Problem 51 - College Algebra with Corequisite Support

Question

For the following exercises, solve each system in terms of \(A, B, C, D, E, \quad\) and \(F\) where \(A-F\) are nonzero numbers. Note that \(A \neq B\) and \(A E \neq B D\). \(x+y=A\) \(x-y=B\)

Step-by-Step Solution

Answer

\( x = \frac{A + B}{2}, \quad y = \frac{A - B}{2} \).

Step 1: Add the Equations

We have two equations: 1. \( x + y = A \) 2. \( x - y = B \). To eliminate \( y \), add both equations:\[(x + y) + (x - y) = A + B\]This simplifies to:\[2x = A + B\].

Step 2: Solve for x

From the equation obtained in Step 1, solve for \( x \):\[2x = A + B\]Divide both sides by 2 to solve for \( x \):\[x = \frac{A + B}{2}\].

Step 3: Subtract the Equations

Using the original equations again, subtract the second equation from the first equation:\[(x + y) - (x - y) = A - B\]This simplifies to:\[2y = A - B\].

Step 4: Solve for y

From the equation obtained in Step 3, solve for \( y \):\[2y = A - B\]Divide both sides by 2 to solve for \( y \):\[y = \frac{A - B}{2}\].

Step 5: Write the Solution

The solution for the system of equations is found as:\[x = \frac{A + B}{2}, \quad y = \frac{A - B}{2}\]This is the solution expressed in terms of \( A \) and \( B \).

Key Concepts

Simultaneous Equations

Simultaneous equations are sets of equations with multiple variables that are solved together. They are crucial in mathematics because they allow us to find precise values for variables that work in all given equations. To tackle these, we need to find a common solution that solves each equation in the system simultaneously. This can be done using various methods, such as substitution, elimination, or graphical methods.
- **Substitution Method**: Solve one equation for one variable and substitute that solution in the other equations.
- **Elimination Method**: This involves eliminating one variable by adding or subtracting equations, which helps simplify the system. In our exercise, we used elimination to solve for both variables.
- **Graphical Method**: Each equation is plotted on a graph, and the point(s) where all lines intersect are the solutions.
These methods highlight the versatility and applicability of simultaneous equations in real-world problem-solving.

Linear Equations

Linear equations are equations of the first degree. This means that the variables in the equation are raised to the power of one. They form a line on a graph, which is why they are called 'linear.' The general form of a linear equation in two variables is expressed as: \[ax + by = c\] Where \(a\), \(b\), and \(c\) are constants. In our exercise, both given equations, \(x+y=A\) and \(x-y=B\), are linear equations. These types of equations are straightforward to work with due to their simple structure.
- **Characteristics of Linear Equations**:
- They have constant rates of change, represented in their graphs as straight lines.
- They can have one solution (intersect at one point), no solutions (parallel lines), or infinitely many solutions (coinciding lines).
Understanding how linear equations work makes solving them more intuitive and allows for their application in various fields including physics, economics, and engineering.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions and equations to achieve a desired form or solution. It is a core skill in solving all types of algebraic problems, including systems of equations. Key techniques include:
- **Simplification**: Combine like terms or use basic arithmetic operations to simplify expressions, as shown when the equations \((x+y) + (x-y)\) and \((x+y) - (x-y)\) were simplified.
- **Isolation of Variables**: Often involves moving terms across the equals sign by performing inverse operations to solve for one variable at a time. For example, solving \(2x = A + B\) for \(x\) by dividing both sides by 2.
- **Substitution**: Substitute expressions obtained from one equation into another to find solutions.
These techniques are interwoven in solving systems of equations and are essential in reducing complex problems into more manageable ones. Mastery of algebraic manipulation can significantly ease the process of finding solutions efficiently.