Solving Linear Equations

Example 1 It’s obvious what the answer is. However, we need to start with the basics and work our way up because we need to make sure that we have GREAT fundamentals in math. In other words, we need to know WHY we do things other than “That’s what my teacher told me to do.” Why do we subtract the three? Next question: why do we subtract the three?

Solving Linear Equations Example 2 Most people know that the first thing you should do on this problem is add five to both sides. But why? Why not divide by the two first? Why not move the nine? Why is the five first? We move the five rather than the nine because the five is on the same side of the equals sign as the variable. We move the five before we move the two because of the order of operations. When you are solving an equation, you are undoing the problem. Therefore you use the order of operations in reverse order. Let me explain…

Solving Linear Equations ORDER OF OPERATIONS PEMDASPEMDAS ARENTHESES XPONENTS ULTIPLICATION IVISION DDITION UBTRACTION

Solving Linear Equations ORDER OF OPERATIONS PARENTHESES EXPONENTS MULTIPLICATION DIVISION ADDITION SUBTRACTION PEMDAS is used for evaluating expressions with no variables, like this:

Solving Linear Equations ORDER OF OPERATIONS SUBTRACTION ADDITION DIVISION MULTIPLICATION EXPONENTS PARENTHESES However, since we are solving an equation, we are UNDOING the order of operations. Therefore, we need to UNDO things in reverse order.

Solving Linear Equations Example 2 We add five to both sides since it is subtracted from the variable. We move the two by division because it is multiplied times the variable.

Solving Linear Equations Example 2 One last thing before we move on. When I say that x = 7, I am saying TWO things. What are the two things? The first thing that I am saying is that 7 works. That means when I plug 7 in for the x, I get a TRUE statement.

Solving Linear Equations Example 2 One last thing before we move on. When I say that x = 7, I am saying TWO things. What are the two things? What else am I saying? I am saying that NOTHING ELSE works.

Solving Rational Equations Example 3 This one is for you to try. Can you get a fraction for an answer?

Solving Rational Equations Example 4 Why is this problem different than the others we’ve seen so far? Because there’s a variable on both sides of the equation. Can you check your answers?

Solving Rational Equations Example 4 There are three rules when checking your answers: 1)Always plug the answers back into the ORIGINAL problem, 2)Never cross the equals sign, and 3)Use the order of operations on each side of the equation.

Solving Rational Equations Example 5 Before working this problem, get rid of the parentheses by distributing the outside numbers. If there are like terms on the same side of the equals sign, just combine them as they are.

Solving Rational Equations Example 5 Can you check these?

Solving Rational Equations Example 6 Don’t forget to multiply by the number and the sign in front of the number.

Solving Rational Equations Example 7 Fractions!!! You only need to know one “trick” to solve equations with fractions. To multiply a fraction times a whole number, divide the bottom into the whole number, then multiply what’s left. Let’s practice a couple of those before we work this equation.

Solving Rational Equations Example 7 Back to the problem. Remember, to multiply fractions times whole numbers, divide the bottom into the whole number, then multiply what’s left. ( ) 12 Find the common denominator of all of the fractions and multiply everything by that.

Solving Rational Equations Example 8 Remember, to multiply fractions times whole numbers, divide the bottom into the whole number, then multiply what’s left. ( ) 15

Solving Rational Equations Example 10 What happened that is different from the other problems? If the variables go away, then the answer is either “no solutions” or “all real numbers”. It’s “no solutions” if what’s leftover is a false statement. It’s “all real numbers” if what’s leftover is a true statement. O

Solving Rational Equations Example 11 Remember, if the variables go away, then the answer is either “no solutions” or “all real numbers”. It’s “no solutions” if what’s leftover is a false statement. It’s “all real numbers” if what’s leftover is a true statement. ALL REAL NUMBERS

Solving Rational Equations Example 12 Is this “no solutions” or “all real numbers”? Neither. Remember, it’s only one of those if the VARIABLES go away. Just keep on working this problem.

Solving Rational Equations Pair Practice: Solve these problems with a partner. 1) 2) 3) 4) 5)

Solving Rational Equations Answers to worksheet #1:

It’s time for… HEY STUPID!!! The Rules are very simple. Be the first person to identify the mistake made in the following problems, raise your hand and hollar “Hey, Stupid!” The first correct answer wins you five bonus points on the Chapter Test. Once you win once, you cannot win again until next chapter.

Solving Rational Equations HEY STUPID!!! #1 Didn’t distribute the 3 in the first step.

Solving Rational Equations HEY STUPID!!! #2 Supposed to subtract the 5 instead of the 3.

Solving Rational Equations HEY STUPID!!! #3 Should be no solutions, because the statement that is leftover is NOT true. ALL REAL NUMBERS

Solving Rational Equations HEY STUPID!!! #4 Didn’t distribute the negative in the first step.

Solving Rational Equations Hey, Stupid! #5 You have to multiply EVERYTHING by the common denominator. They didn’t multiply the four by six. ( ) 6