Select The Correct Answer.Which Set Of Coordinates Satisfies The Equations $3x - 2y = 15$ And $4x - Y = 20$?

Introduction

In mathematics, a system of linear equations is a set of two or more linear 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. We will use the method of substitution and elimination to find the solution.

The Problem

We are given two linear equations:

1. $3x - 2y = 15$
2. $4x - y = 20$

Our goal is to find the values of $x$ and $y$ that satisfy both equations.

The Method of Substitution

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

Let's start by solving the second equation for $y$:

$$4x - y = 20$$

$$y = 4x - 20$$

Now, substitute this expression for $y$ into the first equation:

$$3x - 2(4x - 20) = 15$$

Expand and simplify:

$$3x - 8x + 40 = 15$$

$$-5x + 40 = 15$$

Subtract 40 from both sides:

$$-5x = -25$$

Divide both sides by -5:

$$x = 5$$

Now that we have found the value of $x$, we can substitute it into one of the original equations to find the value of $y$. Let's use the second equation:

$$4x - y = 20$$

$$4(5) - y = 20$$

$$20 - y = 20$$

Subtract 20 from both sides:

$$-y = 0$$

Divide both sides by -1:

$$y = 0$$

The Solution

Therefore, the solution to the system of linear equations is:

$$x = 5$$

$$y = 0$$

The Method of Elimination

Another way to solve a system of linear equations is to use the method of elimination. This involves adding or subtracting the equations to eliminate one of the variables.

Let's start by multiplying the first equation by 2 and the second equation by 1:

$$6x - 4y = 30$$

$$4x - y = 20$$

Now, add the two equations:

$$(6x + 4x) - (4y + y) = 30 + 20$$

Combine like terms:

$$10x - 5y = 50$$

Now, multiply the second equation by 5:

$$20x - 5y = 100$$

Subtract the first equation from the second equation:

$$(20x - 10x) - (5y - 5y) = 100 - 50$$

Combine like terms:

$$10x = 50$$

Divide both sides by 10:

$$x = 5$$

Now that we have found the value of $x$, we can substitute it into one of the original equations to find the value of $y$. Let's use the second equation:

$$4x - y = 20$$

$$4(5) - y = 20$$

$$20 - y = 20$$

Subtract 20 from both sides:

$$-y = 0$$

Divide both sides by -1:

$$y = 0$$

The Solution

Therefore, the solution to the system of linear equations is:

$$x = 5$$

$$y = 0$$

Conclusion

$$$$$$$$

Q: What is a system of linear equations?

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

Q: What are the two main methods for solving a system of linear equations?

A: The two main methods for solving a system of linear equations are the method of substitution and the method of elimination.

Q: What is the method of substitution?

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

Q: What is the method of elimination?

A: The method of elimination 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 method of substitution is often easier to use when one of the equations is already solved for one variable. The method of elimination is often easier to use when the coefficients of the variables are the same.

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

A: You can use the same methods, but you will need to use more equations and variables. You can also use other methods, such as graphing or matrices.

Q: What if I have a system of linear equations with fractions or decimals?

A: You can use the same methods, but you will need to multiply or divide the equations by the least common multiple (LCM) of the denominators to eliminate the fractions or decimals.

Q: How do I check my answer?

A: You can check your answer by plugging the values of the variables back into the original equations to make sure they are true.

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

A: If you get a system of linear 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 get a system of linear equations with infinitely many solutions?

A: If you get a system of linear 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 calculator to solve a system of linear equations?

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

Q: Can I use a computer program to solve a system of linear equations?

A: Yes, you can use a computer program to solve a system of linear equations. Many computer programs, such as MATLAB or Python, have built-in functions for solving systems of linear equations.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of linear equations. We have covered the basics of solving systems of linear equations, including the method of substitution and the method of elimination. We have also covered some common mistakes and how to check your answer.