Problem 21 - Precalculus

Question

Solve each system of linear equations by elimination. $$\begin{aligned} &x-y=-3\\\ &x+y=7 \end{aligned}$$

Step-by-Step Solution

Answer

The solution to the system of equations is \( x = 2, y = 5 \).

Step 1: Identify the Equations

We are given two linear equations: \( x - y = -3 \) and \( x + y = 7 \). These form a system of equations that we will solve using the elimination method.

Step 2: Add the Equations

To eliminate one of the variables, add the two equations together. \( (x - y) + (x + y) = -3 + 7 \) Simplify to get: \( 2x = 4 \)

Step 3: Solve for x

Divide both sides of the equation by 2 to solve for \( x \): \( x = \frac{4}{2} \) \( x = 2 \)

Step 4: Substitute x into One Equation

Substitute \( x = 2 \) back into one of the original equations to find \( y \). Using \( x + y = 7 \): \( 2 + y = 7 \)

Step 5: Solve for y

Subtract 2 from both sides of \( 2 + y = 7 \): \( y = 7 - 2 \) \( y = 5 \)

Step 6: Check the Solution

Substitute \( x = 2 \) and \( y = 5 \) into the other equation to verify the solution:\( x - y = -3 \) becomes \( 2 - 5 = -3 \), which is correct. The solution \( (x, y) = (2, 5) \) satisfies both equations.

Key Concepts

Elimination Method

The elimination method is a powerful technique used to solve systems of linear equations. The key idea is to eliminate one of the variables by combining the equations. This simplifies the system and makes it easier to find solutions. Here's a simple breakdown of the process:

• Identify the Equations: Start with two linear equations that you are given.
• Combine the Equations: Add or subtract the equations to eliminate one variable. For example, in our original system, we added the equations to cancel out the variable "y".
• Solve for the Remaining Variable: Once you've eliminated one variable, solve the remaining equation to find its value.
• Substitute Back: Plug the value back into one of the original equations to solve for the other variable.
• Verify the Solution: Check that the solution satisfies both original equations. This confirms accuracy.

By mastering the elimination method, you'll efficiently and accurately solve systems of equations, which is a fundamental skill in algebra.

Solving Equations

Solving equations involves finding values for the variables that make the equation true. When working with a system of linear equations, our goal is to find a common solution for all equations involved. Let's explore the steps:

• Understand the Equation Structure: Linear equations typically take the form of ax + by = c, where a, b, and c are constants. Recognizing this format helps in manipulating and solving them.
• Choose Your Method of Solving: There are several methods available such as substitution, graphing, and elimination. Choose one that suits the problem at hand.
• Perform Algebraic Operations: Use basic algebraic operations like addition, subtraction, multiplication, and division to isolate variables and simplify the equation.
• Check Your Solution: Always substitute the solution back into the original equations to ensure they hold true.

Understanding each step when solving equations builds confidence and enables you to tackle more complicated systems with ease.

Linear Algebra

Linear algebra focuses on linear equations, matrices, and vector spaces—key components in mathematical modeling and problem-solving in various fields.

• Linear Equations: These are equations involving one or more variables with constant coefficients. Solving these is often one of the first applications learned in algebra.
• Systems of Equations: Linear algebra excels in solving systems where multiple equations interact with common variables, as seen in the given exercise.
• Matrix Representation: Systems of linear equations can be represented using matrices. This allows for more elegant and computationally efficient solutions, especially with complexity increases.
• Applications: Linear algebra is not just an academic exercise; it has applications in science, engineering, economics, and many other disciplines, making it a versatile tool in both theoretical and practical contexts.

By grasping the core principles of linear algebra, you will enhance your ability to solve complex problems and interpret a broad range of mathematical situations.