Solve The System Of Equations To Find The Values Of X X X And Y Y Y .${ \begin{align*} 2x + 3y &= 53 \ 3x - Y &= 19 \end{align*} }$

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables, $x$ and $y$. We will use the given system of equations as an example to demonstrate the steps involved in solving such systems.

The System of Equations

The given system of equations is:

{ \begin{align*} 2x + 3y &= 53 \\ 3x - y &= 19 \end{align*} \}

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

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 Second Equation for $y$

We can solve the second equation for $y$ by isolating $y$ on one side of the equation.

{ \begin{align*} 3x - y &= 19 \\ -y &= 19 - 3x \\ y &= 3x - 19 \end{align*} \}

Step 2: Substitute the Expression for $y$ into the First Equation

Now that we have an expression for $y$, we can substitute it into the first equation.

{ \begin{align*} 2x + 3y &= 53 \\ 2x + 3(3x - 19) &= 53 \end{align*} \}

Step 3: Simplify the Equation

We can simplify the equation by distributing the $3$ to the terms inside the parentheses.

{ \begin{align*} 2x + 9x - 57 &= 53 \\ 11x - 57 &= 53 \end{align*} \}

Step 4: Add $57$ to Both Sides of the Equation

To isolate the term with $x$, we can add $57$ to both sides of the equation.

{ \begin{align*} 11x - 57 + 57 &= 53 + 57 \\ 11x &= 110 \end{align*} \}

Step 5: Divide Both Sides of the Equation by $11$

Finally, we can divide both sides of the equation by $11$ to solve for $x$.

{ \begin{align*} \frac{11x}{11} &= \frac{110}{11} \\ x &= 10 \end{align*} \}

Step 6: Substitute the Value of $x$ into the Expression for $y$

Now that we have the value of $x$, we can substitute it into the expression for $y$.

{ \begin{align*} y &= 3x - 19 \\ y &= 3(10) - 19 \\ y &= 30 - 19 \\ y &= 11 \end{align*} \}

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 variable.

Step 1: Multiply the Two Equations by Necessary Multiples

To eliminate one variable, we can multiply the two equations by necessary multiples such that the coefficients of $y$ in both equations are the same.

{ \begin{align*} 2x + 3y &= 53 \\ 3x - y &= 19 \end{align*} \}

We can multiply the first equation by $1$ and the second equation by $3$.

{ \begin{align*} 2x + 3y &= 53 \\ 9x - 3y &= 57 \end{align*} \}

Step 2: Add the Two Equations

Now that the coefficients of $y$ are the same, we can add the two equations to eliminate $y$.

{ \begin{align*} (2x + 3y) + (9x - 3y) &= 53 + 57 \\ 11x &= 110 \end{align*} \}

Step 3: Solve for $x$

We can solve for $x$ by dividing both sides of the equation by $11$.

{ \begin{align*} \frac{11x}{11} &= \frac{110}{11} \\ x &= 10 \end{align*} \}

Step 4: Substitute the Value of $x$ into One of the Original Equations

Now that we have the value of $x$, we can substitute it into one of the original equations to solve for $y$.

{ \begin{align*} 2x + 3y &= 53 \\ 2(10) + 3y &= 53 \\ 20 + 3y &= 53 \end{align*} \}

Step 5: Subtract $20$ from Both Sides of the Equation

To isolate the term with $y$, we can subtract $20$ from both sides of the equation.

{ \begin{align*} 20 + 3y - 20 &= 53 - 20 \\ 3y &= 33 \end{align*} \}

Step 6: Divide Both Sides of the Equation by $3$

Finally, we can divide both sides of the equation by $3$ to solve for $y$.

{ \begin{align*} \frac{3y}{3} &= \frac{33}{3} \\ y &= 11 \end{align*} \}

Conclusion

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Introduction

Solving systems of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. In our previous article, we demonstrated two methods for solving a system of linear equations with two variables, $x$ and $y$. In this article, we will answer some frequently asked questions about solving systems of linear equations.

Q: What is a system of linear equations?

A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is a statement that two expressions are equal, and the variables are the unknown values that we are trying to find.

Q: How do I know which method to use to solve a system of linear equations?

There are two main methods for solving systems of linear equations: the substitution method and the elimination method. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable. You can choose the method that you prefer, or you can use a combination of both methods.

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

If you have a system of linear equations with three or more variables, you can use the same methods that we discussed earlier. However, you may need to use a combination of both methods, or you may need to use a different method altogether. For example, you can use the elimination method to eliminate one variable, and then use the substitution method to solve for the remaining variables.

Q: How do I know if a system of linear equations has a unique solution, no solution, or infinitely many solutions?

To determine if a system of linear equations has a unique solution, no solution, or infinitely many solutions, you can use the following criteria:

• If the system of equations is consistent (i.e., the equations are true for some values of the variables), and the number of equations is equal to the number of variables, then the system has a unique solution.
• If the system of equations is inconsistent (i.e., the equations are not true for any values of the variables), then the system has no solution.
• If the system of equations is consistent, but the number of equations is greater than the number of variables, then the system has infinitely many solutions.

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

If you have a system of linear equations with fractions or decimals, you can use the same methods that we discussed earlier. However, you may need to use a combination of both methods, or you may need to use a different method altogether. For example, you can use the elimination method to eliminate one variable, and then use the substitution method to solve for the remaining variables.

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

Yes, you can use a calculator to solve a system of linear equations. Many calculators have built-in functions for solving systems of linear equations, such as the "solve" function. You can also use a computer algebra system (CAS) to solve a system of linear equations.

Q: What are some common mistakes to avoid when solving systems of linear equations?

Some common mistakes to avoid when solving systems of linear equations include:

• Not checking if the system of equations is consistent or inconsistent before trying to solve it.
• Not using the correct method for solving the system of equations.
• Not checking if the solution is unique, or if there are infinitely many solutions.
• Not using a calculator or computer algebra system to check the solution.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics, and it is an essential skill for anyone who wants to succeed in algebra and geometry. In this article, we have answered some frequently asked questions about solving systems of linear equations, and we have provided some tips and tricks for solving these types of problems. We hope that this article has been helpful, and we encourage you to practice solving systems of linear equations to become proficient in this skill.