Algebra Lesson (Veritas and Dominate the GMAT Videos)
MULTIPLY BY 1 SUMMARY
"Keep it Simple"
SUBSTITUTION METHOD
- Always works but can take LONGER TO DO!
ELIMINATION METHOD
- Crafty way to do it! I prefer this METHOD! 1. Stack up Equations 2. Get one variable as the EQUAL but OPPOSITE sign, so that when you add it they will cancel each other out. That way you can solve for 1 variable, then plug in that info, and solve for the second!
SECTION 2: EXPONENTS and ROOTS (The GMATS bread and butter testing strategy)
- Exponents and Roots are all about Abstraction and Large/Weird Numbers...just what the GMAT loves to Tests! How Do I Solve these Problems? Strategy to Use for Exponent Problems 1. Find Common Bases 2. Multiply/Factor 3. Find Patterns
SECTION 2: COMBINING LIKE TERMS/FACTOR (CRUCIAL)
- Factoring is ESSENTIAL when you need to make your math look like their math!
RP: Multiplying by 1 (3 Flashcard Set)
- Goal is to SIMPLIFY - See Multiple Denominators - TIME TO MULTIPLY BY A STRATEGIC 1
INEQUALTIES (STRATEGY: MAKE UP NUMBERS)
- INEQUALITES 1. The MOST IMPORTANT thing to remember is that the SIGN switches when you multiply/divide by a NEGATIVE NUMBER!!!
3. FIND PATTERNS
- Many numbers on the GMAT will be far to large to calculate! So use Pattern Recognition to solve the problems!
What is a good reminder to do when you "do the same to both sides??"
- Multiply both sides of the term by the number, and more specifically each term on EACH side. (notice picture) - Heres how to handle it: Put the term on the side in PARANTHESIS, before you multiple it. This will remind me that both terms get multiple. (in this case 21 goes into 4 and -x/3) -The GMAT will reward me for doing this.
TOOL 1: Multiply by 1
- On the GMAT, My GOAL is to ELIMINATE DENOMINATORS! - Do this by Multiplying by a STRATEGIC 1 to knockout the DENOMINATORS. - Dont forget: Goal is to SIMPLIFY so Keep it simple when multiplying by 1
4. ELIMINATE VARIABLES
- Remember, when problems involve MULTIPLE VARIABLES, my job is often to eliminate variables to simplify! -Use the Elimination Method and Substitution Method
Most Advanced EXPONENT Type Question will revolve around what? (MUST KNOW)
- The GMAT will raise a power, to another base with a power. THIS LOOKS CONFUSING. Just like in the previous problem. - Just Multiply it out and use the standard Exponent rules to Solve. THIS IS CRITICAL.
COMBINING INEQUALITIES - doing this will give me a competitive advantage! LEARN TO DO THIS!
- Use ELIMINATION METHOD - Make sure the signs are flipped the same
Algebra doesn't have to be Scary, but you should avoid whenever possible by?
- Using non-standard techniques, word problem tricks, etc. - Working Backwords (Backsolving), Picking Numbers, etc
1. FIND COMMON BASES
- on the GMAT you will almost always want to break down big COMPOSITE BASES down into their PRIME FACTOR BASES! - Once this is done, THEN you can employ the Exponent Rules we know so well! CRUCIAL!
EXPONENT Rules Review
-Know these Cold! - If I can't remember them on test day, just prove the rule to myself using small numbers.
Slide 2 - Solution
1.
3 Most Important QUADRATIC EQUATIONS (MEMORIZE)
1. DIFFERENCE OF SQUARES 2. PERFECT SQUARES
DOMINATE THE GMAT - ALGEBRA COURSE PART 2
1. Exponents 2. Roots and Radicals 3. Symplifying Algebraic Expressions 4. Examples 5. Strategies
Quadratic Formula - DONT NEED TO KNOW IT!
