What value of satisfies the equation
For linear equations with several fractions, the fastest and safest approach on the SAT is to clear the denominators immediately by multiplying every term by the least common multiple of the denominators. Once you have an equation with only integers, distribute carefully—especially across subtractions—then combine like terms to reduce it to a simple form like number = coefficient · x + number. Finally, use inverse operations (usually add/subtract, then divide) to isolate , and quickly double-check any addition or subtraction with negatives, since tiny arithmetic slips are the most common source of wrong answers.
Hints
Get rid of the fractions first
Dealing with three different denominators is messy. What single number can you multiply every term by to clear the denominators 6, 3, and 2 all at once?
Be careful with distribution and subtraction
After you clear the denominators, you will have an expression like on the left. Make sure the multiplies both and .
Combine like terms, then isolate
Once you distribute, combine all the terms and constants on each side to get a simple equation with a number on one side and an expression like on the other. Then use inverse operations (add, subtract, divide) to solve for .
Check your arithmetic with negatives
When you combine the constants on the left, you are adding two negative numbers. Be careful with signs so that you don't make a small arithmetic mistake that changes the final value of .
Desmos Guide
Enter the two sides of the equation as separate functions
In Desmos, type y = (4x - 7)/6 - (2x + 5)/3 on one line and y = (x - 11)/2 on the next line to graph the left-hand side and the right-hand side as two separate lines.
Find the intersection point
Use your mouse or finger to click on the point where the two lines intersect. The x-coordinate of this intersection is the solution to the equation (the value of that makes both sides equal).
Alternative: Use a single expression
You can also type y = (4x - 7)/6 - (2x + 5)/3 - (x - 11)/2. Then look for the x-intercept (where the graph crosses the x-axis, i.e., where ). That x-value is the solution.
Step-by-step Explanation
Clear the denominators
The denominators are 6, 3, and 2. Their least common multiple is 6.
Multiply every term in the equation by 6:
This simplifies the fractions:
Distribute and simplify each side
Distribute the numbers outside the parentheses:
- On the left: .
- On the right: .
So the equation becomes:
Combine like terms on the left:
Solve the linear equation for
Start from
Add 33 to both sides:
Now divide both sides by 3:
So the value of that satisfies the equation is .