[PSAT] 3-MONTH FINAL PREP

3-Month Final Prep

SATashkent

System of Equations

Real-life application of system of equations

Systems of equations solve problems with multiple conditions, like determining quantities or costs in real-world scenarios.

They find values for several variables that satisfy all conditions at once.

Multiple Choice

You buy 3 notebooks and 2 pens for 10 dollars, while 2 notebooks and 3 pens cost 11 dollars. How much is each item?

• Notebook: 2.6 dollars,
  Pen: 1.6 dollars

• Notebook: 1.6 dollars,
  Pen: 2.6 dollar

• Notebook: 3 dollars,
  Pen: 2 dollars

• Notebook: 1 dollar,
  Pen: 3 dollars

Multiple Choice

A gym charges 30 dollars per month plus 2 dollars per class, or 20 dollars per month plus 3 dollars per class. When are the costs equal?

• 5 classes
• 10 classes
• 15 classes
• 20 classes

How to solve with “Substitution”,”Addition and subtraction”

​Substitution: Isolate one variable in an equation and plug it into the other to solve.

Addition/Subtraction: Add or subtract equations to cancel a variable, then solve for the rest.

Multiple Choice

Solve:

x + 2y = 5

x - y = 2

• x = 1, y = 2
• x = 4, y = 0
• x = 2, y = 1
• x = 3, y = 1

Multiple Choice

Solve:

3x + 4y = 17

3x - 4y = 1

• x = 2, y = 3
• x = 3, y = 2
• x = 4, y = 1
• x = 1, y = 4

Being able to identify which way is suitable for the question

Use substitution when a variable is easy to isolate;

choose addition/subtraction when coefficients match or can cancel out easily.

Multiple Choice

For y = 3x - 1 and 6x - 2y = 4, which method works best and why?

• Elimination method works best.
• Graphing method is the most efficient.
• Substitution method works best.
• Matrix method is the simplest approach.

Multiple Choice

For 5x + 2y = 10 and 5x - 2y = 2, which method is easier and why?

• Substitution method is easier.
• Graphing method is easier.
• Matrix method is easier.
• Elimination method is easier.

Making the question easier by giving common denominator or
making the format the same

Convert equations to have matching coefficients or clear fractions with a common denominator to simplify solving.

Multiple Choice

Solve:
$$\left(\frac{1}{3}\right)x+\left(\frac{1}{2}\right)y=4\ \ and\ \ \ \left(\frac{1}{6}\right)x-\left(\frac{1}{4}\right)y=1$$

• (9, 2)
• (12, 1)
• (6, 4)
• (3, 8)

Multiple Choice

Solve:
0.2x + 0.5y = 1
0.4x - 0.5y = 2

• x = 5, y = 0
• x = 10, y = -1
• x = 0, y = 2
• x = 3, y = 1

Solving systems of linear equations word problems

Define variables for unknowns, write equations from the problem’s conditions, and solve them together.

Multiple Choice

A pet store has 8 cats and dogs with 26 legs total. How many of each animal are there?

• 4 cats and 4 dogs
• 3 cats and 5 dogs
• 2 cats and 6 dogs
• 1 cat and 7 dogs

Multiple Choice

A trip upstream takes 4 hours at 10 km, and downstream takes 2 hours. What’s the boat’s speed and current’s speed?

• Boat speed: 3.5 km/h; Current speed: 1.5 km/h

• Boat speed: 2 km/h;

  Current speed: 3 km/h

• Boat speed: 4 km/h;
  Current speed: 0.5 km/h

• Boat speed: 5 km/h;
  Current speed: 2 km/h

Identifying the easier ways of solving the system of linear equations

Spot simple coefficients or relationships to pick the fastest method, like substitution for isolated variables or addition for opposite coefficients.

Multiple Choice

Which algebraic method is the most computationally efficient for solving the following system?

0.75x - (1/3)y = 5

2x + (2/9)y = -10

• Substitution, because isolating 'x' or 'y' from either equation leads to simpler expressions.

• Addition, because a simple multiplication of one equation can create coefficients for 'y' that sum to zero.

• Addition, because a simple multiplication of one equation can create coefficients for 'x' that sum to zero.

• Substitution, because both equations are already in a form highly conducive to direct substitution without prior manipulation.

Multiple Choice

For

x - 3y = 5

-x + y = -3,

what’s the quickest way to solve and why?

