Intermediate Algebra/Solving Equations

Equations

Introduction

Equations, essentially, are two expressions set to be equal to each other. To be equal is to have the same value, so $4=4$ and $4=8/2$. We can express many word problems as equations. For example, the square of the sum of a number and 6 is equal to 49, which would be shown as $(x+6{)}^2=49$. Another example would be the sum of the square of a number and 6 is equal to 7, which would like this $x^2+6=7$.

Properties of Equality

The equal sign that depicts the fact that both sides of it are equal is a very strange symbol with many properties. It tells you various traits of each side, and it allows you to manipulate each side in specific ways. Here are the different properties of that sign:

Property Name	Definition	Example
Reflexive	a = a	7 = 7
Symmetric	If a = b, then b = a	If (3)(2) = 6, then 6 = (3)(2)
Transitive	If a = b & b = c, then a = c	If 8 = (4)(2) and (4)(2) = (2)(4), then 8 = (2)(4)
Substitution	If a = b, then one can replace a with b or vice versa	If a = b and 1 + a = 3, then 1 + b = 3
Addition	You can add one number to both sides of the equation.	x - 6 = 14 x - 6 + 6 = 14 + 6 x = 20
Subtraction	You can subtract one number from both sides of the equation.	x + 6 = 14 x + 6 - 6 = 14 - 6 x = 8
Multiplication	You can multiply both sides of the equation by a number.	x/6 = 18 (6)(x/6) = (18)(6) x = 108
Division	You can divide both sides of the equation by a number.	6x = 18 6x/6 = 18/6 x = 3

Practice Problems

Decide whether these following problems are expressions or equations.

1. $2x+6$

2. $3(x-14{)}^2=12$

3. $4+6(2x+16)$

Identify which properties are being used in the following problems.

1. $a=b$ and $3-a=4$, so $3-b=4$.

2. $x+9=12$, then $x=3$.

3. $x-9y=4y$, then $x=13y$.

Answers

1. Expression

2. Equation

3. Expression

1. Substitution Property of Equality

2. Subtraction Property of Equality

3. Addition Property of Equality (combine like-terms!)

Solving Equations

Although we have already solved a few equations, we will now discuss the formal idea of solving equations. To solve an equation, you are finding the value of any variables within the equation. To find the value of a variable, you have to manipulate the equation to state $*insertvariablehere*=*somenumber*$. Then you know the value of the variable! You will use the Properties of Equality to manipulate the equation into the desired form.

Practice Problems

Solve for x in the following equations.

1. $x+7=12$

2. $-\frac{2}{3}x+9=15$

3. $3x-15=8x$

4. $c(x-4m)=d(n-9)$

5. $20x+12=4(5x+3)$

Answers

1. $\begin{array}{ccc}x+7& =& 12\\ x& =& 5\end{array}$ Subtract 7

2. $\begin{array}{ccc}-\frac{2}{3}x+9& =& 15\\ -\frac{2}{3}x& =& 6\\ x& =& -9\end{array}$ Subtract 9, then multiply by -3/2 (Inverse Property of Multiplication)

3. $\begin{array}{ccc}3x-15& =& 8x\\ -15& =& 5x\\ -3& =& x\end{array}$ Subtract 3x, then divide by 5

4. No Solution, because there are too many variables to find a single number for x.

5. $\begin{array}{ccc}20x+12& =& 4(5x+3)\\ 20x+12& =& 20x+12\end{array}$ All Real Numbers. Perform Distributive Property, and you'll get the same equation on both sides. Thus, any number would work!

Lesson Review

Equations are two expressions that are equal to each other, and they are expressed by putting each of them on one side of the equal sign. You can add, subtract, multiply, or divide both sides of an equation while keeping it equal (For example, we know that 7 = 7, correct? What if we subtracted 2 from each side? We'd still have a true statement: 5 = 5). There are other properties of equality, such as the Reflexive, Symmetric, Transitive, and Substitution. You will be using all of these properties to solve (find the value of) variables in equations.

Lesson Quiz

1. What property is expressed here? $a=b$ and $b=c$, then $a=c$.

2. If I divided both sides of an equation by 4, would it still be equal on both sides? If so, why?

3. Solve for y. $5(\frac{y}{6}-2)=2y+4$

Quiz Answers

1. Transitive Property of Equality

2. Yes, due to the Division Property of Equality.

3. -12