Solve The Following System Of Linear Equations 2x2 By Substituting The Values Of X And Y To Identify The Correct Solution: −2x+9y=−19 And 3x+10y=5

In the world of mathematics, solving systems of linear equations is a fundamental skill with applications across various fields. Whether you're calculating the trajectory of a rocket or balancing a budget, understanding how to find the values of unknowns is crucial. In this comprehensive guide, we'll dive into the process of solving a system of two linear equations with two variables. Our focus will be on the given system: −2x + 9y = −19 and 3x + 10y = 5. We'll explore the underlying principles, step-by-step methods, and provide clear explanations to empower you with the knowledge to tackle similar problems confidently. By the end of this guide, you'll be able to identify the correct solution by substituting values of x and y back into the equations, ensuring a thorough understanding of the solution process.

Understanding Linear Equations

Before we delve into solving the system, let's establish a solid understanding of linear equations. A linear equation is an algebraic expression in which the highest power of any variable is 1. In a two-variable linear equation, we typically have two unknowns, often denoted as x and y. These equations can be represented graphically as straight lines on a coordinate plane. The solution to a system of two linear equations represents the point (x, y) where the two lines intersect. This point satisfies both equations simultaneously, making it the unique solution to the system, provided the lines are not parallel or coincident. Grasping this geometric interpretation helps visualize the solution process and reinforces the concept of finding values that make both equations true.

Methods for Solving Linear Equations

Several methods exist for solving systems of linear equations, each with its strengths and applications. We will focus on two common methods: substitution and elimination. The substitution method involves solving one equation for one variable and substituting that expression into the other equation, effectively reducing the system to a single-variable equation. This method is particularly useful when one equation can be easily solved for one variable. The elimination method, also known as the addition or subtraction method, involves manipulating the equations so that the coefficients of one variable are opposites. Adding the equations then eliminates that variable, leading to a single-variable equation. This method is efficient when the coefficients of one variable are easily made opposites by multiplication.

Solving the System: −2x + 9y = −19 and 3x + 10y = 5

Now, let's apply these methods to our given system of equations: −2x + 9y = −19 and 3x + 10y = 5. We'll demonstrate both the substitution and elimination methods to provide a comprehensive understanding of the solution process.

Method 1: Elimination

The elimination method aims to eliminate one variable by manipulating the equations. To do this, we look for ways to make the coefficients of either x or y opposites. In this case, we can eliminate x by multiplying the first equation by 3 and the second equation by 2. This will make the coefficients of x -6 and 6, respectively.

Multiply the first equation (−2x + 9y = −19) by 3:

3(−2x + 9y) = 3(−19)

This simplifies to:

-6x + 27y = -57

Next, multiply the second equation (3x + 10y = 5) by 2:

2(3x + 10y) = 2(5)

This simplifies to:

6x + 20y = 10

Now, we have the modified system:

-6x + 27y = -57 6x + 20y = 10

Add the two equations together. Notice that the -6x and 6x terms cancel each other out:

(-6x + 27y) + (6x + 20y) = -57 + 10

This simplifies to:

47y = -47

Now, solve for y by dividing both sides by 47:

y = -47 / 47 y = -1

Now that we have the value of y, we can substitute it back into either of the original equations to solve for x. Let's use the first original equation, −2x + 9y = −19:

-2x + 9(-1) = -19 -2x - 9 = -19

Add 9 to both sides:

-2x = -19 + 9 -2x = -10

Divide both sides by -2:

x = -10 / -2 x = 5

Thus, the solution to the system using the elimination method is x = 5 and y = -1.

Method 2: Substitution

The substitution method involves solving one equation for one variable and substituting that expression into the other equation. Let's start by solving the second equation (3x + 10y = 5) for x:

3x = 5 - 10y

Divide both sides by 3:

x = (5 - 10y) / 3

Now, substitute this expression for x into the first equation (−2x + 9y = −19):

-2((5 - 10y) / 3) + 9y = -19

Multiply both sides of the equation by 3 to eliminate the fraction:

3(-2((5 - 10y) / 3) + 9y) = 3(-19) -2(5 - 10y) + 27y = -57

Distribute the -2:

-10 + 20y + 27y = -57

Combine like terms:

47y - 10 = -57

Add 10 to both sides:

47y = -57 + 10 47y = -47

Divide both sides by 47:

y = -47 / 47 y = -1

Now that we have the value of y, substitute it back into the expression for x:

x = (5 - 10(-1)) / 3 x = (5 + 10) / 3 x = 15 / 3 x = 5

Thus, the solution to the system using the substitution method is x = 5 and y = -1.

Verifying the Solution

After obtaining a solution, it's crucial to verify its correctness. This involves substituting the values of x and y back into the original equations to ensure they hold true. This step is a safeguard against errors made during the solution process and confirms that the values satisfy both equations simultaneously.

Substituting into the First Equation (−2x + 9y = −19)

Let's substitute x = 5 and y = -1 into the first equation:

-2(5) + 9(-1) = -19 -10 - 9 = -19 -19 = -19

The equation holds true, confirming that the values satisfy the first equation.

Substituting into the Second Equation (3x + 10y = 5)

Now, let's substitute x = 5 and y = -1 into the second equation:

3(5) + 10(-1) = 5 15 - 10 = 5 5 = 5

The equation also holds true, confirming that the values satisfy the second equation.

Since the values x = 5 and y = -1 satisfy both equations, we can confidently conclude that this is the correct solution to the system.

Common Mistakes to Avoid

When solving systems of linear equations, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them and improve your accuracy.

One frequent mistake is making errors in arithmetic, especially when dealing with negative numbers or fractions. Careful attention to detail and double-checking calculations can prevent these errors. Another common mistake is incorrectly applying the distributive property when multiplying equations by a constant. Ensure that you multiply every term in the equation by the constant, not just some of them. A third pitfall is not properly substituting the expression for one variable into the other equation when using the substitution method. Make sure to replace the variable entirely with the expression, and use parentheses to avoid sign errors. Lastly, forgetting to verify the solution by substituting the values back into the original equations is a critical oversight. Always check your solution to ensure accuracy.

Applications of Linear Equations

Systems of linear equations are not just abstract mathematical concepts; they have numerous real-world applications. They are used extensively in fields like physics, engineering, economics, and computer science. For example, in physics, linear equations can describe the motion of objects or the flow of electrical current in a circuit. In economics, they can model supply and demand curves, helping to determine market equilibrium. In computer graphics, linear equations are used to perform transformations on objects, such as rotations and scaling. Understanding how to solve these systems is therefore a valuable skill with broad applicability.

In this comprehensive guide, we've explored the process of solving systems of linear equations, focusing on the system −2x + 9y = −19 and 3x + 10y = 5. We've covered the fundamental concepts of linear equations, discussed the substitution and elimination methods, and provided a step-by-step solution using both approaches. We've also emphasized the importance of verifying the solution and highlighted common mistakes to avoid. By mastering these techniques, you'll be well-equipped to tackle a wide range of problems involving linear equations. Remember, practice is key to developing proficiency, so continue to apply these methods to various systems and solidify your understanding.

Solving systems of linear equations is more than just a mathematical exercise; it's a valuable skill that enhances your problem-solving abilities and opens doors to numerous applications in the real world. So, embrace the challenge, hone your skills, and confidently navigate the world of linear equations!