/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 72 The sum of two numbers is 7 and ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 72

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

Short Answer

Expert verified
The numbers are 5 and 2.

Step by step solution

01

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: \[ x^2 - y^2 = 21 \]
02

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: \[ 21 = 7(x - y) \]
03

Solve for the difference of the numbers

Simplify the equation to find \( x - y \): \[ 21 = 7(x - y) \] \[ x - y = 3 \]
04

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: \[ 5 + y = 7 \] \[ y = 2 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Write a brief paragraph outlining your strategy for solving a system of two linear equations containing two variables.

Verify that the values of the variables listed are solutions of the system of equations. $$ \begin{array}{l} \left\\{\begin{array}{l} x-y &=3 \\ -3 x+y &=1 \\ \end{array}\right.\\\ x=-2, y =-5 ;(-2,-5) \end{array} $$

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{rr} 2 x+y-3 z= & 0 \\ -2 x+2 y+z= & -7 \\ 3 x-4 y-3 z= & 7 \end{array}\right. $$

Challenge Problem Solve for \(x\) and \(y,\) assuming \(a \neq 0\) and \(b \neq 0\) $$ \left\\{\begin{array}{l} a x+b y=a+b \\ a b x-b^{2} y=b^{2}-a b \end{array}\right. $$

Add: \(\frac{x+1}{x-3}+\frac{4}{x+3}\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.