Question

The sum of two numbers is 7 and the difference of their squares is \(21 .\) Find the numbers.

Short answer

The numbers are 5 and 2.

Step-by-step solution

Set up the equations

Let's denote the two numbers as x and y. According to the problem, we have the following two equations: 1. The sum of the numbers: \[ x + y = 7 \] 2. The difference of their squares: \[ x^2 - y^2 = 21 \]

Use the difference of squares formula

We know that the difference of squares can be factored as follows: \[ x^2 - y^2 = (x + y)(x - y) \] Substitute the sum of x and y from the first equation: \[ 21 = (x + y)(x - y) \] Since we know that \( x + y = 7 \), substitute this in: \[ 21 = 7(x - y) \]

Solve for the difference of the numbers

Simplify the equation to find \( x - y \): \[ 21 = 7(x - y) \] \[ x - y = 3 \]

Solve the system of equations

Now we have the system of linear equations: \[ x + y = 7 \] \[ x - y = 3 \] Add the equations: \[ (x + y) + (x - y) = 7 + 3 \] \[ 2x = 10 \] \[ x = 5 \] Substitute \( x = 5 \) back into the first equation: \[ 5 + y = 7 \] \[ y = 2 \]

Key concepts

algebra

Algebra is a branch of mathematics that uses symbols to represent numbers and express relationships between them. This problem involves setting up and solving equations involving unknown variables. Here, we use algebraic expressions and equations to find the unknown numbers, denoted as x and y. By setting up equations based on the given conditions in the problem, we apply algebra to solve for these variables. This process includes forming and manipulating equations, using substitution and factoring techniques, and solving for unknowns step-by-step.

linear equations

Linear equations are equations that form straight lines when graphed. They have variables raised to the power of 1 and do not have products of variables. In this problem, the two linear equations are:

• \(x + y = 7\)
• \(x - y = 3\)

To solve these, we use techniques like substitution or addition. In our case, adding the equations helps:

• \( (x + y) + (x - y) = 7 + 3 \)
• \( 2x = 10 \)
• \( x = 5 \)

Then, substituting back to find y:

• \( 5 + y = 7 \)
• \( y = 2 \)

Linear equations are fundamental because they represent simple and direct relationships between variables.

difference of squares

The difference of squares is a specific algebraic expression where two square terms are subtracted. It's represented as \(a^2 - b^2\). This can be factored into \( (a - b)(a + b) \). In our problem, the difference of squares is crucial: \[ x^2 - y^2 = 21 \] We use factorization: \[ (x + y)(x - y) = 21 \] With \( x + y = 7 \), we substitute: \[ 21 = 7(x - y) \] From here, solving for \( x - y \):

• \( 21 = 7(x - y) \)
• \( x - y = 3 \)

This simplifies the expression and allows us to solve the system of equations. Thus, the difference of squares plays a key role in transforming our equation into a solvable form.