ISEE Upper Level

Systems of Equations

Solving for two or more unknowns simultaneously using substitution, elimination, or graphing

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Imagine you are a math detective on a top-secret mission! 🕵️‍♂️ You need to figure out the cost of a burger and fries, but the restaurant menu only gives you combo prices. The menu says: 1 Burger + 1 Fries = $10. It also says: 1 Burger - 1 Fries = $4. How much is each item?

Welcome to the world of Systems of Equations! A 'system' is just a fancy way of saying 'two math clues working together to solve a mystery.' If you only have one clue (like $x + y = 10$ ), there are too many possibilities. The burger could be $9 and the fries $1, or the burger could be $6 and the fries $4. But when you get a second clue, there is only one perfect answer that works for both! 🍔🍟

On the ISEE, you will see questions where you have to find the value of two mystery letters, usually $x$ and $y$ . Don't panic! You have two superpowers to solve these: 'Substitution' (swapping things out) and 'Elimination' (adding clues together to make a letter disappear). And remember, the ISEE is a multiple-choice test. If you ever get stuck, you can just plug the answer choices into the equations to see which one makes both clues true. Let's crack the case! 🔍

Key Formula

The Elimination Rule: Stack your equations and add them together to eliminate a variable! If

x + y = 10

and

x - y = 4

, adding them gives

2 x = 14

, so

x = \frac{14}{2} = 7

Practice Questions

4 practice questions for ISEE Upper Level

Q1 Hard

A theater sells adult tickets for $10 and child tickets for $5. A family buys 8 tickets for a total of $55.

Column A

The number of child tickets bought

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 $a$ represent the number of adult tickets and $c$ represent the number of child tickets. We can write a system of two equations: $a + c = 8$ (total tickets) and $10 a + 5 c = 55$ (total cost). From the first equation, we know $a = 8 - c$ . Substituting this into the second equation gives $10 (8 - c) + 5 c = 55$ . Distributing the 10 gives $80 - 10 c + 5 c = 55$ , which simplifies to $80 - 5 c = 55$ . Subtracting 80 from both sides gives $- 5 c = - 25$ , and dividing by -5 yields $c = \frac{- 25}{- 5} = 5$ . The family bought 5 child tickets. Column A is 5 and Column B is 4. Since 5 is greater than 4, the quantity in Column A is greater.

Answer: A

Q2 Hard

A baker makes a total of 45 blueberry and chocolate chip muffins. There are 4 times as many blueberry muffins as chocolate chip muffins. How many blueberry muffins did the baker make?

A 9

B 27

C 36

D 40

Show Solution

Let $b$ be the number of blueberry muffins and $c$ be the number of chocolate chip muffins. The problem gives the system of equations: $b + c = 45$ and $b = 4 c$ . Substitute $4 c$ for $b$ in the first equation: $4 c + c = 45$ , which simplifies to $5 c = 45$ . Dividing both sides by 5 gives $c = 9$ . To find the number of blueberry muffins, substitute $c = 9$ back into the second equation: $b = 4 (9) = 36$ . The baker made 36 blueberry muffins.

Answer: C

Q3 Hard

2 x + y = 11

and

x - 2 y = 3

, what is the value of

x + y

A 4

B 6

C 8

D 14

Show Solution

Solve the system of equations. From the second equation, isolate $x$ : $x = 2 y + 3$ . Substitute this expression for $x$ into the first equation: $2 (2 y + 3) + y = 11$ . Distribute the 2 to get $4 y + 6 + y = 11$ , which simplifies to $5 y + 6 = 11$ . Subtract 6 from both sides to get $5 y = 5$ , which means $y = 1$ . Now substitute $y = 1$ back into the equation for $x$ : $x = 2 (1) + 3 = 5$ . The question asks for the value of $x + y$ , which is $5 + 1 = 6$ .

Answer: B

Q4 Hard

5 a + 10 = ab + 2 b

and

a \neq = - 2

, what is the value of

b

A 2

B 5

C 10

D 12

Show Solution

Factor both sides of the equation. On the left side, factor out the common term of 5: $5 (a + 2)$ . On the right side, factor out the common term of $b$ : $b (a + 2)$ . The equation becomes $5 (a + 2) = b (a + 2)$ . Since the problem states that $a \neq = - 2$ , we know that $a + 2 \neq = 0$ . Therefore, we can divide both sides of the equation by $(a + 2)$ to get $5 = b$ .

Answer: B

Tips & Strategies

Plug in the answer choices! If you're stuck, just try the choices (A, B, C, D) in BOTH equations. If a choice works for both clues, you found the winner! 🏆
Look for opposites. If you see a $+ y$ in one equation and a $- y$ in the other, stack them up and add! They will cancel out like magic.

Common Mistakes

Watch out for finding the wrong variable! If the question asks for $y$ , don't accidentally stop after you find $x$ . Always double-check what the question is asking for before bubbling your answer. 🛑
Don't forget to do the same thing to both sides! If you multiply one side of an equation by 2, you MUST multiply the other side by 2 as well.

Frequently Asked Questions

What if I forget how to do elimination or substitution during the ISEE?

Don't panic! The ISEE is multiple-choice. You can always use the 'guess and check' method by plugging the answer choices into the equations. There is no penalty for guessing, so pick your favorite letter if you're totally stuck!

Will I have to graph these on the test?

Usually not! While graphing is a way to solve systems of equations in school, the ISEE focuses more on algebra and word problems. Stick to substitution, elimination, or plugging in the choices!

What is a Quantitative Comparison question?

It's a special ISEE question type! You get Column A and Column B. You have to decide if A is bigger, B is bigger, they are equal, or if it's impossible to tell. Solve the math system first, then compare your answer to the other column!

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