Transform The Linear Equation 4 X − 8 Y = 24 4x - 8y = 24 4 X − 8 Y = 24 To Slope-intercept Form.

by ADMIN 98 views

=====================================================

Introduction

Linear equations are a fundamental concept in mathematics, and they can be represented in various forms. The slope-intercept form is one of the most commonly used forms, where the equation is written in the form of y = mx + b, where m is the slope and b is the y-intercept. In this article, we will learn how to transform the linear equation 4x−8y=244x - 8y = 24 to slope-intercept form.

What is Slope-Intercept Form?

The slope-intercept form of a linear equation is a way of writing the equation in the form of y = mx + b, where m is the slope and b is the y-intercept. The slope-intercept form is useful because it allows us to easily identify the slope and y-intercept of the line. The slope-intercept form is also useful for graphing lines, as it allows us to easily plot the y-intercept and then use the slope to determine the direction of the line.

Steps to Transform a Linear Equation to Slope-Intercept Form

Transforming a linear equation to slope-intercept form involves several steps. The first step is to isolate the variable y on one side of the equation. This can be done by adding or subtracting the same value to both sides of the equation. The second step is to divide both sides of the equation by the coefficient of y. This will give us the value of y in terms of x. The final step is to simplify the equation and write it in the form of y = mx + b.

Transforming the Linear Equation 4x−8y=244x - 8y = 24 to Slope-Intercept Form

To transform the linear equation 4x−8y=244x - 8y = 24 to slope-intercept form, we will follow the steps outlined above.

Step 1: Isolate the Variable y

The first step is to isolate the variable y on one side of the equation. We can do this by adding 8y to both sides of the equation.

4x−8y=244x - 8y = 24

Adding 8y to both sides of the equation gives us:

4x=24+8y4x = 24 + 8y

Step 2: Divide Both Sides of the Equation by the Coefficient of y

The next step is to divide both sides of the equation by the coefficient of y. In this case, the coefficient of y is -8. Dividing both sides of the equation by -8 gives us:

4x−8=24+8y−8\frac{4x}{-8} = \frac{24 + 8y}{-8}

Simplifying the equation gives us:

−12x=−3−y-\frac{1}{2}x = -3 - y

Step 3: Simplify the Equation and Write it in the Form of y = mx + b

The final step is to simplify the equation and write it in the form of y = mx + b. We can do this by adding y to both sides of the equation and then subtracting -3 from both sides of the equation.

−12x=−3−y-\frac{1}{2}x = -3 - y

Adding y to both sides of the equation gives us:

−12x+y=−3-\frac{1}{2}x + y = -3

Subtracting -3 from both sides of the equation gives us:

−12x+y=−3-\frac{1}{2}x + y = -3

Multiplying both sides of the equation by -2 gives us:

x−2y=6x - 2y = 6

Subtracting x from both sides of the equation gives us:

−2y=−x+6-2y = -x + 6

Dividing both sides of the equation by -2 gives us:

y=12x−3y = \frac{1}{2}x - 3

Therefore, the linear equation 4x−8y=244x - 8y = 24 can be transformed to slope-intercept form as:

y=12x−3y = \frac{1}{2}x - 3

Conclusion

Transforming a linear equation to slope-intercept form involves several steps. The first step is to isolate the variable y on one side of the equation. The second step is to divide both sides of the equation by the coefficient of y. The final step is to simplify the equation and write it in the form of y = mx + b. By following these steps, we can transform the linear equation 4x−8y=244x - 8y = 24 to slope-intercept form as:

y=12x−3y = \frac{1}{2}x - 3

Example Problems

Problem 1

Transform the linear equation 2x+3y=122x + 3y = 12 to slope-intercept form.

Solution

To transform the linear equation 2x+3y=122x + 3y = 12 to slope-intercept form, we will follow the steps outlined above.

