Solve The System Of Equations:1. 4 − 4.8 X = 1.7 Y 4 - 4.8x = 1.7y 4 − 4.8 X = 1.7 Y 2. 12.8 + 1.7 Y = − 13.2 X 12.8 + 1.7y = -13.2x 12.8 + 1.7 Y = − 13.2 X

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. In this article, we will focus on solving a system of two linear equations with two variables. The given system of equations is:

1. $4 - 4.8x = 1.7y$
2. $12.8 + 1.7y = -13.2x$

Understanding the System of Equations

To solve this system of equations, we need to understand the concept of linear equations and how to manipulate them to find the values of the variables. A linear equation is an equation in which the highest power of the variable(s) is 1. In this case, we have two linear equations with two variables, x and y.

Method 1: Substitution Method

One way to solve this system of equations is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Step 1: Solve the First Equation for y

We can start by solving the first equation for y:

$4 - 4.8x = 1.7y$

To solve for y, we can divide both sides of the equation by 1.7:

$y = \frac{4 - 4.8x}{1.7}$

Step 2: Substitute the Expression for y into the Second Equation

Now that we have an expression for y, we can substitute it into the second equation:

$12.8 + 1.7y = -13.2x$

Substituting the expression for y, we get:

$12.8 + 1.7\left(\frac{4 - 4.8x}{1.7}\right) = -13.2x$

Step 3: Simplify the Equation

Simplifying the equation, we get:

$12.8 + 4 - 4.8x = -13.2x$

Combine like terms:

$16.8 - 4.8x = -13.2x$

Step 4: Solve for x

Now we can solve for x by isolating it on one side of the equation:

$16.8 - 4.8x + 13.2x = 0$

Combine like terms:

$16.8 + 8.4x = 0$

Subtract 16.8 from both sides:

$8.4x = -16.8$

Divide both sides by 8.4:

$x = -2$

Step 5: Find the Value of y

Now that we have the value of x, we can find the value of y by substituting x into one of the original equations. We will use the first equation:

$4 - 4.8x = 1.7y$

Substituting x = -2, we get:

$4 - 4.8(-2) = 1.7y$

Simplifying the equation, we get:

$4 + 9.6 = 1.7y$

Combine like terms:

$13.6 = 1.7y$

Divide both sides by 1.7:

$y = 8$

Method 2: Elimination Method

Another way to solve this system of equations is by using the elimination method. This method involves adding or subtracting the equations to eliminate one of the variables.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of x or y in both equations are the same.

First, we will multiply the first equation by 13.2 and the second equation by 4.8:

1. $4 - 4.8x = 1.7y$
2. $12.8 + 1.7y = -13.2x$

Multiplying the first equation by 13.2, we get:

$52.8 - 63.36x = 22.04y$

Multiplying the second equation by 4.8, we get:

$61.44 + 8.16y = -63.36x$

Step 2: Add or Subtract the Equations

Now that we have the equations multiplied by necessary multiples, we can add or subtract them to eliminate one of the variables. We will add the equations:

$52.8 - 63.36x + 61.44 + 8.16y = 22.04y - 63.36x$

Combine like terms:

$114.24 + 8.16y = 22.04y - 63.36x + 63.36x$

Simplify the equation:

$114.24 + 8.16y = 22.04y$

Step 3: Solve for y

Now we can solve for y by isolating it on one side of the equation:

$114.24 = 22.04y - 8.16y$

Combine like terms:

$114.24 = 13.88y$

Divide both sides by 13.88:

$y = 8.25$

Step 4: Find the Value of x

Now that we have the value of y, we can find the value of x by substituting y into one of the original equations. We will use the first equation:

$4 - 4.8x = 1.7y$

Substituting y = 8.25, we get:

$4 - 4.8x = 1.7(8.25)$

Simplifying the equation, we get:

$4 - 4.8x = 14.025$

Subtract 4 from both sides:

$-4.8x = 10.025$

Divide both sides by -4.8:

$x = -2.09$

Conclusion

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

Q: What are the two methods used to solve a system of equations?

A: The two methods used to solve a system of equations are the substitution method and the elimination method.

Q: What is the substitution method?

A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one of the variables.

Q: How do I know which method to use?

A: You can use either method, but the substitution method is often easier to use when one of the equations is already solved for one variable. The elimination method is often easier to use when the coefficients of the variables in both equations are the same.

Q: What if I have a system of equations with three or more variables?

A: If you have a system of equations with three or more variables, you can use the substitution method or the elimination method to solve for two variables, and then use the values of those variables to solve for the third variable.

Q: Can I use a calculator to solve a system of equations?

A: Yes, you can use a calculator to solve a system of equations. Many calculators have built-in functions for solving systems of equations.

Q: What if I have a system of equations with no solution?

A: If you have a system of equations with no solution, it means that the equations are inconsistent and there is no value of the variables that can satisfy both equations.

Q: What if I have a system of equations with infinitely many solutions?

A: If you have a system of equations with infinitely many solutions, it means that the equations are dependent and there are many values of the variables that can satisfy both equations.

Q: Can I use a graphing calculator to solve a system of equations?

A: Yes, you can use a graphing calculator to solve a system of equations. Graphing calculators can graph the equations and find the intersection points, which can be used to solve the system of equations.

Q: What are some common mistakes to avoid when solving a system of equations?

A: Some common mistakes to avoid when solving a system of equations include:

• Not checking if the equations are consistent or inconsistent
• Not checking if the equations are dependent or independent
• Not using the correct method for solving the system of equations
• Not checking if the solution is correct

Q: How do I check if the solution is correct?

A: To check if the solution is correct, you can plug the values of the variables back into the original equations and check if they are true.

Q: What if I get a different solution using a different method?

A: If you get a different solution using a different method, it means that the solution is not unique and there are many values of the variables that can satisfy the system of equations.