Equations

Question 1

What should you do to prepare for SAT equation problems?

Answer 1

Clean off the rust.

Get back to a place where you can recognize the inverse operations quickly, follow the correct sequence of steps, and handle the toughest operations, like equations with negative fractional exponents.

Question 2

What do all equations express as a number sentence?

2x + 7 = 15

Answer 2

Equations express number forms as the same value.

It’s like presenting a scale that’s balanced.

The equal sign, greater than, and less than (=, >, < ) could all be thought of as positions of the scale.

Question 3

How do you use the concept of equality in the equation to find the values of unknowns?

Answer 3

You use the concept to look for ways to do math operations to both sides of the equation until you see the value of the unknown.

You will decide to add, subtract, multiply, divide, expand (using exponents) or root (using radicals) in order to reveal the unknown value.

Question 4

What do you need to add to both sides of this equation to find the unknown?

x - 3 = 10

Answer 4

Add 3 to both sides.

x - 3 +3 = 10 +3

so, x = 13

Question 5

How should you decide which operations should be done to both sides of an equation in order to solve?

Answer 5

You look for inverse operations, which will isolate the unknown.

If you see “6x”, you think the opposite of 6 times the unknown. So, divide by 6.

If you see “u +13”, you think the opposite of adding 13. So, subtract 13.

If you see y under a square root, you think the opposite of rooting. So, square the values on both sides.

Question 6

What do you do if you have more than one thing affecting the unknown?

(x - 4)^{<span><span>2</span></span>} / 3 = 4

Answer 6

Do them step by step one at a time working outside in.

(x - 4)^{<span>2</span>} / 3 = 3, (x3)

(x - 4)^{<span>2</span>} = 9, (root)

• x* - 4 = 3, or x - 4 = -3, (+4)
• x* = 7, or x = 1

Question 7

What does it mean “to isolate the variable” in an equation?

Answer 7

Isolating the variable is the process which results in an equation with an unknown on one side and a value on the other. It is good process for conditions where the unknown is expressed in an identical exponent.

To isolate the variable, use inverse operations.

Example:

12 - 14x = 26 OR 7 + x³ = -20

• 14x = 26 - 12 / x³ = -20 - 7
• 14x = 14 / x³ = -27
• x* = -1 / x = -3

Question 8

Consider this problem:

If 4x + 3 = 9, then what is 4x + 10 =

How can using a principle like “isolating the variable” lead to a “time trap”?

Answer 8

Using “isolating the variable” in a step-by-step manner wastes time.

This problem presents three solution paths: solve for x and substitute, solve for “4x” and substitute, and observe to use “rule of algebra”.

solve for x:

4x + 3 = 9, (-3 both sides)

4x = 6, (/4 both sides)

x = 6/4 reduce to 3/2

substitute into 4(3/2) + 10 cancel twos, becomes 6 + 10

solve for 4x: very similar to above, but at “4x = 6” substitute for “4x” in the phrase “4x + 10” and directly to 16.

Observation using rule of alegebra: What changed from the given statement to the incomplete number sentence. “+3” became “+10”. The difference is “+7”, so by the “rule of algebra” since 7 was added to one side of the scale, to keep it equal, add 7 to 9.

The difference in time is only about 6 seconds, but if you do every problem six seconds longer, for every ten problems you have one less minute.

Question 9

How do you solve an equation with one variable?

6x - 8 = -3x + 10

Answer 9

Isolate the variable by using inverse operations

6x - 8 = -3x + 10

Get all the x’s on one side by adding 3x to both sides:

9x - 8 = 10.

Then, add 8 to both sides:

9x = 18.

Then, divide both sides by 9:

x = 2

Question 10

What are the inverse operations that you will use in answering algebraic process questions?

Answer 10

“Inverse operations” means basically “do the opposite to isolate the unknown.”

You see, “+7”, you do -7

you see, “5n” you do /5

you see “p²”, you have to root

when you see square root, you must square.

Question 11

How would you use inverse operations to solve the following equations?

z + 15 = 32

5x = 65

Answer 11

z + 15 = 32

15 is added to z. To find z, subtract 15 from 32.

5x = 65

5 and x are multiplied. To find x, divide 65 by 5.

x = 13

Question 12

What should you Un-learn from your math classes in order to handle the equation questions more efficiently?

Answer 12

You should unlearn a process where you “show all your work”.

American math classes make you show your work primarily to ward off cheating on homework, but also to allow the teacher to offer partial credit.

You should edit the process to eliminate time counsuming scribbling, but not so much that you risk errors.

If you’ve always shown your work, it may be a hard habit to change, but if you don’t, you effectively have 1-2 minutes less for the test than someone who has an edited process.

Question 13

How would you edit your process on an equation like this?

• If x is not equal to 3, and*
• x* / (x - 3) = * 2, then what is the value of * x?