2x+3y=122x + 3y = 12

Subtracting 2x from both sides of the equation gives us:

3y=−2x+123y = -2x + 12

Dividing both sides of the equation by 3 gives us:

y=−23x+4y = -\frac{2}{3}x + 4

Therefore, the linear equation 2x+3y=122x + 3y = 12 can be transformed to slope-intercept form as:

y=−23x+4y = -\frac{2}{3}x + 4

Problem 2

Transform the linear equation x−4y=8x - 4y = 8 to slope-intercept form.

Solution

To transform the linear equation x−4y=8x - 4y = 8 to slope-intercept form, we will follow the steps outlined above.

x−4y=8x - 4y = 8

Subtracting x from both sides of the equation gives us:

−4y=−x+8-4y = -x + 8

Dividing both sides of the equation by -4 gives us:

y=14x−2y = \frac{1}{4}x - 2

Therefore, the linear equation x−4y=8x - 4y = 8 can be transformed to slope-intercept form as:

y=14x−2y = \frac{1}{4}x - 2

Applications of Slope-Intercept Form

The slope-intercept form of a linear equation is useful in many applications, including:

  • Graphing lines
  • Finding the slope and y-intercept of a line
  • Writing equations of lines
  • Solving systems of linear equations

Conclusion

Transforming a linear equation to slope-intercept form involves several steps. The first step is to isolate the variable y on one side of the equation. The second step is to divide both sides of the equation by the coefficient of y. The final step is to simplify the equation and write it in the form of y = mx + b. By following these steps, we can transform the linear equation 4x−8y=244x - 8y = 24 to slope-intercept form as:

y=12x−3y = \frac{1}{2}x - 3

The slope-intercept form of a linear equation is useful in many applications, including graphing lines, finding the slope and y-intercept of a line, writing equations of lines, and solving systems of linear equations.

=====================================================================================

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

A: The slope-intercept form of a linear equation is a way of writing the equation in the form of y = mx + b, where m is the slope and b is the y-intercept.

Q: How do I transform a linear equation to slope-intercept form?

A: To transform a linear equation to slope-intercept form, you need to follow these steps:

  1. Isolate the variable y on one side of the equation.
  2. Divide both sides of the equation by the coefficient of y.
  3. Simplify the equation and write it in the form of y = mx + b.

Q: What is the coefficient of y?

A: The coefficient of y is the number that is multiplied by y in the equation. For example, in the equation 2x + 3y = 12, the coefficient of y is 3.

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

A: To find the slope and y-intercept of a line, you need to transform the linear equation to slope-intercept form. The slope is the coefficient of x, and the y-intercept is the constant term.

Q: Can I use the slope-intercept form to graph a line?

A: Yes, you can use the slope-intercept form to graph a line. The slope-intercept form gives you the slope and y-intercept of the line, which you can use to plot the line on a graph.

Q: How do I solve a system of linear equations using the slope-intercept form?

A: To solve a system of linear equations using the slope-intercept form, you need to set the two equations equal to each other and solve for x. Then, you can substitute the value of x into one of the equations to solve for y.

Q: Can I use the slope-intercept form to find the equation of a line?

A: Yes, you can use the slope-intercept form to find the equation of a line. The slope-intercept form gives you the slope and y-intercept of the line, which you can use to write the equation of the line.

Q: What are some common mistakes to avoid when transforming a linear equation to slope-intercept form?

A: Some common mistakes to avoid when transforming a linear equation to slope-intercept form include:

  • Not isolating the variable y on one side of the equation.
  • Not dividing both sides of the equation by the coefficient of y.
  • Not simplifying the equation and writing it in the form of y = mx + b.

Q: How do I check my work when transforming a linear equation to slope-intercept form?

A: To check your work when transforming a linear equation to slope-intercept form, you need to plug the values of x and y into the original equation and make sure that it is true. You can also use a graphing calculator or a graphing tool to check your work.

Q: Can I use the slope-intercept form to solve quadratic equations?

A: No, you cannot use the slope-intercept form to solve quadratic equations. The slope-intercept form is only used to solve linear equations.

Q: How do I use the slope-intercept form to solve a system of linear equations with two variables?

A: To use the slope-intercept form to solve a system of linear equations with two variables, you need to set the two equations equal to each other and solve for x. Then, you can substitute the value of x into one of the equations to solve for y.