1. Instead just back solve for X to see if they work.
Strategy Summary: Algebra (CRUCIAL)
1. Multiplying by 1 2. FACTORING (make my math look like theirs) 3. Combining like Terms 4. Use the ANSWER CHOICES as a guide when choosing how to solve a problem. 5. Doing the Same To Both Sides (for equations) 6. Eliminate Variables. 7. When I see Exponent Problems with Addition/Subtraction, Always FACTOR. 8. Pattern Recognition - Parallel Problems (smaller problems with the same principles)
Dealing with Quadratics: The Practice Approach
1. Set equation to 0 2. Then unfoil (standard method) 3. Don't Forget Last Step! Set each term equal to zero to get either of the possible answers!
DOMINATE THE GMAT - ALGEBRA COURSE PART 1
1. Simultaneous Equations 2. Quadratic Equations 3. Inequalities 4. Examples 5. Misc Strategies and Considerations.
1. Simultaneous Equations: Two Methods for solving variables. (Also refer to Algebra Skillbuilder notes)
2. Elimination Method - Easy once you learn, and FASTER Steps: 1. Make the coefficient of either variable the same in both equations, but opposite signs 2. Place one equation above the other 3. Add the two equations to eliminate one variable 4. Solve for the remaining variable. 1. Substitution Method - Often easier, but takes LONGER Steps: 1. Solve for the variables in terms of the other variable 2. Plug the result into the second equation, creating one equation with one variable 3. Solve for that missing Variable. * Remember 2 equations and 2 variables means you can solve. For the most part. *Not all simultaneous equations are solvable if they equal each other. (see attached picture. One you solve using elimination method you will see that both equal 0) Tip: Don't make snap decisions on data sufficiency questions. Chase it down the rabbit hole to make sure you can solve. On difficult questions, its worth my time to prove to myself that it is not solvable.
VERITAS - ALGEBRA LESSON
ALEGRAB ON THE GMAT IS ALL ABOUT MAKING THE "INCONVENIENT MATH" CONVENIENT - Algebra is CORE to what the GMAT test. - Training my self to think like the testmaker. - Algebra Toolkit + Strategies = Solve Problems - Exponents and Roots
*Prime Base Rule to REMEMBER!
If we have Prime Bases^Variable = Prime Bases^Exponent, we can solve. The EXPONENTS MUST MATCH UP. Ex. We know that 3^a does = 3^5, or a=5 because 3 is a PRIME base!
Remember: What will Multi Variable Questions will ALWAYS have?
Multi Variable Questions will ALWAYS have a trap answer of the value of the other Variable! DONT FORGET TO DOUBLE CHECK IM ANSWERING THE RIGHT QUESTION
- Slide #2 from Previous Problem.
Remember: 3^5(2^5) = (2*3)^5 = 6^5 - The GMAT will try to HIDE THE ANSWER (Remember to Think like the testmaker)
STEP 1
STEP 1: MULTIPLY BY A STRATEGIC 1
STEP 2
STEP 2: LEVERAGE THE OTHER INFO GIVEN AND SOLVE.
LESSON OUTLINE
Section 1: Algebra Toolkit Section 2: Exponents and Roots Section 3: Quadratic Equations Section 4: Common Algebraic Equat. Section 5: Inequalities
RP: ELIMINATING VARIABLES (Word Problem Format)
Steps to Solve 1. Take Word Problem format and setup into my 2 Equations. 2. Solve for Variables using either method I prefer! - Elimination is fastest, but tougher to picture at first. 3. IT IS WAY FASTER TO REALIZE WHICH VARIABLE THE ANSWER IS ASKING FOR, AND TO SOLVE IMMEDIATELY FOR THAT ONE. (example refers to the First Life total) (refer to picture)
What are my EXPONENT STRATEGIES to solve?
Strategy to Use for Exponent Problems 1. Find Common Bases (Prime Numbers if possible) 2. Multiply/Factor 3. Find Patterns
ALGEBRA LESSON SUMMARY (REVIEW)
Takeaways 1. Most important Subject for Quant! 2. Inequalities: When you multiply or divide an inequality by a NEGATIVE, you MUST FLIP THE SIGN 3. Common Algebraic Terms: Difference of Squares and Perfect Squares!
