Graph A Line With A Slope Of 4 3 \frac{4}{3} 3 4 ​ That Contains The Point ( 3 , − 1 (3, -1 ( 3 , − 1 ].

Introduction

Graphing a line with a given slope and point is a fundamental concept in mathematics, particularly in algebra and geometry. In this article, we will discuss how to graph a line with a slope of $\frac{4}{3}$ that contains the point $(3, -1)$. We will use the point-slope form of a linear equation, which is a powerful tool for graphing lines.

The Point-Slope Form

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

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line, and $m$ is the slope of the line. In our case, we are given the point $(3, -1)$ and the slope $m = \frac{4}{3}$.

Substituting the Given Values

We can substitute the given values into the point-slope form of the equation:

$$y - (-1) = \frac{4}{3}(x - 3)$$

Simplifying the equation, we get:

$$y + 1 = \frac{4}{3}(x - 3)$$

Simplifying the Equation

To simplify the equation further, we can multiply both sides by 3 to eliminate the fraction:

$$3(y + 1) = 4(x - 3)$$

Expanding the equation, we get:

$$3y + 3 = 4x - 12$$

Rearranging the Equation

To put the equation in the standard form of a linear equation, we can rearrange the terms:

$$4x - 3y - 15 = 0$$

Graphing the Line

Now that we have the equation of the line, we can graph it. To graph the line, we can use the point $(3, -1)$ and the slope $\frac{4}{3}$.

Using the Point-Slope Form to Graph the Line

To graph the line, we can use the point-slope form of the equation. We can start by plotting the point $(3, -1)$ on the coordinate plane. Then, we can use the slope $\frac{4}{3}$ to find another point on the line.

Finding Another Point on the Line

To find another point on the line, we can use the slope $\frac{4}{3}$. We can start by drawing a line through the point $(3, -1)$ with a slope of $\frac{4}{3}$. Then, we can find another point on the line by moving 1 unit to the right and 4 units up from the point $(3, -1)$.

Plotting the Line

Once we have found another point on the line, we can plot the line on the coordinate plane. We can use a ruler or a straightedge to draw a line through the two points.

Conclusion

In this article, we discussed how to graph a line with a given slope and point. We used the point-slope form of a linear equation to find the equation of the line, and then we graphed the line using the point and the slope. We also used the point-slope form to find another point on the line and plot the line on the coordinate plane.

Example Problems

Here are some example problems that you can try to practice graphing lines with a given slope and point:

• Graph a line with a slope of $\frac{2}{3}$ that contains the point $(2, 1)$.
• Graph a line with a slope of $-\frac{3}{4}$ that contains the point $(4, -2)$.
• Graph a line with a slope of $\frac{5}{2}$ that contains the point $(1, 3)$.

Tips and Tricks

Here are some tips and tricks that you can use to graph lines with a given slope and point:

• Make sure to use the point-slope form of the equation to find the equation of the line.
• Use the slope to find another point on the line.
• Plot the line on the coordinate plane using a ruler or a straightedge.
• Check your work by plugging in the point and the slope into the equation.

Common Mistakes

Here are some common mistakes that you can avoid when graphing lines with a given slope and point:

• Make sure to use the correct slope and point in the equation.
• Make sure to plot the line on the correct coordinate plane.
• Make sure to use the point-slope form of the equation to find the equation of the line.

Conclusion

Introduction

Graphing a line with a given slope and point is a fundamental concept in mathematics, particularly in algebra and geometry. In our previous article, we discussed how to graph a line with a given slope and point using the point-slope form of a linear equation. In this article, we will answer some frequently asked questions about graphing lines with a given slope and point.

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

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

$$y - y_1 = m(x - x_1)$$

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

Q: How do I find the equation of a line with a given slope and point?

A: To find the equation of a line with a given slope and point, you can use the point-slope form of the equation. Simply substitute the given values into the equation and simplify.

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 by dividing the change in y-coordinates by the change in x-coordinates.

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

A: To graph a line with a given slope and point, you can use the point-slope form of the equation to find the equation of the line, and then plot the line on the coordinate plane using a ruler or a straightedge.

Q: What is the difference between the slope and the y-intercept?

A: The slope of a line is a measure of how steep the line is, while the y-intercept is the point where the line intersects the y-axis.

Q: How do I find the y-intercept of a line?

A: To find the y-intercept of a line, you can set x = 0 in the equation of the line and solve for y.

Q: What is the equation of a horizontal line?

A: The equation of a horizontal line is given by:

$$y = c$$

where c is a constant.

Q: What is the equation of a vertical line?

A: The equation of a vertical line is given by:

$$x = c$$

where c is a constant.

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

A: To graph a line with a given slope and point using a graphing calculator, you can enter the equation of the line into the calculator and use the graphing function to plot the line.

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

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

• Using the wrong slope or point in the equation
• Plotting the line on the wrong coordinate plane
• Not using the point-slope form of the equation to find the equation of the line

Conclusion

Graphing a line with a given slope and point is a fundamental concept in mathematics, particularly in algebra and geometry. We answered some frequently asked questions about graphing lines with a given slope and point, and provided some tips and tricks for avoiding common mistakes. With practice and patience, you can become proficient in graphing lines with a given slope and point.

Example Problems

Here are some example problems that you can try to practice graphing lines with a given slope and point:

• Graph a line with a slope of $\frac{2}{3}$ that contains the point $(2, 1)$.
• Graph a line with a slope of $-\frac{3}{4}$ that contains the point $(4, -2)$.
• Graph a line with a slope of $\frac{5}{2}$ that contains the point $(1, 3)$.

Tips and Tricks

Here are some tips and tricks that you can use to graph lines with a given slope and point:

• Make sure to use the point-slope form of the equation to find the equation of the line.
• Use the slope to find another point on the line.
• Plot the line on the coordinate plane using a ruler or a straightedge.
• Check your work by plugging in the point and the slope into the equation.

Common Mistakes

Here are some common mistakes that you can avoid when graphing lines with a given slope and point:

• Make sure to use the correct slope and point in the equation.
• Make sure to plot the line on the correct coordinate plane.
• Make sure to use the point-slope form of the equation to find the equation of the line.