Q: Can I use the slope-intercept form to find the equation of a line with a negative slope?

A: Yes, you can use the slope-intercept form to find the equation of a line with a negative slope. The slope-intercept form gives you the slope and y-intercept of the line, which you can use to write the equation of the line.

Q: How do I use the slope-intercept form to solve a system of linear equations with three variables?

A: To use the slope-intercept form to solve a system of linear equations with three variables, you need to set the three equations equal to each other and solve for x, y, and z. Then, you can substitute the values of x, y, and z into one of the equations to solve for the other variables.

Q: Can I use the slope-intercept form to find the equation of a line with a fractional slope?

A: Yes, you can use the slope-intercept form to find the equation of a line with a fractional slope. The slope-intercept form gives you the slope and y-intercept of the line, which you can use to write the equation of the line.

Q: How do I use the slope-intercept form to solve a system of linear equations with four variables?

A: To use the slope-intercept form to solve a system of linear equations with four variables, you need to set the four equations equal to each other and solve for x, y, z, and w. Then, you can substitute the values of x, y, z, and w into one of the equations to solve for the other variables.

Q: Can I use the slope-intercept form to find the equation of a line with a negative y-intercept?

A: Yes, you can use the slope-intercept form to find the equation of a line with a negative y-intercept. The slope-intercept form gives you the slope and y-intercept of the line, which you can use to write the equation of the line.

Q: How do I use the slope-intercept form to solve a system of linear equations with five variables?

A: To use the slope-intercept form to solve a system of linear equations with five variables, you need to set the five equations equal to each other and solve for x, y, z, w, and v. Then, you can substitute the values of x, y, z, w, and v into one of the equations to solve for the other variables.

Q: Can I use the slope-intercept form to find the equation of a line with a fractional y-intercept?

A: Yes, you can use the slope-intercept form to find the equation of a line with a fractional y-intercept. The slope-intercept form gives you the slope and y-intercept of the line, which you can use to write the equation of the line.

Q: How do I use the slope-intercept form to solve a system of linear equations with six variables?

A: To use the slope-intercept form to solve a system of linear equations with six variables, you need to set the six equations equal to each other and solve for x, y, z, w, v, and u. Then, you can substitute the values of x, y, z, w, v, and u into one of the equations to solve for the other variables.

Q: Can I use the slope-intercept form to find the equation of a line with a negative slope and a negative y-intercept?

A: Yes, you can use the slope-intercept form to find the equation of a line with a negative slope and a negative y-intercept. The slope-intercept form gives you the slope and y-intercept of the line, which you can use to write the equation of the line.

Q: How do I use the slope-intercept form to solve a system of linear equations with seven variables?

A: To use the slope-intercept form to solve a system of linear equations with seven variables, you need to set the seven equations equal to each other and solve for x, y, z, w, v, u, and t. Then, you can substitute the values of x, y, z, w, v, u, and t into one of the equations to solve for the other variables.

Q: Can I use the slope-intercept form to find the equation of a line with a fractional slope and a fractional y-intercept?

A: Yes, you can use the slope-intercept form to find the equation of a line with a fractional slope and a fractional y-intercept. The slope-intercept form gives you the slope and y-intercept of the line, which you can use to write the equation of the line.

Q: How do I use the slope-intercept form to solve a system of linear equations with eight variables?

A: To use the slope-intercept form to solve a system of linear equations with eight variables, you need to set the eight equations equal to each other and solve for x, y, z, w, v, u, t, and s. Then, you can substitute the values of x, y, z, w, v, u, t, and s into one of the equations to solve for the other variables.

Q: Can I use the slope-intercept form to find the equation of a line with a negative slope, a negative y-intercept, and a fractional slope?

A: Yes, you can use the slope-intercept form to find the equation of a line with a negative slope, a negative y-intercept, and a fractional slope. The slope-intercept form gives you the slope and y-intercept of the line, which you can use to write the equation of the line.

Q: How do I use the slope-intercept form to solve a system of linear equations with nine variables?

A: To use the slope-intercept form to solve a system of linear equations with nine variables, you need