# Author Archive | Michael Schwartz

## Manipulating Algebraic Equations and Expressions with Fractions: Systems of Equations

A student writes, Sometimes the answer explanation suggests that I clear fractions by multiplying an entire equation by a common denominator. Sometimes the answer explanation suggests that I manipulate just one side of an equation. How do I know which is better? This is a surprisingly subtle question. It turns out that the answer is […]

## Manipulating Algebraic Equations and Expressions with Fractions: A Word Problem

A student writes, Sometimes the answer explanation suggests that I clear fractions by multiplying an entire equation by a common denominator. Sometimes the answer explanation suggests that I manipulate just one side of an equation. How do I know which is better? This is a surprisingly subtle question. It turns out that the answer is […]

## Manipulating Algebraic Equations and Expressions with Fractions: Equations

A student writes, Sometimes the answer explanation suggests that I clear fractions by multiplying an entire equation by a common denominator. Sometimes the answer explanation suggests that I manipulate just one side of an equation. How do I know which is better? This is a surprisingly subtle question. It turns out that the answer is […]

## Manipulating Algebraic Equations and Expressions with Fractions: Another Example

A student writes, Sometimes the answer explanation suggests that I clear fractions by multiplying an entire equation by a common denominator. Sometimes the answer explanation suggests that I manipulate just one side of an equation. How do I know which is better? This is a surprisingly subtle question. It turns out that the answer is […]

## Manipulating Algebraic Equations and Expressions with Fractions: Expressions

A student writes, Sometimes the answer explanation suggests that I clear fractions by multiplying an entire equation by a common denominator. Sometimes the answer explanation suggests that I manipulate just one side of an equation. How do I know which is better? This is a surprisingly subtle question. It turns out that the answer is […]

## Manipulating Algebraic Equations and Expressions with Fractions: A Quick Quiz

A student writes, Sometimes the answer explanation suggests that I clear fractions by multiplying an entire equation by a common denominator. Sometimes the answer explanation suggests that I manipulate just one side of an equation. How do I know which is better? This is a surprisingly subtle question. It turns out that the answer is […]

## Tricky Data Sufficiency Questions: Explanations #7 and #8

Today, we’re going to review the last two questions from the Tricky Data Sufficiency Questions practice post. If you’re just tuning in, give the questions a shot, check your answers here, and then review the explanations to questions #1-2, #3-4, and #5-6. Ready? Let’s move along. Question #7 7. If p + q = 2r […]

## Tricky Data Sufficiency Questions: Explanations #5 and #6

Today, we present the explanations for questions #5-6 from our Tricky Data Sufficiency Questions challenge. Be sure to check out the explanations for questions #1-2, and questions #3-4 before moving on to today’s post. Let’s get started.   Question #5 5. What is the value of m+n? (1) mn = -8 (2) -2m = n […]

## Tricky Data Sufficiency Questions: Explanations #3 and #4

Today, we present the explanations to questions #3-4 from our Tricky Data Sufficiency Questions challenge post. You can find the explanations to questions #1-2 here. Let’s get started.   Question #3 3. If 2x = 2y – 3z, what is the value of z? (1) y = x + 2 (2) x = y – […]