SSAT Upper Level

Linear Expressions & Equations

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)

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Imagine you have a mystery box of donuts. You know the box plus 2 extra donuts equals 12 donuts total. How many are in the box? If you guessed 10, congratulations! You just did algebra! 🍩

In math, a linear equation is basically just a fun mystery puzzle. Instead of a physical box, mathematicians use letters like $x$ or $y$ to hide the secret number. The equal sign $=$ is like a perfectly balanced seesaw on a playground. Whatever is sitting on the left side of the seesaw must weigh exactly the same as whatever is sitting on the right side. ⚖️

On the SSAT, you get to be the ultimate math detective! Your main job is to get the mystery letter all by itself so you can figure out its true identity. To do this, you just use the exact opposite math of what you see. If the equation shows addition, you use subtraction to undo it! If you see multiplication, you use division! It's like walking backward to retrace your steps. Just remember the most important rule of the playground: Whatever you do to one side of the seesaw, you absolutely MUST do to the other side to keep it perfectly balanced. Let's grab our magnifying glasses and crack some secret codes! 🕵️‍♂️

Key Formula

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

a \cdot x = b

, you isolate

x

by dividing both sides by

a

x = \frac{b}{a}

Practice Questions

5 practice questions for SSAT Upper Level

Q1 Hard

A local bakery sells chocolate, vanilla, and strawberry cupcakes. On Saturday, they sold 4 times as many chocolate cupcakes as vanilla cupcakes, and they sold 3 times as many vanilla cupcakes as strawberry cupcakes. If the bakery sold a total of 320 cupcakes, how many chocolate cupcakes did they sell?

A 20

B 60

C 120

D 240

E 280

Show Solution

Let $s$ represent the number of strawberry cupcakes sold. Since the bakery sold 3 times as many vanilla cupcakes, the number of vanilla cupcakes is $3 s$ . They sold 4 times as many chocolate cupcakes as vanilla cupcakes, so the number of chocolate cupcakes is $4 (3 s) = 12 s$ . The total number of cupcakes sold is 320, which gives the equation: $s + 3 s + 12 s = 320$ . Combining like terms, we get $16 s = 320$ . Dividing by 16, we find $s = 20$ . The question asks for the number of chocolate cupcakes, which is $12 s$ . Substituting $s = 20$ into the expression gives $12 (20) = 240$ chocolate cupcakes.

Answer: D

Q2 Hard

What is the solution to the equation

3 (x + 4) - 2 x = 5 (x - 2)

- \frac{11}{2}

- 1

\frac{1}{2}

\frac{11}{2}

11

Show Solution

First, distribute the constants on both sides of the equation: $3 x + 12 - 2 x = 5 x - 10$ . Next, combine the like terms on the left side: $x + 12 = 5 x - 10$ . To isolate $x$ , subtract $x$ from both sides: $12 = 4 x - 10$ . Then, add 10 to both sides: $22 = 4 x$ . Finally, divide by 4 to solve for $x$ : $x = \frac{22}{4}$ , which simplifies to $\frac{11}{2}$ .

Answer: D

Q3 Hard

What is the value of

\frac{3 a ^{2} - 2 b}{c}

when

a = 4

b = - 3

, and

c = 6

A 7

B 8

C 9

D 10

E 12

Show Solution

Substitute the given values into the expression: $\frac{3 ( 4 ) ^{2} - 2 ( - 3 )}{6}$ . First, evaluate the exponent: $4^{2} = 16$ . The expression becomes $\frac{3 ( 16 ) - 2 ( - 3 )}{6}$ . Next, perform the multiplication in the numerator: $3 (16) = 48$ and $- 2 (- 3) = 6$ . The expression is now $\frac{48 + 6}{6}$ . Add the terms in the numerator: $48 + 6 = 54$ . Finally, divide by the denominator: $\frac{54}{6} = 9$ .

Answer: C

Q4 Hard

Chloe is saving money to buy a new bicycle. She saves $15 in her first week, $22 in her second week, and $29 in her third week. If she continues to increase her weekly savings by the same amount each week, in which week will Chloe first save more than $100 in a single week?

A The 12th week

B The 13th week

C The 14th week

D The 15th week

E The 16th week

Show Solution

Chloe's savings form an arithmetic sequence where the first term is 15 and the common difference is $22 - 15 = 7$ . The formula for the $n$ th term of an arithmetic sequence is $a_{n} = a_{1} + d (n - 1)$ . We want to find the smallest integer $n$ such that $15 + 7 (n - 1) > 100$ . Subtract 15 from both sides: $7 (n - 1) > 85$ . Divide by 7: $n - 1 > \frac{85}{7}$ , which is approximately $12.14$ . Add 1 to both sides: $n > 13.14$ . Since $n$ must be a whole number, the first week she saves more than $100 is the 14th week. (Checking the 13th week: $15 + 7 (12) = 15 + 84 = 99$ . Checking the 14th week: $15 + 7 (13) = 15 + 91 = 106$ .)

Answer: C

Q5 Hard

A mother is currently four times as old as her daughter. In 10 years, the mother will be twice as old as her daughter. How old is the daughter currently?

A 5

B 10

C 15

D 20

E 25

Show Solution

Let $d$ be the daughter's current age. The mother's current age is $4 d$ . In 10 years, the daughter's age will be $d + 10$ and the mother's age will be $4 d + 10$ . The problem states that in 10 years, the mother will be twice as old as the daughter, which gives the equation: $4 d + 10 = 2 (d + 10)$ . Distribute the 2 on the right side: $4 d + 10 = 2 d + 20$ . Subtract $2 d$ from both sides: $2 d + 10 = 20$ . Subtract 10 from both sides: $2 d = 10$ . Divide by 2: $d = 5$ . The daughter is currently 5 years old.

Answer: A

Tips & Strategies

Use the answers to cheat (legally)! If you're stuck on an equation, try plugging the multiple-choice answers back into the $x$ spot to see which one makes the equation perfectly balanced.
Write down every single step. Trying to juggle numbers in your head is like trying to carry 10 pizzas without a box—something will drop! Write your steps neatly on your scratch paper.

Common Mistakes

Watch out for doing the opposite operation on only ONE side! If you add $5$ to the left, you MUST add $5$ to the right, or your math seesaw will crash.
Don't forget to answer the actual question! Sometimes you'll find $x = 2$ , but the SSAT asks for $x + 10$ . Always read the final question one last time before bubbling!

Frequently Asked Questions

Do I need to be able to do this in my head?

Nope! In fact, doing algebra in your head is a trap. The SSAT gives you plenty of scratch paper, so always write out your steps to avoid silly mistakes.

What if I forget the opposite of multiplication or division?

Just think of them as best friends! Multiplication and division undo each other. Addition and subtraction undo each other.

Are linear equations on the Lower, Middle, or Upper level SSAT?

You'll see simple one-step equations on the Lower level, and more two-step equations on the Middle and Upper levels. They are a huge part of the test!

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