Question 98·Medium·Linear Equations in One Variable
If , what is the value of ?
When an equation has the same expression (like ) on both sides, treat that whole expression as a single variable to avoid unnecessary distribution and reduce mistakes. Rewrite the equation in terms of this new variable, solve the resulting linear equation step by step (collect like terms, then isolate the variable), and then translate back to the original expression to pick the matching answer choice quickly.
Hints
Notice the repeated expression
Look at both sides of the equation. What expression inside parentheses appears on both sides?
Use a substitution to simplify
Consider letting a single letter, like , stand for the expression . How would the equation look then?
Solve the simpler linear equation
Once you have an equation like , collect like terms and isolate using addition/subtraction and then multiplication/division.
Desmos Guide
Graph both sides as functions of a new variable
In one line, type y = 7x - 4 and in another line type y = 3x + 20. Here, represents the expression from the original problem.
Find the intersection
Look for the point where the two lines intersect and tap that point. The x-coordinate of the intersection is the value of in the original equation.
Step-by-step Explanation
Treat the repeated expression as a single variable
Notice that appears on both sides of the equation:
Let . Then the equation becomes:
Now you only need to solve this simpler linear equation for .
Isolate the variable term
Start with:
Subtract from both sides to get all the terms on one side:
Next, add to both sides to move the constant term:
Now is almost isolated.
Solve for the expression and match the answer choice
From
divide both sides by :
Remember that , so
The value of is , which corresponds to answer choice C) 6.