• Substitution

• Subtraction

• Addition

• No solution

Exponents

All exponents rules mentioned in presentations

Exponent rules simplify expressions:

- Multiply like bases: x⁴ · x⁵ = x⁹

- Divide like bases: x⁸ ÷ x⁵ = x³

- Power to a power: (x²)³ = x⁶

- Distribute exponent: (xy)² = x²y²

Multiple Choice

Simplify: 3x² · 2x⁵

• 3x⁸
• 6x⁷
• 6x⁵
• 5x⁷

Multiple Choice

Simplify: ((2y³)²) ÷ (4y⁵)

• y
• 4y⁴
• y²
• 2y

Power 0 meaning

Any non-zero number to the power of 0 equals 1.

Multiple Choice

What is 7⁰ ?

• 1.5
• 7
• 1
• 0

Multiple Choice

Simplify: (4x⁰y³)²

• 16y⁶
• 4x⁰y⁶
• 4y⁶
• 8y³

Negative power meaning

A negative exponent means you need to take the number and flip it upside down (find its reciprocal). Then, you just use the positive version of that exponent.

Multiple Choice

Simplify: 5⁻²

• 5/2
• -25
• 1/5
• 1/25

Multiple Choice

Simplify: (a⁻³b²) ÷ (a²b⁻⁴)

• a⁵b²
• b⁴/a²
• a⁻⁵b⁶
• b⁶/a

Identifying the link between“ what’s given”and “what is asked”

Spot which exponent rule to apply from the equation.

Multiple Choice

If y³ · yᵏ = y⁷, what is k?

• 4
• 6
• 5
• 3

Multiple Choice

Given x⁹ ÷ x⁴ = xᵐ, find m.

• 7
• 6
• 4
• 5

Converting all bases into same number and solving with exponents

Rewrite numbers as powers of the same base to apply exponent rules.

Multiple Choice

Solve for x: 4ˣ = 2ˣ⁺³

• 2
• 5
• 3
• 4

Multiple Choice

Simplify: 8²ˣ · 2ˣ⁻²

• 2^(7ˣ - 2)
• 4^(3ˣ - 1)
• 2^(6ˣ - 2)
• 8^(2ˣ + 1)

Radicals

Radicals meaning

Radicals represent roots of numbers.

- Example: √x means a number that, when squared, gives x.

- Cube roots: ∛x is the number that cubed equals x.

Multiple Choice

Which of the following expressions is equal to ∛(8x⁶y³)?

• 3x²y
• 4xy
• x³y²
• 2x²y

Multiple Choice

Simplify the expression:

$$\ \ \ \frac{\left(\sqrt[]{50}+2\sqrt[]{18}\right)}{\sqrt[]{2}}$$

• 11
• 13
• 9
• 7

Proof of every single radicals concept(available on presentation)

Multiple Choice

Which radical identity is used to simplify √18 = √9 · √2?

• √(a·b) = √a · √b

• √(x²) = x

• √(a/b) = √a / √b

• √(a+b) = √a + √b

Multiple Choice

Simplify √(5x)² using radical identities.

• 5x
• (5x)²
• 5√x
• √(5x)

Converting exponents into radicals and radicals to exponents

Multiple Choice

Rewrite the following expression as a simplified radical expression, assuming all variables represent positive real numbers:

$$(16x^5y^{-\frac{2}{3}})^{\frac{3}{4}}​$$

• $$8x^3\sqrt[4]{x^3}\cdot y^2\sqrt[4]{x^3}$$

• $$\ \ \frac{8x^3\sqrt[4]{x^3}}{\sqrt[]{y}}$$

• $$\ \ \ \frac{8x^3\sqrt[4]{x^3}}{\sqrt[3]{y^2}}$$

• $$\ \ \ \frac{8x^3\sqrt[4]{x^3}}{y^2}$$

Multiple Choice

Convert ⁴√(x³) into exponent form.

• x^(4/3)
• x^(3/2)
• x^(1/4)
• x^(3/4)

Identifying the link between“ what’s given”and “what is asked”

Match the question to the correct radical/exponent rule.

Multiple Choice

Given ∛(x²) = 4, convert to exponent form and solve for x.

• ±8
• ±2
• ±16
• ±4

Multiple Choice

If √x · √y = √36, what rule applies? Find √(xy).

• 12
• 18
• 6
• 9

Identifying the whole part of radicals

Estimate the integer closest to the square root.

- Example: √65 ≈ 8.06, so whole part is 8.

Multiple Choice

Estimate the whole part of √50 ?

• 7
• 9
• 8
• 6

Multiple Choice

What is the whole part of 17⁰.⁵?

• 5
• 3
• 4
• 6

Rationalizing denominator

Multiply top and bottom by a term that removes the radical.

