Solving Systems Of Equations Finding The Value Of X

In the realm of mathematics, systems of equations play a crucial role in solving problems involving multiple variables and constraints. These systems often arise in various scientific and engineering applications, as well as in everyday scenarios. This article delves into the process of solving a system of equations to determine the value of an unknown variable, x. We will explore the steps involved in substituting values, combining like terms, and applying algebraic properties to arrive at the solution. Understanding these techniques is fundamental for anyone seeking to excel in mathematics and related fields. Let's embark on this journey of mathematical exploration and uncover the value of x in the given system of equations.

Understanding Systems of Equations

Before diving into the specific problem, let's first grasp the concept of a system of equations. A system of equations is a set of two or more equations containing the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. In other words, we seek a solution that makes each equation in the system true. Systems of equations can be linear, quadratic, or involve other types of functions, and they can have one solution, no solutions, or infinitely many solutions. The methods for solving systems of equations vary depending on the type of equations involved. For linear systems, common techniques include substitution, elimination, and graphing. For nonlinear systems, more advanced methods may be required. In this article, we will focus on a linear system and utilize the substitution method to find the value of x. Grasping the essence of systems of equations sets the stage for tackling more complex mathematical challenges.

The Power of Substitution

The substitution method is a powerful technique for solving systems of equations. It involves solving one equation for one variable and then substituting that expression into another equation. This process eliminates one variable, resulting in a single equation with one unknown, which can then be easily solved. Once the value of one variable is found, it can be substituted back into either of the original equations to find the value of the other variable. The substitution method is particularly useful when one equation is already solved for one variable or when it is easy to isolate one variable. This method allows us to systematically reduce the complexity of the system and arrive at a solution. In the context of our problem, we will leverage the substitution method to find the value of x in the given system of equations.

The Given System of Equations

We are presented with the following system of equations:

3x + y = 9
y = -4x + 10

This system consists of two linear equations with two variables, x and y. Our objective is to determine the value of x that satisfies both equations simultaneously. Notice that the second equation is already solved for y, making the substitution method an ideal approach for solving this system. The first equation, 3x + y = 9, represents a linear relationship between x and y. The second equation, y = -4x + 10, also represents a linear relationship, but it explicitly expresses y in terms of x. This explicit expression for y is the key to applying the substitution method. By substituting this expression into the first equation, we can eliminate y and obtain an equation solely in terms of x. This equation can then be solved for x, providing us with the desired solution.

Step 1: Substitute the Value of y in the First Equation

As the first step in solving the system, we substitute the expression for y from the second equation into the first equation. The second equation states that y = -4x + 10. We replace the y in the first equation, 3x + y = 9, with this expression. This substitution yields the following equation:

3x + (-4x + 10) = 9

This equation now contains only one variable, x, making it solvable. The substitution process has effectively eliminated y, allowing us to focus on finding the value of x. It's crucial to perform the substitution carefully, ensuring that the entire expression for y is correctly placed within the first equation. Parentheses are used to maintain the correct order of operations and prevent any sign errors. The resulting equation, 3x + (-4x + 10) = 9, is a linear equation in x, and we can proceed to simplify and solve it using algebraic techniques. The substitution step is a critical step in the substitution method, as it transforms the system of equations into a single equation that can be readily solved.

Step 2: Combine Like Terms

Now that we have the equation 3x + (-4x + 10) = 9, our next step is to combine like terms. This involves simplifying the equation by grouping together terms that have the same variable and exponent. In this case, we have two terms with x: 3x and -4x. Combining these terms, we get:

3x - 4x + 10 = 9
-x + 10 = 9

The 3x and -4x terms are combined to give -x. The constant term, 10, remains unchanged. This simplification reduces the equation to a more manageable form, -x + 10 = 9. Combining like terms is a fundamental algebraic technique that simplifies expressions and equations, making them easier to solve. It involves identifying terms with the same variable and exponent and then adding or subtracting their coefficients. This process helps to consolidate the equation and isolate the variable of interest. In our case, combining like terms has brought us closer to solving for x. The simplified equation, -x + 10 = 9, is now ready for the next step in the solution process.

Step 3: Apply the Subtraction Property of Equality

To isolate x and solve for its value, we need to apply the subtraction property of equality. This property states that if we subtract the same value from both sides of an equation, the equation remains balanced. In our equation, -x + 10 = 9, we want to isolate the -x term. To do this, we subtract 10 from both sides of the equation:

-x + 10 - 10 = 9 - 10
-x = -1

Subtracting 10 from both sides cancels out the +10 on the left side, leaving us with -x on the left and -1 on the right. The equation is now simplified to -x = -1. The subtraction property of equality is a cornerstone of algebraic manipulation. It allows us to move terms from one side of the equation to the other while maintaining the equality. By applying this property, we can systematically isolate the variable we are trying to solve for. In our case, subtracting 10 from both sides has successfully isolated the -x term. The resulting equation, -x = -1, is a simple equation that can be easily solved for x. We are now just one step away from finding the value of x.

Step 4: Solve for x

We have arrived at the equation -x = -1. To solve for x, we need to get rid of the negative sign in front of x. We can do this by multiplying both sides of the equation by -1:

(-1) * (-x) = (-1) * (-1)
x = 1

Multiplying -x by -1 gives us x, and multiplying -1 by -1 gives us 1. Therefore, we have found that x = 1. This is the solution for x in the given system of equations. Multiplying both sides of an equation by -1 is a common technique for changing the sign of a variable or term. It is based on the multiplicative property of equality, which states that if we multiply both sides of an equation by the same nonzero value, the equation remains balanced. In our case, multiplying by -1 has successfully eliminated the negative sign in front of x, allowing us to solve for its value. The solution, x = 1, is the value of x that satisfies both equations in the system. We have successfully navigated the steps of substitution, combining like terms, and applying algebraic properties to arrive at the answer.

Conclusion

In conclusion, by employing the substitution method and carefully applying algebraic principles, we have successfully determined that the value of x in the given system of equations is 1. This process involved substituting the expression for y from one equation into the other, combining like terms, and utilizing the subtraction property of equality to isolate x. The solution, x = 1, represents the value that satisfies both equations simultaneously. Understanding and mastering these techniques is essential for solving a wide range of mathematical problems, particularly those involving systems of equations. The ability to manipulate equations and isolate variables is a fundamental skill in mathematics and its applications. This example illustrates the power and versatility of algebraic methods in finding solutions to mathematical challenges. The value of x being 1 is not just a numerical answer; it represents a point of intersection between the two lines represented by the given equations, showcasing the geometric interpretation of solving systems of equations.