A guide to isolating quantities on the digital SAT
What are isolating quantities problems?
Many real-world scenarios can be described using equations and formulas with multiple variables. For example, the formula for the area, , of a rectangle with length and width is . In this form, area () is isolated: it is alone on one side of the equation.
If we want to find the equivalent equation, but with isolated, we can divide both sides of the equation by , giving us .
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How do I isolate quantities?
Manipulating formulas: temperature
Like solving equations, but with more variables
Isolating quantities is similar to solving equations. However, instead of ending up with a variable equal to a constant, we end up with a variable equal to an expression containing other variables. In fact, if you've rewritten a linear equation like in slope-intercept form, then you've already isolated by rearranging a linear equation!
The most important thing to remember is that the rearranged equation will remain equivalent to the original equation only if we always treat both sides equally: whenever we do something to one side, we must do the exact same thing to the other side.
To isolate a quantity in an equation or formula:
- Write down the original equation. If needed, translate the word problem or given context into an equation.
- Perform the same operation on both sides of the equation to begin isolating the desired quantity.
- Repeat Step 2 until the desired quantity is isolated.
Let's look at some examples!
A physics student uses the formula to calculate the kinetic energy in joules, , of an object with a mass of kilograms traveling at a speed of meters per second. What is in terms of and ?
The distance traveled by an object moving at constant velocity is found by multiplying the velocity of the subject by time . Write an equation that gives the time in terms of and .
try: identify the steps to isolate a variable
The equation above shows the volume of a right cylinder with radius and height .
To rewrite the equation to isolate , we must first isolate . To do so, we
both sides of the equation by .
Next, to isolate , we
both sides of the equation.
Which of the following correctly expresses ?
practice: isolate a quantity in one step
Giselle uses the formula to calculate , the simple interest in dollars, she collects from a loan of dollars at an interest rate of per month over a period of months. Which of the following expresses in terms of the other variables?
practice: write an equation, then isolate a quantity
The perimeter of a rectangle is found by multiplying the sum of the rectangle's length and width by . Which of the following equations gives the width in terms of and ?
practice: isolate a quantity with exponents
The equation above shows the , the gravitational force between two objects, in terms of , the gravitational constant, and , the masses of the two objects, and , the distance between the two objects. Which of the following correctly expresses in terms of , , , and ?