# 11.3 Solving equations of the form x+a=b and x-a=b

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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 $x+a=b$ and $x-a=b$ .

## Section overview

• Equations
• Solutions and Equivalent Equations
• Solving Equations

## Equation

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

The following are examples of equations:

$\underset{\text{expression}}{\underset{\text{This}}{\underbrace{x+6}}}\underset{=}{\underset{}{\underset{}{=}}}\underset{\text{expression}}{\underset{\text{This}}{\underbrace{10}}}$ $\underset{\text{expression}}{\underset{\text{This}}{\underbrace{x-4}}}\underset{=}{\underset{}{\underset{}{=}}}\underset{\text{expression}}{\underset{\text{This}}{\underbrace{-11}}}$ $\underset{\text{expression}}{\underset{\text{This}}{\underbrace{3y-5}}}\underset{=}{\underset{}{\underset{}{=}}}\underset{\text{expression}}{\underset{\text{This}}{\underbrace{-2+2y}}}$

Notice that $x+6$ , $x-4$ , and $3y-5$ 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.

## 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 $x+7=\text{10}$ .

When $x=3$ ,

Verify that $-6$ is a solution to $5y+8=-\text{22}$

When $y=-6$ ,

Verify that 5 is not a solution to $a-1=2a+3$ .

When $a=5$ ,

Verify that -2 is a solution to $3m-2=-4m-\text{16}$ .

When $m=-2$ ,

## Practice set a

Verify that 5 is a solution to $m+6=\text{11}$ .

Substitute 5 into $m+6=\text{11}$ . Thus, 5 is a solution.

Verify that $-5$ is a solution to $2m-4=-\text{14}$ .

Substitute -5 into $2m-4=-\text{14}$ . Thus, -5 is a solution.

Verify that 0 is a solution to $5x+1=1$ .

Substitute 0 into $5x+1=1$ . Thus, 0 is a solution.

Verify that 3 is not a solution to $-3y+1=4y+5$ .

Substitute 3 into $-3y+1=4y+5$ . Thus, 3 is not a solution.

Verify that -1 is a solution to $6m-5+2m=7m-6$ .

Substitute -1 into $6m-5+2m=7m-6$ . Thus, -1 is a solution.

## Equivalent equations

Some equations have precisely the same collection of solutions. Such equations are called equivalent equations. For example, $x-5=-1$ , $x+7=11$ , and $x=4$ are all equivalent equations since the only solution to each is $x=4$ . (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 ↓ ↓ ↓ $x$ = 4 $x+7$ = 11 $x-5$ = -1

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There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
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da
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I only see partial conversation and what's the question here!
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please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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Professor
I think
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if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
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analytical skills graphene is prepared to kill any type viruses .
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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write examples of Nano molecule?
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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Is there any normative that regulates the use of silver nanoparticles?
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biomolecules are e building blocks of every organics and inorganic materials.
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how did you get the value of 2000N.What calculations are needed to arrive at it
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