ISEE Upper Level

Linear Expressions & Equations

Q: What if there are variables on both sides of the equation?

No problem! First, move all the `x` terms to one side by adding or subtracting. For example, in `5x + 2 = 3x + 10`, subtract `3x` from both sides to get `2x + 2 = 10`, then solve as usual.

Writing, simplifying, and evaluating algebraic expressions, and solving single-variable linear equations — excludes systems (see systems-of-equations), inequalities (see inequalities), and polynomial/radical expressions (see polynomials-quadratics, radical-rational-expressions)

Generate Unlimited Practice Questions

Start Free Practice

Learn This Topic

Imagine you are a secret agent trying to crack a safe. In math, solving equations is just like being a detective! You are looking for a mystery number, usually hiding behind a letter like $x$ . 🕵️‍♂️

Think of an equation like a perfectly balanced seesaw at the playground. The equal sign $=$ is the exact middle of the seesaw. Whatever you do to one side, you absolutely must do to the other side to keep it from crashing down. If you take away 5 slices of pizza from the left side, you have to take away 5 slices of pizza from the right side! 🍕⚖️

On the ISEE, you will see questions called "Linear Equations." That is just a fancy math term for finding that missing number. Your mission is to get $x$ all by itself on one side of the equal sign. How? By using opposite actions! If a number is added to $x$ , you subtract it. If a number is multiplied by $x$ , you divide it. You are basically peeling away the extra numbers layer by layer, just like peeling an onion (but without the tears!). 🧅

Whether you are taking the Mathematics Achievement section or looking at Column A and Column B in the Quantitative Reasoning section, just remember your golden rule: Keep the seesaw balanced! You've got this! Let's solve some mysteries. 🚀

Key Formula

The Golden Rule of Equations: Whatever you do to one side, do to the other! To get

x

alone in

a \cdot x + b = c

, first subtract

b

, then divide by

a

x = \frac{c - b}{a}

Practice Questions

4 practice questions for ISEE Upper Level

Q1 Hard

5 p - 2 q = 8

, then which expression is equal to

q

\frac{5 p - 8}{2}

\frac{5 p + 8}{2}

\frac{- 5 p - 8}{2}

\frac{8 - 5 p}{2}

Show Solution

To solve the equation for $q$ , first isolate the term containing $q$ . Subtract $5 p$ from both sides to get $- 2 q = 8 - 5 p$ . Next, divide both sides of the equation by $- 2$ to get $q = \frac{8 - 5 p}{- 2}$ . To simplify this fraction and remove the negative from the denominator, multiply the numerator and the denominator by $- 1$ , which yields $q = \frac{5 p - 8}{2}$ .

Answer: A

Q2 Hard

If the sum of all even integers from 2 to 50, inclusive, is

S

, then which expression represents the sum of all even integers from 2 to 46, inclusive?

S - 98

S - 96

S + 98

S - 50

Show Solution

The original sum $S$ represents the addition of all even numbers from 2 up to 50 (i.e., $2 + 4 + ... + 46 + 48 + 50$ ). The new sequence only asks for the sum of even integers from 2 to 46. This means the new sum is exactly the same as $S$ , but missing the last two terms: 48 and 50. Therefore, you must subtract 48 and 50 from $S$ . Since $48 + 50 = 98$ , the new sum is represented by the expression $S - 98$ .

Answer: A

Q3 Hard

The sum of 4 consecutive integers is 130.

Column A

The least of the 4 consecutive integers

Column B

A The quantity in Column A is greater.

B The quantity in Column B is greater.

C The two quantities are equal.

D The relationship cannot be determined from the information given.

Show Solution

Let the 4 consecutive integers be represented as $x$ , $x + 1$ , $x + 2$ , and $x + 3$ . Their sum is $x + (x + 1) + (x + 2) + (x + 3) = 130$ , which simplifies to $4 x + 6 = 130$ . Subtracting 6 from both sides gives $4 x = 124$ . Dividing by 4 gives $x = 31$ . The four integers are 31, 32, 33, and 34. The least of these integers is 31 (Column A). Since 31 is less than 32 (Column B), the quantity in Column B is greater.

Answer: B

Q4 Hard

At a movie theater, child tickets cost $5 and adult tickets cost $8. If a group of 10 people paid $62 for tickets, how many child tickets did they buy?

A 4

B 5

C 6

D 8

Show Solution

Let $c$ represent the number of child tickets and $a$ represent the number of adult tickets. We are given two equations: $c + a = 10$ (total people) and $5 c + 8 a = 62$ (total cost). From the first equation, we can express $a$ in terms of $c$ : $a = 10 - c$ . Substitute this into the cost equation: $5 c + 8 (10 - c) = 62$ . Distribute the 8 to get $5 c + 80 - 8 c = 62$ . Combine like terms to get $- 3 c + 80 = 62$ . Subtract 80 from both sides: $- 3 c = - 18$ . Divide by -3 to find $c = 6$ . The group bought 6 child tickets.

Answer: C

Tips & Strategies

For Quantitative Comparison questions, don't just guess! Actually solve both mini-equations for $x$ and $y$ before comparing them. It only takes a few extra seconds.
Plug your answer back in! If you find that $x = 5$ , put 5 back into the original equation to see if it makes sense. It's a built-in way to check your work on the ISEE Mathematics Achievement section!
Remember PEMDAS in reverse! When solving for $x$ , undo addition and subtraction first, then undo multiplication and division. Peel the onion from the outside in!

Common Mistakes

Watch out for negative signs! If you have $- 2 x = 8$ and divide by $- 2$ , the answer is $- 4$ , not $4$ . Dropping a negative sign is the most common trap on the ISEE.
Don't forget the Golden Rule: If you add or subtract a number on the left side, you MUST do it to the right side too. Don't let your seesaw crash!

Frequently Asked Questions

What if I get stuck and can't solve the equation?

Since the ISEE is multiple choice, you can use a trick called 'plugging in'! Just take the answer choices (A, B, C, D) and plug them into the equation for $x$ to see which one works perfectly.

Is there a penalty if I guess wrong on the ISEE?

Nope! The ISEE does not have a guessing penalty. If you are running out of time or completely stuck, make sure to pick an answer for every single question.

What if there are variables on both sides of the equation?

No problem! First, move all the $x$ terms to one side by adding or subtracting. For example, in $5 x + 2 = 3 x + 10$ , subtract $3 x$ from both sides to get $2 x + 2 = 10$ , then solve as usual.

What's the difference between Quantitative Reasoning and Mathematics Achievement?

Quantitative Reasoning asks you to think logically, like comparing Column A and Column B. Mathematics Achievement is more like the math tests you take in school, where you calculate a specific answer to a problem.

Generate Unlimited Practice Questions

Learn This Topic

Practice Questions

Tips & Strategies

Common Mistakes

Frequently Asked Questions

Related Topics

More Algebra

Other Levels

Browse

Generate Unlimited Practice Questions