Problem 44 - Algebra Form and Function

Question

Consider two numbers \(x\) and \(y\) satisfying the equations \(x+y=4\) and \(x-y=2\) (a) Describe in words the conditions that each equation places on the two numbers. (b) Find two numbers \(x\) and \(y\) satisfying both equations.

Step-by-Step Solution

Answer

Question: Find the values of x and y that satisfy the given system of equations: 1. \(x + y = 4\) 2. \(x - y = 2\) Answer: The values of x and y that satisfy the given system of equations are x=3 and y=1.

Step 1: Understanding the Equations

The given system of equations can be written as: 1. \(x + y = 4\) 2. \(x - y = 2\) (a) Equation 1 states that the sum of the two numbers x and y is 4. Equation 2 states that the difference between the two numbers x and y is 2.

Step 2: Adding the Equations

A common technique to solve these types of problems is to add or subtract the equations to eliminate one unknown variable. In this case, we will add Equation 1 and Equation 2 to eliminate the variable y: \((x+y) + (x-y) = 4 + 2\)

Step 3: Solving for x

Now, simplify the left side of the equation: \(x+y+x-y = 2x = 6\) Then, divide both sides by 2: \(2x = 6 \Rightarrow x=3\)

Step 4: Substituting x into an Equation

Now that we have the value for x, we can substitute it back into either Equation 1 or Equation 2 to find the value for y. We will use Equation 1: \(3 + y = 4\)

Step 5: Solving for y

Subtract 3 from both sides of the equation: \(y = 4 - 3\) Now, find the value of y: \(y = 1\)

Step 6: Conclusion

(b) The two numbers, x and y, satisfying both equations are x=3 and y=1.

Key Concepts

Variables

In mathematics, variables are symbols used to represent unknown or changing values. In the context of simultaneous equations, like the ones in our exercise, variables such as \(x\) and \(y\) denote the numbers we are trying to find. Variables play a crucial role because they provide flexibility, allowing us to generalize problems across different scenarios. Here, \(x\) and \(y\) are the two values that must satisfy both equations:

• \(x + y = 4\)
• \(x - y = 2\)

Both conditions must hold true simultaneously, hence the name 'simultaneous equations'. These variables help form equations that provide structured conditions for finding specific solutions. Think of them as placeholders that will be replaced by actual numbers once the problem is solved.

Elimination Method

The elimination method is a systematic way to solve simultaneous equations by removing one of the variables. The goal is to transform the system into one that allows for straightforward calculation of the remaining variable.Here's how it works:

• By either adding or subtracting the equations, one variable is eliminated, simplifying the problem.
• In our exercise, we add the equations \(x + y = 4\) and \(x - y = 2\), which cancels out \(y\), leaving \(2x = 6\). This simplification happens because \(+y\) and \(-y\) sum to zero.

Once one variable is eliminated, you solve for the remaining variable easily. For instance, \(2x = 6\) becomes \(x = 3\) when both sides are divided by 2. This method is efficient and particularly useful for linear equations as it reduces the number of steps needed to reach a solution.

Substitution Method

After determining one variable using elimination, the substitution method allows us to find the other variable. Substitution involves taking the value or expression for one variable derived from one equation and substituting it into another equation. Let's break it down:

• We start with the known value of \(x = 3\).
• Take this value and substitute it back into either original equation. In our exercise, we chose \(x + y = 4\).
• Replace \(x\) with 3, leading to the equation \(3 + y = 4\).

From here, solving for \(y\) is direct: subtract 3 from both sides to find \(y = 1\). The substitution method complements elimination by effectively using derived information to solve completely for all variables. It ensures you have consistent, accurate solutions for the system of equations.