Rewrite The Equation In The Form { Ax + By = C $}$. Use Integers For { A, B $}$, And { C $} . . . { Y = \frac{1}{4}x + 4 \}
Introduction
In mathematics, equations are often written in various forms to facilitate problem-solving and analysis. One common form is the standard form, which is represented as { Ax + By = C $}$. In this article, we will focus on rewriting the given equation in the standard form using integers for { A, B $}$, and { C $}$. The given equation is { y = \frac{1}{4}x + 4 $}$.
Understanding the Given Equation
The given equation is a linear equation in slope-intercept form, which is represented as { y = mx + b $}$, where { m $}$ is the slope and { b $}$ is the y-intercept. In this case, the slope is { \frac{1}{4} $}$ and the y-intercept is { 4 $}$.
Rewriting the Equation in Standard Form
To rewrite the equation in standard form, we need to multiply both sides of the equation by the least common multiple (LCM) of the coefficients of { x $}$ and { y $}$. In this case, the LCM is { 4 $}$, which is the denominator of the slope.
# Import necessary modules
import math
slope = 1/4
y_intercept = 4
lcm = 4
new_slope = slope * lcm
new_y_intercept = y_intercept * lcm
After multiplying both sides of the equation by { 4 $}$, we get:
{ 4y = x + 16 $}$
Simplifying the Equation
To simplify the equation, we need to isolate the term with the variable { x $}$ on one side of the equation. We can do this by subtracting { 16 $}$ from both sides of the equation.
# Subtract 16 from both sides of the equation
new_equation = "4y = x + 16"
new_equation = new_equation.replace("16", "-16")
After subtracting { 16 $}$ from both sides of the equation, we get:
{ 4y = x - 16 $}$
Rewriting the Equation in Standard Form
Now that we have isolated the term with the variable { x $}$ on one side of the equation, we can rewrite the equation in standard form.
# Rewrite the equation in standard form
standard_form = "4y - x = -16"
The final answer is { 4y - x = -16 $}$.
Conclusion
In this article, we have rewritten the given equation in standard form using integers for { A, B $}$, and { C $}$. We have also provided step-by-step instructions on how to rewrite the equation in standard form. The final answer is { 4y - x = -16 $}$.
Additional Resources
For more information on rewriting equations in standard form, please refer to the following resources:
FAQs
Q: What is the standard form of an equation? A: The standard form of an equation is represented as { Ax + By = C $}$, where { A, B $}$, and { C $}$ are integers.
Q: How do I rewrite an equation in standard form? A: To rewrite an equation in standard form, you need to multiply both sides of the equation by the least common multiple (LCM) of the coefficients of { x $}$ and { y $}$.
Q: What is the standard form of an equation?
A: The standard form of an equation is represented as { Ax + By = C $}$, where { A, B $}$, and { C $}$ are integers. This form is useful for solving systems of linear equations and for graphing linear equations.
Q: How do I rewrite an equation in standard form?
A: To rewrite an equation in standard form, you need to multiply both sides of the equation by the least common multiple (LCM) of the coefficients of { x $}$ and { y $}$. This will eliminate any fractions and make it easier to solve the equation.
Q: What is the least common multiple (LCM)?
A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of 2 and 3 is 6, because 6 is the smallest number that both 2 and 3 can divide into evenly.
Q: How do I find the LCM of two numbers?
A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest multiple that they have in common. Alternatively, you can use the following formula:
LCM(a, b) = (a × b) / GCD(a, b)
where GCD(a, b) is the greatest common divisor of a and b.
Q: What is the greatest common divisor (GCD)?
A: The greatest common divisor (GCD) is the largest number that two or more numbers have in common. For example, the GCD of 12 and 18 is 6, because 6 is the largest number that both 12 and 18 can divide into evenly.
Q: How do I find the GCD of two numbers?
A: To find the GCD of two numbers, you can list the factors of each number and find the largest factor that they have in common. Alternatively, you can use the following formula:
GCD(a, b) = (a × b) / LCM(a, b)
Q: Can I use a calculator to find the LCM and GCD?
A: Yes, you can use a calculator to find the LCM and GCD of two numbers. Most calculators have a built-in function for finding the LCM and GCD.
Q: Why is it important to rewrite an equation in standard form?
A: Rewriting an equation in standard form is important because it makes it easier to solve the equation and to graph the equation. It also helps to eliminate any fractions and to make the equation more manageable.
Q: Can I use the standard form of an equation to solve a system of linear equations?
A: Yes, you can use the standard form of an equation to solve a system of linear equations. By rewriting each equation in standard form, you can then use substitution or elimination to solve the system.
Q: What are some common mistakes to avoid when rewriting an equation in standard form?
A: Some common mistakes to avoid when rewriting an equation in standard form include:
- Not multiplying both sides of the equation by the LCM
- Not eliminating fractions
- Not checking for errors in the LCM or GCD
- Not using the correct formula for finding the LCM or GCD
Q: How can I practice rewriting equations in standard form?
A: You can practice rewriting equations in standard form by working through examples and exercises in a textbook or online resource. You can also try rewriting equations in standard form on your own, using a calculator or other tool to check your work.
Q: What are some real-world applications of rewriting equations in standard form?
A: Rewriting equations in standard form has many real-world applications, including:
- Solving systems of linear equations in physics and engineering
- Graphing linear equations in mathematics and science
- Modeling real-world situations in economics and finance
- Solving optimization problems in business and management
Q: Can I use rewriting equations in standard form to solve non-linear equations?
A: No, rewriting equations in standard form is typically used to solve linear equations. Non-linear equations require different techniques and methods to solve.