# Solving linear equations: the addition property

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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

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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Sherica
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Sherica
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Tamia
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Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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Kristine 2*2*2=8
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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J, combine like terms 7x-4y
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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