If , what is the value of ?
When you see the exact same complicated expression repeated in an equation (like in two different terms), treat that expression as a single variable to simplify the problem. Rewrite the equation in terms of this new variable, combine like terms to isolate the variable, clear any fractions by multiplying both sides by the least common denominator, and solve. At the end, translate back to the original expression the variable represents; this is much faster and less error-prone than expanding everything in terms of .
Hints
Simplify the repeated part
The quantity appears in both fractions. Try replacing it with a single variable, like , to make the equation easier to see.
Form a simple linear equation in one variable
After you let , your equation should look like . How can you move the terms to one side and the constant numbers to the other?
Deal with the fraction coefficient
Once you get an equation of the form , think about what you can multiply both sides by to turn into just .
Desmos Guide
Enter each side of the equation as a line
In Desmos, let stand for . Type y1 = (2/5)x + 11 on one line and y2 = (3/5)x - 1 on another; these represent the left and right sides of the equation.
Use the intersection to read the value
Zoom or pan until you see where the two lines intersect, then tap the intersection point. The x-coordinate shown by Desmos is the value of , which is the same as the value of .
Step-by-step Explanation
Replace the repeated expression with a single variable
Notice that the expression appears on both sides of the equation.
Let .
Then the equation becomes
Collect like terms to isolate the -term
Start from
Subtract from both sides:
Now add 1 to both sides:
Clear the fraction and solve, then substitute back
To remove the fraction, multiply both sides of by 5:
We defined , so this means