Question 163·Easy·Equivalent Expressions
Which of the following is equivalent to ?
For algebraic equivalence questions, first fully simplify the given expression by distributing and combining like terms. Then either match that simplified form to a choice or factor it to see if it can be written in the same factored form as one of the options; avoid expanding every choice unless necessary, since working from the original expression is usually faster and less error-prone.
Hints
Expand the parentheses first
Focus on the part . How can you use the distributive property to rewrite this without parentheses?
Combine like terms
After you expand , you will have an expression with terms and a constant. Which terms can you combine?
Look for a common factor
Once you have a simplified expression in the form , check if you can factor out a greatest common factor to match one of the answer choices.
Desmos Guide
Enter the original expression
Type y = 3(x - 4) + x into Desmos to graph the original expression as a line.
Enter each answer choice
On new lines, type y = 4(x - 3), y = 4(x + 3), y = 2(x - 6), and y = x - 12. You will now see several lines on the same coordinate plane.
Compare the graphs
Look for which answer choice’s line lies exactly on top of the line for y = 3(x - 4) + x for all x-values. That choice is the expression equivalent to the original.
Step-by-step Explanation
Distribute the 3 across the parentheses
Start by expanding using the distributive property:
So the whole expression becomes:
Combine like terms
Combine the terms and :
Now the expression is simplified to .
Factor the simplified expression
Now factor by taking out the greatest common factor, 4:
So is equivalent to .