STEP 1 - FINDING COMMON BASES
Takeaways 1. Notice that 5^4 matches the 5^2y on the other side, so y must = 2. - Then distribute the 2 throughout all Y's to solve.
FACTORING is a CRUCIAL skill on the GMAT. (With Review Problem)
Takeaways 1. Notice the Sucker Answer is 11! 2. Factor out the FACTORIALS - Don't do the math!
RP - MULTIPY/FACTOR! (2 Part Slide)
Takeaways 1. When you see Addition/Subtraction problems with Exponents ALWAYS FACTOR! - This will turn them into multiplication problem, and thus you have Exponent Rules to work with. 2. On difficult Algebra problems, when in doubt, GLANCE AT THE ANSWER choices. This will give me a good idea into what form I need my answer in.
EXPONENTS SUMMARY
Takeaways. 1. ALWAYS FIRST find Common Bases. 2. Exponent Problems are multiplication problems. Factor to get them into this Form. 3. Always Look for Patterns. And don't be afraid of Big Numbers because of this.
RP: FIND PATTERNS (Exponents and Units Digits. 2 Part Slide)
Takeaways: 1. DONT DO THE MATH! 2. Look for the Pattern. 3. Establish your pattern by using Small numbers. 4. Find my Anchor, which is the number with the LAST unique units digit. (this problems actor is to the 4th Power.)
RP: ADVANCED EXPONENTS #1 (must review!)
Takeaways: 1. Do a small, simple PARALLEL PROBLEM off to the side, to understand the principles. 2. Don't be thrown off by the Abstract/Large numbers. 3.
QUADRATIC EQUATION #1 - (SHORTCUT)
Takeaways: 1. I GOT THIS RIGHT! YES
RP: PERFECT SQUARES - SOLUTION
Takeaways: 1. I got this right the first time!! YAY!
RP: ADVANCE EXPONENTS #2
Takeaways: 1. I solved this using the non-algebraic method of Back Solving (Working Backwards. 2. I knew that I had to make the Left Side look just like the Right side and picked a few numbers! Worked out. 3. Veritas used Factoring and Pattern thinking.
Data Sufficiency REMINDER regarding QUADRATIC EQUATIONS -
Takeaways: 1. If the C term is a positive PERFECT SQAURE (example is 16). Then when you factor, you will more than likely will produce 2 roots that are the SAME...meaning you could find a sufficient answer WITH a quadratic Equation in problem (see below RP)
RP - FACTORING
Takeaways: 1. It is NOT 4^32. I thought it was at first, but the 4^8's aren't being multiplied to each other, they are being added, so that exponent rule doesn't hold true. 2. Factor out a 4^8 and get (1+1+1+1)= 4 -> so 4(4^8)= 4^1+8 = 4^9
RP: QUADRATIC EQUATION (Data Sufficiency)
Takeaways: 1. Normally, I have a DS question involving Quadratic Equations, the answer will be INSUFFICIENT because the Quadratic will produce 2 different roots...making it insufficient. 2. Unless the C term is a perfect square! Double check always.
RP: INEQUALITIES
Takeaways: 1. Plug in values for each variable that follows the given rules. (Picking Number Strategy) 2. I just lined up the rules give, with the Inequality sign pointing the same way on all. This made solving easy. Solution: (B) W > X > Z > Y
RP: QUADRATIC EQUATIONS
Takeaways: 1. Recognize the DIFFERENCE OF SQUARES equation in the question! Look for the squared term if I didn't see it. 2. Always manipulate the answer into the form the question is looking for (ex. (x-y)=50 ). Solution 2. Input what the Difference of Squares equation equals. 3. Then plug in the value for (x+y) = 2. Now I have 2(x-y)=100 and solve for (x-y) as 50 (D)! *dont forget to leave the (x-y) intact!
