: What is the point-slope form of a linear equation?

A: The point-slope form of a linear equation is given by:

y−y1=m(x−x1)y - y_1 = m(x - x_1)

where {(x_1, y_1)$}$ is a point on the line, and {m$}$ is the slope of the line.

Q: How do I use the point-slope form to graph a line?

A: To graph a line with a given point and slope, follow these steps:

  1. Write the point-slope form of the equation using the given point and slope.
  2. Simplify the equation.
  3. Plot the given point on the coordinate plane.
  4. Use the slope to find another point on the line.
  5. Draw the line by connecting the points with a straight line.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Q: How do I calculate the slope of a line?

A: To calculate the slope of a line, use the formula:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

where {(x_1, y_1)$}$ and {(x_2, y_2)$}$ are 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 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, set x equal to 0 in the equation of the line and solve for y.

Q: Can I graph a line with a negative slope?

A: Yes, you can graph a line with a negative slope. A negative slope means that the line slopes downward from left to right.

Q: Can I graph a line with a zero slope?

A: Yes, you can graph a line with a zero slope. A zero slope means that the line is horizontal and does not change in the y-direction.

Q: Can I graph a line with a vertical slope?

A: No, you cannot graph a line with a vertical slope. A vertical slope means that the line is undefined and does not have a slope.

Q: How do I graph a line with a given point and slope using a graphing calculator?

A: To graph a line with a given point and slope using a graphing calculator, follow these steps:

  1. Enter the point-slope form of the equation into the calculator.
  2. Simplify the equation using the calculator's simplify function.
  3. Plot the given point on the coordinate plane using the calculator's graphing function.
  4. Use the calculator's slope function to find the slope of the line.
  5. Draw the line by connecting the points with a straight line using the calculator's graphing function.

Q: What are some common mistakes to avoid when graphing a line with a given point and slope?

A: Some common mistakes to avoid when graphing a line with a given point and slope include:

  • Not using the point-slope form of a linear equation.
  • Not simplifying the equation.
  • Not using a graphing calculator.
  • Not plotting the given point on the coordinate plane.
  • Not using the slope to find another point on the line.
  • Not drawing the line by connecting the points with a straight line.

Q: How do I check my work when graphing a line with a given point and slope?

A: To check your work when graphing a line with a given point and slope, follow these steps:

  1. Verify that the equation is in the point-slope form.
  2. Verify that the equation is simplified.
  3. Verify that the given point is plotted on the coordinate plane.
  4. Verify that the slope is used to find another point on the line.
  5. Verify that the line is drawn by connecting the points with a straight line.

Q: What are some real-world applications of graphing a line with a given point and slope?

A: Some real-world applications of graphing a line with a given point and slope include:

  • Linear regression: Graphing a line with a given point and slope is a key step in linear regression, which is a statistical technique used to model the relationship between two variables.
  • Physics: Graphing a line with a given point and slope is used to model the motion of objects in physics, such as the trajectory of a projectile.
  • Engineering: Graphing a line with a given point and slope is used to design and optimize systems in engineering, such as the design of a bridge.