Solving the Given System of Equations

Oct 13, 2025 by windailys.com 11 views

**Solving a System of Equations: A Step-by-Step Guide** =====================================================

What is a System of Equations?

A system of equations is a set of two or more equations that contain multiple variables. In this article, we will focus on solving a system of two linear equations with two variables.

Why Do We Need to Solve Systems of Equations?

Solving systems of equations is an essential skill in mathematics, science, and engineering. It helps us to find the values of variables that satisfy multiple equations simultaneously. This is particularly useful in real-world applications, such as:

Finding the intersection point of two lines
Determining the values of unknowns in a physical system
Solving optimization problems

How to Solve a System of Equations?

There are several methods to solve a system of equations, including:

Substitution Method: This involves solving one equation for one variable and substituting the expression into the other equation.
Elimination Method: This involves adding or subtracting the equations to eliminate one variable.
Graphical Method: This involves graphing the equations on a coordinate plane and finding the intersection point.

Solving the Given System of Equations

Let's solve the system of equations:

$\begin{align*} 5x + 2y &= 7 \\ -2x + 6y &= 9 \end{align*}$

Step 1: Multiply the Two Equations by Necessary Multiples

To eliminate one variable, we need to multiply the two equations by necessary multiples such that the coefficients of $x$ 's in both equations are the same.

$\begin{align*} (5x + 2y) \times 2 &= 7 \times 2 \\ -2x + 6y &= 9 \end{align*}$

$\begin{align*} 10x + 4y &= 14 \\ -2x + 6y &= 9 \end{align*}$

Step 2: Add the Two Equations

Now, we can add the two equations to eliminate the variable $x$ .

$\begin{align*} (10x + 4y) + (-2x + 6y) &= 14 + 9 \\ 8y &= 23 \end{align*}$

Step 3: Solve for $y$

Now, we can solve for $y$ by dividing both sides of the equation by 8.

$\begin{align*} y &= \frac{23}{8} \\ y &= 2.875 \end{align*}$

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

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

$\begin{align*} 5x + 2(2.875) &= 7 \\ 5x + 5.75 &= 7 \end{align*}$

Step 5: Solve for $x$

Now, we can solve for $x$ by subtracting 5.75 from both sides of the equation and then dividing both sides by 5.

$\begin{align*} 5x &= 7 - 5.75 \\ 5x &= 1.25 \\ x &= \frac{1.25}{5} \\ x &= 0.25 \end{align*}$

Conclusion

In this article, we solved a system of two linear equations with two variables using the elimination method. We multiplied the two equations by necessary multiples, added the two equations to eliminate one variable, solved for the other variable, and then substituted the value of the other variable into one of the original equations to solve for the first variable.

Frequently Asked Questions

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that contain multiple variables.

Q: Why do we need to solve systems of equations?

A: Solving systems of equations is an essential skill in mathematics, science, and engineering. It helps us to find the values of variables that satisfy multiple equations simultaneously.

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

A: There are several methods to solve a system of equations, including the substitution method, elimination method, and graphical method.

Q: How do I choose the correct method to solve a system of equations?

A: The choice of method depends on the type of equations and the variables involved. For example, if the equations are linear, the elimination method may be the most efficient way to solve the system.

Q: What is the importance of solving systems of equations in real-world applications?

A: Solving systems of equations is essential in real-world applications, such as finding the intersection point of two lines, determining the values of unknowns in a physical system, and solving optimization problems.

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. However, it's essential to understand the underlying concepts and methods to ensure accurate results.

Q: How do I check my answer when solving a system of equations?

A: To check your answer, substitute the values of the variables back into the original equations and verify that they are true.