Answer 13

Never recopy the equation. The first step is multiply 2 by (x - 3).

write, x = 2x - 6

6 goes left, -x goes right

grid in the answer “6”

Mental math saves precious time.

Question 14

What is the best way to solve this SAT math problem?

If 2x + 4 = 3/4, then what does 8x + 16 equal?

Answer 14

Use the “rule of algebra”.

Observe the change from the left side of the equation “2x + 4” to the phrase “8x + 16”. The left side was multiplied by 4, so multiply 3/4 by 4. The fours cancel. Look for the answer “3” among the distractors.

Solving for x takes longer.

Question 15

What is the simplest equation trap within SAT math?

If 2x² - 30 =* * 6x² - 14…

Answer 15

The “expectation of solving for the unknown” trap

The rest of the problem reads “…what is the value of x²?”.

Since many students are conditioned that when they see an equation, they solve for the unknown reflexively, they often don’t look at the question within the problem.

This is called an “attention to detail” error. It produces a confident, but wrong, answer.

Question 16

How do you improve your speed at doing problems without exposing yourself to avoidable errors?

Answer 16

Think slowly and deliberately as you examine the given and the question that you need to answer. Go quickly through computation and edit step-by-step processes.

Question 17

What kinds of operations in computation should you avoid editing process too much?

Answer 17

Distributing negatives and dealing with absolute values and inequalities you should probably do carefully and step-by-step. Many common wrong answers are based in errors within these processes.

Question 18

How should you prepare to use factoring on SAT math?

Answer 18

Since factoring takes time, you should do as little pure factoring as possible.

Certain problems involving Least Common Multiple will use factor tree processes. However, the SAT test uses factoring as a time management advantage. Most often, the factoring you have to do on the test is direct eye recognition. Such as…

x² - 9 factors to (x + 3)(x - 3)

You probably learned “the difference of two squares” in math, but if you need a refresher, look in the deck called “Time Savers”.

Question 19

What acronym helps you remember how to multiply two binomials?

(a + b)(c + d)

Answer 19

To multiply two binomials, use the FOIL method.

• (a + b)(c + d) =*
• ac + ad + bc + bd*

FOIL is an acronym for:

F - product of the FIRST termsO - product of the OUTERMOST termsI - product of the INNERMOST termsL - product of the LAST terms

Question 21

What’s the trap in this presentation of multiplying binomials?

(x - 5)² = x² + 35

Answer 21

The trap in this problem is more a “common error”.

The right hand side of the equation has two values and it creates a suggestion that multiplying the left will result in two values as well.

Proper FOILing leads to x² - 10x + 25, x = -1

Question 22

How does observation of an equation open a path that is much more efficient on a question like this?

If 7x - 5 = -1, then what is 7x + 19?

Answer 22

Observation shows us an equation with a partial equation as a question.

Reflex based in classroom math should be ignored. You should just observe that the left side of the equation changed and notice that if you add “24” to the “-5” you can see that would make the “19” in the partial equation. So, you’d add “24” to the “-1” on the right side to make 23 the answer. That ony takes 3 or 4 seconds.

Question 23

What is a “linear equation”?

Answer 23

A linear equation has two variables that express a relation of equality that can be represented as a line on a plane.

For example: 2x - 3y = 15. As the x value changes the y value changes.

Question 24

What is the slope intercept form for linear equations?

Answer 24

y = mx + b

Where “m” represents the slope presented as a fraction or a whole number, and “b” represents the value of “y” when “x” is zero.

Question 28

What is a system of equations?

Answer 28

A system of equations involving

Question 33

Find the greatest common factor (GCF) of the expressions below.

ca + cb = ?

ca - cb = ?

Answer 33

Factor out c as shown below:

ca + cb = c(a + b)

ca - cb = c(a - b)

Question 34

How do you factor out the following expression?

a² - b²

Answer 34

Difference of squares formula:

a² - b² = (a + b)(a - b)

Example:

36 - 4x² = 6² - 2²x² = (6 + 2x) (6 - 2x)

Question 35

How do you factor perfect square trinomials?

• a² + 2ab + b²*
• a² - 2ab + b²*

Answer 35

Factor perfect square trinomials as shown below:

a² + 2ab + b² = (a + b)(a + b)

= (a + b)²

a² - 2ab + b² = (a - b)( a - b)

= (a - b)²

Question 36

How do you use factoring method to solve a quadratic equation?

ax² + bx + c = 0

Answer 36

• If possible, factor the left side of the equation
• Make each factor equal to zero and solve two linear equations
• Example:*
• x² + 15 = 8x* ⇒ x² - 8x + 15 = 0

(x - 3)(x - 5) = 0

• x* - 3 = 0 or x - 5 = 0
• x* = 3 or x = 5