Question 121·Easy·Equivalent Expressions
Which expression is equivalent to ?
For expression-equivalence questions with parentheses, first apply the distributive property carefully to each group, treating any minus sign as part of the factor (for example, think of −2 instead of 2). After expanding, line up like terms (same variable and power) and combine them, paying close attention to signs. If you are unsure, you can quickly verify by plugging in an easy value (like or ) into the original expression and each choice to see which one always matches.
Hints
Start by removing parentheses
Look at each set of parentheses. What property lets you multiply the number outside the parentheses by each term inside to remove the parentheses?
Be careful with the negative sign on the 2
Think of the second part as multiplying by , not just 2. How does that affect both and inside the parentheses?
Combine like terms after distributing
Once you have expanded both products, you should have several terms. Which terms have in them, and which are just numbers? Combine each group separately.
Desmos Guide
Enter the original expression as a function
In Desmos, type y1 = 5(2x - 3) - 2(3x - 4). This graphs the original expression as a line in terms of (you can treat as the same kind of variable as ).
Enter each answer choice as its own function
On new lines, enter:
y2 = 4x - 7y3 = 16x - 23y4 = 4x + 7y5 = 22x - 7
Compare graphs or tables to find the match
Look at the graphs: the correct choice will have a line that lies exactly on top of the graph of y1 for all . Alternatively, use a table for each function (click the gear icon and select 'Table') and compare the -values for several -values (like 0, 1, and 2); the equivalent expression will always have the same -values as the original.
Step-by-step Explanation
Apply the distributive property to each set of parentheses
Use the distributive property: multiply the number outside the parentheses by each term inside.
- For : multiply 5 by and 5 by to get .
- For : treat the factor as and multiply by and by to get .
So the original expression becomes
Write the expression without parentheses
Now that you have distributed both factors, rewrite the expression as a sum of terms without parentheses:
You will next combine like terms (the terms with and the constant terms).
Combine like terms
Group the terms with together and the constant terms together:
So the simplified expression is
which is equivalent to the original expression.