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 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 . 🕵️♂️
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 all by itself on one side of the equal sign. How? By using opposite actions! If a number is added to , you subtract it. If a number is multiplied by , 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. 🚀
Practice Questions
5 practice questions for ISEE Middle Level
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- David's speed is , so Leo's speed is . After 45 minutes, David's distance is and Leo's distance is . Since they are 27 miles apart, the difference between their distances is 27: . By adding 27 to both sides and subtracting from both sides, this equation can be rewritten as .
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- Multiply each fraction by the least common denominator (or cross-multiply): . Expand the left side: . Subtract from both sides to get , and then add 10 to both sides to find .
| Hours | Total Cost |
|---|---|
| 1 | $85.00 |
| 2 | $130.00 |
| 4 | $220.00 |
| 5 | $265.00 |
What is the fixed house call fee?
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- First, find the hourly rate by calculating the difference in cost for 1 additional hour. From 1 hour to 2 hours, the cost increases from $85.00 to $130.00. , so the hourly rate is $45.00. To find the fixed house call fee, subtract the cost of 1 hour ($45.00) from the total cost of a 1-hour job ($85.00). .
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- First, combine the constants on the left side of the equation: . The equation becomes . To isolate , subtract from both sides to get . Finally, subtract 5 from both sides to find .
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- First, evaluate the square roots in the expression: and . Substitute these values back into the expression to get . Divide the outside coefficient 8 by the denominator 4 to get 2, resulting in the equivalent expression .
Tips & Strategies
- For Quantitative Comparison questions, don't just guess! Actually solve both mini-equations for and before comparing them. It only takes a few extra seconds.
- Plug your answer back in! If you find that , 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 , 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 and divide by , the answer is , not . 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 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 terms to one side by adding or subtracting. For example, in , subtract from both sides to get , 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.