Select The Correct Answer.Given The Formula Below, Solve For $x$.$y - Y_1 = M(x - X_1)$A. $x = \frac{y - Y_1 + X_1}{m}$ B. \$x = \frac{m(y - Y_1)}{x_1}$[/tex\] C. $x = \frac{y - Y_1}{m} - X_1$

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Introduction

The point-slope form of a linear equation is a fundamental concept in mathematics, particularly in algebra and geometry. It is used to represent a line in the coordinate plane and is essential in solving problems involving linear equations. In this article, we will focus on solving for x in the point-slope form of a linear equation, which is given by the formula: $y - y_1 = m(x - x_1)$. We will analyze each option and determine the correct solution.

Understanding the Point-Slope Form

The point-slope form of a linear equation is a way to represent a line in the coordinate plane. It is given by the formula: $y - y_1 = m(x - x_1)$. In this formula, (x1, y1) is a point on the line, and m is the slope of the line. The point-slope form is useful because it allows us to find the equation of a line that passes through a given point and has a given slope.

Solving for x

To solve for x in the point-slope form of a linear equation, we need to isolate x on one side of the equation. Let's analyze each option and determine the correct solution.

Option A: $x = \frac{y - y_1 + x_1}{m}$

To determine if this option is correct, let's start by multiplying both sides of the equation by m to eliminate the fraction:

m(y−y1)=m(x−x1)m(y - y_1) = m(x - x_1)

Next, we can expand the right-hand side of the equation:

my−my1=mx−mx1my - my_1 = mx - mx_1

Now, we can add mx1 to both sides of the equation to get:

mx+mx1=my−my1mx + mx_1 = my - my_1

Subtracting mx from both sides gives us:

mx1=my−my1−mxmx_1 = my - my_1 - mx

Dividing both sides by m gives us:

x1=my−my1−mxmx_1 = \frac{my - my_1 - mx}{m}

Simplifying the right-hand side of the equation gives us:

x1=y−y1m−xx_1 = \frac{y - y_1}{m} - x

This is not the correct solution, so option A is incorrect.

Option B: $x = \frac{m(y - y_1)}{x_1}$

To determine if this option is correct, let's start by multiplying both sides of the equation by x1 to eliminate the fraction:

x1(y−y1)=mx1−mxx_1(y - y_1) = mx_1 - mx

Next, we can expand the right-hand side of the equation:

x1y−x1y1=mx1−mxx_1y - x_1y_1 = mx_1 - mx

Now, we can add mx to both sides of the equation to get:

x1y−x1y1+mx=mx1x_1y - x_1y_1 + mx = mx_1

Subtracting mx from both sides gives us:

x1y−x1y1=mx1−mxx_1y - x_1y_1 = mx_1 - mx

Dividing both sides by x1 gives us:

y−y1=mx1−mxx1y - y_1 = \frac{mx_1 - mx}{x_1}

Simplifying the right-hand side of the equation gives us:

y−y1=m(x−x1)y - y_1 = m(x - x_1)

This is the original equation, so option B is not the correct solution.

Option C: $x = \frac{y - y_1}{m} - x_1$

To determine if this option is correct, let's start by multiplying both sides of the equation by m to eliminate the fraction:

m(y−y1)=mx−mx1m(y - y_1) = mx - mx_1

Next, we can expand the right-hand side of the equation:

my−my1=mx−mx1my - my_1 = mx - mx_1

Now, we can add mx1 to both sides of the equation to get:

my−my1+mx1=mxmy - my_1 + mx_1 = mx

Subtracting my from both sides gives us:

−my1+mx1=mx−my-my_1 + mx_1 = mx - my

Dividing both sides by m gives us:

−y1+x1=mx−mym-y_1 + x_1 = \frac{mx - my}{m}

Simplifying the right-hand side of the equation gives us:

−y1+x1=x−y-y_1 + x_1 = x - y

Adding y to both sides gives us:

x−y+y1−x1=0x - y + y_1 - x_1 = 0

Adding x1 to both sides gives us:

x−y+y1=x1x - y + y_1 = x_1

Subtracting y1 from both sides gives us:

x−y=x1−y1x - y = x_1 - y_1

Adding y to both sides gives us:

x=y+x1−y1x = y + x_1 - y_1

Subtracting x1 from both sides gives us:

x−x1=y−y1x - x_1 = y - y_1

Dividing both sides by m gives us:

x−x1m=y−y1m\frac{x - x_1}{m} = \frac{y - y_1}{m}

Multiplying both sides by m gives us:

x−x1=y−y1x - x_1 = y - y_1

Adding x1 to both sides gives us:

x=y−y1+x1x = y - y_1 + x_1

Dividing both sides by m gives us:

x=y−y1+x1mx = \frac{y - y_1 + x_1}{m}

This is not the correct solution, so option C is incorrect.

Conclusion

After analyzing each option, we can conclude that none of the options are correct. However, we can derive the correct solution by following the steps outlined above. The correct solution is:

x=y−y1+x1mx = \frac{y - y_1 + x_1}{m}

This solution is derived by multiplying both sides of the equation by m to eliminate the fraction, expanding the right-hand side of the equation, and simplifying the resulting expression.

Final Answer

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

A: The point-slope form of a linear equation is a way to represent a line in the coordinate plane. It is given by the formula: $y - y_1 = m(x - x_1)$. In this formula, (x1, y1) is a point on the line, and m is the slope of the line.

Q: How do I solve for x in the point-slope form of a linear equation?

A: To solve for x in the point-slope form of a linear equation, you need to isolate x on one side of the equation. This can be done by multiplying both sides of the equation by m to eliminate the fraction, expanding the right-hand side of the equation, and simplifying the resulting expression.

Q: What is the correct solution for x in the point-slope form of a linear equation?

A: The correct solution for x in the point-slope form of a linear equation is:

x=y−y1+x1mx = \frac{y - y_1 + x_1}{m}

Q: Why is it important to solve for x in the point-slope form of a linear equation?

A: Solving for x in the point-slope form of a linear equation is important because it allows you to find the equation of a line that passes through a given point and has a given slope. This is a fundamental concept in mathematics, particularly in algebra and geometry.

Q: What are some common mistakes to avoid when solving for x in the point-slope form of a linear equation?

A: Some common mistakes to avoid when solving for x in the point-slope form of a linear equation include:

  • Not multiplying both sides of the equation by m to eliminate the fraction
  • Not expanding the right-hand side of the equation
  • Not simplifying the resulting expression
  • Not checking the solution for x

Q: How can I practice solving for x in the point-slope form of a linear equation?

A: You can practice solving for x in the point-slope form of a linear equation by working through examples and exercises. You can also use online resources, such as math websites and apps, to practice solving for x in the point-slope form of a linear equation.

Q: What are some real-world applications of solving for x in the point-slope form of a linear equation?

A: Some real-world applications of solving for x in the point-slope form of a linear equation include:

  • Finding the equation of a line that passes through a given point and has a given slope
  • Determining the slope of a line given two points
  • Finding the equation of a line that passes through a given point and has a given slope, and then using that equation to make predictions or forecasts.

Conclusion

Solving for x in the point-slope form of a linear equation is an important concept in mathematics, particularly in algebra and geometry. By following the steps outlined above and practicing solving for x in the point-slope form of a linear equation, you can become proficient in solving this type of equation.