Multiple Choice

Rationalize the denominator: 2 / √5

• √5 / 2
• 2√5 / 5
• 5 / 2√5
• 2 / 5

Multiple Choice

Simplify by rationalizing: 3 / (√2 + 1)

• 3 / (√2 - 1)
• 3√2 + 3
• 3√2 - 3
• 3√2 + 1

Exponential equations

To solve radical equations, isolate the radical and square/cube both sides.

Multiple Choice

Solve: √(x + 4) = 3

• 1
• 9
• 7
• 5

Multiple Choice

Solve: ∛(2x − 1) = 2

• 2.0
• 5
• 3.5
• 4.5

2-Month revision

SATashkent

Expressions

Real-life applications: Model everyday scenarios
(e.g., costs, distances)

Simplify expressions: Combine like terms, expand, and reduce complexity

Shortcut multiplication formulas: Use identities like
(a + b)² = a² + 2ab + b²

Equivalent expressions: Recognize different forms of the same expression

Math vocabulary: Interpret "more than," "less than," etc., for accuracy

Word problems: Translate scenarios into mathematical expressions

Multiple Choice

A moving company charges a $50 base fee and $1.25 per kilometer. Write an expression for the total cost of traveling d kilometers and calculate it for 64 kilometers.

• 1.25+50d;

  130

• 64+1.25d;

  150

• 50+1.25d;

  100

• 50+1.25d;

  130

Multiple Choice

Simplify the expression: 6x − 3(2x − 5) + 4

• 12
• 18
• 19

• 15

Multiple Choice

Simplify (-2x + 3)² -9

• -4x² - 12x
• 8x² - 6x
• -2x² + 6x
• 4x² + 12x

Multiple Choice

Which is equivalent to 4(x + 2) - 3x?

• x + 8
• x + 6
• 4x + 2
• 3x + 8

Multiple Choice

A number is 4 more than twice another number. If the sum of both numbers is 22, what is the smaller number?

• 6
• 12
• 10
• 8

Multiple Choice

A garden's length is 3 meters more than its width w. Write an expression for its area.

• w(w + 3)

• w + 3
• w^2 + 3
• 3w

Linear Equations

Real-life applications: Model costs (e.g., trip expenses)

Solving equations: Isolate variables, balance sides

Make subject questions: Rearrange for a specific variable

Cross-multiplication: Solve proportions diagonally

Word problems: Turn scenarios into equations

Multiple Choice

You spend $40: $20 on Nutella, 10 sweeties at x each. Find x.

• 5
• 4
• 3
• 2

Multiple Choice

Solve 4x + 8 = 16.

• 2
• 3
• 1
• 4

Multiple Choice

A taxi costs $5 plus $0.50 per mile. Write the cost c for m miles.

• c = 5 - 0.50 * m
• c = 5 + 0.25 * m
• c = 10 + 0.50 * m
• c = 5 + 0.50 * m

Multiple Choice

If y = 3x + 5, find x in terms of y.

• (y - 3) / 5
• (y + 5) / 3
• (y - 5) / 3
• 3y + 5

Multiple Choice

Solve 2/x = 4/6.

• 6
• 2
• 3
• 1

Linear Inequalities

Real-life applications: Budgeting (e.g., max purchases within limits)

Solving inequalities: Isolate variable, flip sign with negatives

Inequality signs: >, <, ≥, ≤; strict vs. inclusive

Graphing solutions: Number lines; open/closed circles, shading

Inequality vocabulary: "At least," "no more than," etc.

Word problems: Translate scenarios into inequalities

Multiple Choice

Oxunboy has $100 for games at $14 each. How many can he buy?

• 7
• 10
• 3
• 5

Multiple Choice

Solve 4x + 7 > 19.

• x > 3
• x < 3
• x > 5
• x = 3

Multiple Choice

What does "at most 10" mean?

• It means a value that is greater than 10.
• It means a value that is exactly 10.
• It means a value that can be 10 or more.
• It means a value that is 10 or less.

Multiple Choice

Graph x < 3 on a number line.

• Graph x < 3 with a closed circle at 3 and shading to the left.
• Graph x < 3 with an open circle at 3 and no shading.
• Graph x < 3 with a closed circle at 3 and shading to the right.
• Graph x < 3 with an open circle at 3 and shading to the left.

Multiple Choice

"More than 5" translates to?

• x >= 5
• x < 5
• x = 5
• x > 5

Multiple Choice

A bus holds 45 people max, with 8 teachers. How many students?

• 37 students
• 25 students
• 40 students
• 30 students