Algebraic expressions and solving equations — ISEE Upper practice questions

What this topic is

Algebraic expressions and solving equations on the ISEE Upper Mathematics Achievement section covers simplifying and evaluating expressions, using identities that connect sums and products of two variables to other quantities, isolating a variable after distributing, and turning a short word problem into an equation.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Mid

If $x+y=7$ and $xy=10$, what is the value of $x^2+y^2$?

A24✗ Not this oneB29✓ CorrectC39✗ Not this oneD49✗ Not this one

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Answer: B — 29

Use the identity $(x+y)^2=x^2+2xy+y^2$. Since $x+y=7$ and $xy=10$, we have $49=x^2+20+y^2$, so $x^2+y^2=29$.

Question 2Easier

A bicycle rental costs a one-time fee of 13 dollars plus 9 dollars for each hour. Let $C$ be the total cost for $h$ hours. Which equation represents the total cost?

A$C=13h+9$✗ Not this oneB$C=9(h+13)$✗ Not this oneC$C=22h$✗ Not this oneD$C=9h+13$✓ Correct

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Answer: D — $C=9h+13$

The hourly charge is $9h$ dollars, and the one-time fee adds 13 dollars, so $C=9h+13$.

Question 3Easier

If $3(y+4)=27$, what is the value of $y$?

A5✓ CorrectB13✗ Not this oneC$\dfrac{23}{3}$✗ Not this oneD$-5$✗ Not this one

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Answer: A — 5

Dividing both sides by 3 gives $y+4=9$. Subtracting 4 gives $y=5$.

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