RP: ADVANCED ROOTS (Multiplying by 1) (2 Part Slide)
Takeaways: 1. Roots have identical properties to exponents. 2. When you have addition/subtraction in root problems look to FACTOR (just like EXPONENTS)! 3. Get rid of Denominators by multiplying by 1. 4. Look at the Answer choice to guide how you factor. 5. The bigger factor you can take out of each term, the faster you can solve.
RP: INEQUALITIES#1 (Solution)
Takeaways: 1. Statement 1 2. Statement 2 - IS A MAJOR HINT - Its obvious that this is INSUFFICIENT. When this happens, its more than likely a CLUE to solve! 3. When you're multiplying/dividing by a variable, thats when the tentmakers will catch you napping! 4. When you see an inequality, ALWAYS consider the possibility that its negative!
RP: Combining Inequalities #1 (Solution) (Data Sufficiency)
Takeaways: 1. Statement 1: Both statements obviously insufficient. 2. Statement 2: 3. Combine Inequalities to solve for X and Y 4. Manipulate the equation to make sure the signs point in the right direction AND one of the variables will eliminate itself.
RP: Manipulating Inequalities #1 (Problem) (Data Sufficiency)
Takeaways: 1. Statement 1: Don't forget you can manipulate the statements to work for me. 2. Statement 2: Obviously SUFFICIENT. They give me enough info to solve! (Regardless if the statement is TRUE OR NOT! Don't Forget what Sufficiency means) 3. Make the Question Look just like the QUESTION STEM.
SECTION 4 - COMMON ALGEBRAIC EQUATIONS
Takeaways: 1. There is huge power in knowing these BACKWARDS and FORWARDS! 2. These will SAVE ME HUGE AMOUNTS OF TIME! 3. The GMAT loves to test things that look hard, but by knowing these I can tackle them.
RP: DIFFERENCE OF SQUARES #1
Takeaways: 1. Use the Answer Choices to guide Answer form 2. Dont do the HUGE math! 3. Notice that the "1" in the problem can be a perfect square. Don't let it fool you!
RP: Perfect Squares (HARD) (2 Part Slide)
Takeaways: 1. Use the PERFECT SQUARE RULE! - makes this HARD problem easier!
5 General Rules for INEQUALITIES
Takeaways: 1. WHEN IN DOUBT....MAKE UP NUMBERS!
RP: QUADRATIC EQUATION #1 (Solve by solving for k)
Takeaways: 1. You can take the 4 they give you, plug into for x and solve for k, then put back into the quadratic equation to solve. OR you can notice the Shortcut. (See below)
RP: QUADRATIC EQUATIONS #2
Takeaways: 1. REMEMBER (x+y)^2 IS NOT EQUAL TO x^2 + y^2. This would leave out the middle term 2. It is equal to (x+y)(x+y). Solution: 1. (√7+√7)^2 = (2√7)^2 = (2^2)(√7^2) = (4)(7) = 28!
RP - LIKE TERMS
Takeaways: 1. Use the Answer Choices to find which common factor they are looking for (in this case you know its a √3)! 2. When numbers are large or scary, look to factor, and use the answer choices to guide which factor to go with.
RP: Finding Common Bases
Takeaways: Steps 1. Find Common Bases (Make sure they are down to their Prime Form if possible) 2. Multiply (Factor) 3. Find Pattern
What methods should I use to Eliminate Variables?
The SUBSTITUTION and ELIMINATION Method.
What are my Goals to solve Algebra Problems?
To Do One of the following - Simplify for an expression - Solve for a Variable - Make My Math LOOK Like the Math in the ANSWER CHOICE. (Use the Answer choices as a guide)
All the same rules apply to INEQUALITIES as with equations, EXCEPT...?
When you multiply or divide by a negative number, you flip the sign!
SECTION 2: QUADRATIC EQUATIONS
ax^2 + bx +c = 0 (Standard Form) - Need to reverse foil to get its two Roots
SECTION 3- QUADRATIC EQUATION SUMMARY
x^2 means Quadratic Equation Set to 0 and Factor If I can't figure out factors, BACKSOLVE from answer choices
