Problem 50 - Elementary Algebra Within Reach

Question

The sum of two numbers \(x\) and \(y\) is 50 and the difference of the two numbers is 20 . The system of equations that represents this situation is $$ \left\\{\begin{array}{l} x+y=50 \\ x-y=20 \end{array}\right. \text {. } $$ Solve this system to find the two numbers.

Step-by-Step Solution

Answer

The two numbers are 35 and 15.

Step 1: Identify the system of equations

The system of equations given in the problem is: \(x + y = 50\) and \(x - y = 20\). These equations represent the sum and difference of the two numbers, respectively.

Step 2: Solve by addition or subtraction

A good strategy for this problem could be to add the given two equations in order to eliminate one variable. By adding \(x + y = 50\) and \(x - y = 20\) you get \(2x = 70\).

Step 3: Solve for x

To solve for \(x\), divide both sides of the equation by 2. This gives: \(x = 70 / 2\), so \(x = 35\).

Step 4: Substitute x value into one equation to solve for y

You can substitute \(x = 35\) into either of the original equations. Opt for the simpler one: \(x - y = 20\). Substituting 35 for x gives: \(35 - y = 20\).

Step 5: Solve for y

Rearrange the equation from step 4 to solve for \(y\). This gives: \(y = 35 - 20\), so \(y = 15\).

Key Concepts

Addition Method

The addition method, also known as the elimination method, is a convenient technique to solve systems of linear equations. This method involves adding or subtracting the equations in order to eliminate one of the variables, making it easier to solve for the other variable.

In our given problem, the system of equations is:

• \(x + y = 50\)
• \(x - y = 20\)

After identifying the system, adding these two equations together helps remove the variable \(y\) because:

• \((x + y) + (x - y) = 50 + 20\)
• leads to \(2x = 70\)

By using the addition method, we essentially simplify the problem into a single equation with only one variable, making it straightforward to solve for \(x\). Understanding when and how to effectively use the addition method is vital for making the solution process straightforward and clear.

Linear Equations

Linear equations are fundamental in algebra and represent a relationship between variables with no exponents greater than one. When solving problems, especially systems of equations, linear equations often describe how two variables relate to each other.

In the exercise, two linear equations represent the sum and difference of two numbers:

• \(x + y = 50\)
• \(x - y = 20\)

These equations are called a 'system' because they must be solved together, as more than one variable is involved. Each equation in a system provides unique information that helps determine the values of the variables.

Understanding linear equations is crucial because they appear in many forms in mathematics, and mastering them allows tackling more complex mathematical problems.

Algebraic Substitution

Algebraic substitution is an essential concept used to solve systems of equations, particularly after determining one of the variables. Once one variable is known, substitution involves replacing the variable in another equation with its numerical value to find the remaining variable.

In the step-by-step solution, once \(x = 35\) is determined, this value is substituted into the equation \(x - y = 20\) to solve for \(y\). The substitution process goes as follows:

• Replace \(x\) with 35 in the second equation: \(35 - y = 20\)
• Then solve for \(y\)
• This leads to \(y = 15\)

Substitution is simple yet effective; it reduces the complexity of systems by turning a two-variable equation into a single-variable equation.

Mastering algebraic substitution is a key skill as it is one of the fundamental techniques for solving equations across various branches of mathematics.