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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to solve equations of the form x + a = b and x - a = b . By the end of the module students should understand the meaning and function of an equation, understand what is meant by the solution to an equation and be able to solve equations of the form x + a = b size 12{x+a=b} {} and x a = b size 12{x - a=b} {} .

Section overview

  • Equations
  • Solutions and Equivalent Equations
  • Solving Equations

Equations

Equation

An equation is a statement that two algebraic expressions are equal.

The following are examples of equations:

x + 6 This expression = = 10 This expression x - 4 This expression = = - 11 This expression 3 y - 5 This expression = = - 2 + 2 y This expression

Notice that x + 6 size 12{x+6} {} , x 4 size 12{x - 4} {} , and 3 y 5 size 12{3y - 5} {} are not equations. They are expressions. They are not equations because there is no statement that each of these expressions is equal to another expression.

Solutions and equivalent equations

Conditional equations

The truth of some equations is conditional upon the value chosen for the variable. Such equations are called conditional equations . There are two additional types of equations. They are examined in courses in algebra, so we will not consider them now.

Solutions and solving an equation

The set of values that, when substituted for the variables, make the equation true, are called the solutions of the equation.
An equation has been solved when all its solutions have been found.

Sample set a

Verify that 3 is a solution to x + 7 = 10 size 12{x+7 = "10"} {} .

When x = 3 size 12{x=3} {} ,

x + 7 = 10 becomes 3 + 7 = 10 10 = 10 which is a  true  statement, verifying that 3   is a solution to   x + 7 = 10

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Verify that - 6 is a solution to 5 y + 8 = 22 size 12{5y+8= - "22"} {}

When y = - 6 size 12{y=-6} {} ,

5 y + 8 = - 22 becomes 5 ( - 6 ) + 8 = - 22 - 30 + 8 = - 22 - 22 = - 22 which is a  true  statement, verifying that - is a solution to 5 y + 8 = - 22

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Verify that 5 is not a solution to a 1 = 2 a + 3 size 12{a - 1=2a+3} {} .

When a = 5 size 12{a=5} {} ,

a - 1 = 2 a + 3 becomes 5 - 1 = 2 5 + 3 5 - 1 = 10 + 3 4 = 13 a  false  statement, verifying that   5   is not a solution to  a - 1 = 2 a + 3

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Verify that -2 is a solution to 3 m 2 = 4 m 16 size 12{3m - 2= - 4m - "16"} {} .

When m = - 2 size 12{x=3} {} ,

3 m - 2 = - 4 m - 16 becomes 3 ( - 2 ) - 2 = - 4 ( - 2 ) - 16 - 6 - 2 = 8 - 16 - 8 = - 8 which is a   true  statement, verifying that - 2   is a solution to  3 m - 2 = - 4 m - 16

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Practice set a

Verify that 5 is a solution to m + 6 = 11 size 12{m+6="11"} {} .

Substitute 5 into m + 6 = 11 size 12{m+6="11"} {} . Does 5 plus 6 equal 11? Yes. Thus, 5 is a solution.

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Verify that - 5 is a solution to 2 m 4 = 14 size 12{2m - 4= - "14"} {} .

Substitute -5 into 2 m 4 = 14 size 12{2m - 4= - "14"} {} . does 2 time negative 5 minus 4 equal negative 14? Yes. Thus, -5 is a solution.

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Verify that 0 is a solution to 5 x + 1 = 1 size 12{5x+1=1} {} .

Substitute 0 into 5 x + 1 = 1 size 12{5x+1=1} {} . Does 5 times zero plus one equal 1? Yes. Thus, 0 is a solution.

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Verify that 3 is not a solution to 3 y + 1 = 4 y + 5 size 12{ - 3y+1=4y+5} {} .

Substitute 3 into 3 y + 1 = 4 y + 5 size 12{ - 3y+1=4y+5} {} . Does negative 3 times 3 plus 1 equal 4 times 3 plus 5? No. Thus, 3 is not a solution.

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Verify that -1 is a solution to 6 m 5 + 2 m = 7 m 6 size 12{6m - 5+2m=7m - 6} {} .

Substitute -1 into 6 m 5 + 2 m = 7 m 6 size 12{6m - 5+2m=7m - 6} {} . Does 6 times negative 1 minus 5 plus 2 times negative 1 equal 7 times negative 1 minus 6? Yes. Thus, -1 is a solution.

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Equivalent equations

Some equations have precisely the same collection of solutions. Such equations are called equivalent equations. For example, x - 5 = - 1 size 12{"x - 5 "=" -1"} {} , x + 7 = 11 size 12{"x "+" 7 "=" 11"} {} , and x = 4 size 12{x=4} {} are all equivalent equations since the only solution to each is x = 4 size 12{x=4} {} . (Can you verify this?)

Solving equations

We know that the equal sign of an equation indicates that the number represented by the expression on the left side is the same as the number represented by the expression on the right side.

This number is the same as this number
x size 12{x} {} = 4
x + 7 size 12{x+7} {} = 11
x 5 size 12{x - 5} {} = -1

Questions & Answers

explain and give four Example hyperbolic function
Lukman Reply
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
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Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
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Seidu
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Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
From 1973 to 1979, in the United States, there was an increase of 166.6% of Ph.D. social scien­tists to 52,000. How many were there in 1973?
Khizar Reply
7hours 36 min - 4hours 50 min
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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