How many solutions does the given equation have?
For “how many solutions” questions with linear equations, first simplify both sides (distribute, combine like terms, or divide common factors), then bring all variable terms to one side and constants to the other. If you get a specific value, there is exactly one solution; if the variables cancel and you get a true statement like , there are infinitely many solutions; if the variables cancel and you get a false statement like , there are zero solutions. This approach avoids extra computation and quickly classifies the equation.
Hints
Clear the common factor
Look at . What simple operation can you do to both sides to get rid of the 5?
Compare the simplified sides
After you remove the 5, you should get an equation with on both sides. What happens if you subtract from both sides?
Check if the final statement can be true
When the variable disappears, you are left with a statement involving only numbers. Ask yourself: can that numerical statement ever be true? What does that tell you about the number of solutions?
Desmos Guide
Enter each side as a separate function
In Desmos, type y = 5(x - 3) on one line and y = 5(x + 2) on another line to graph both sides of the equation as separate lines.
Use the graph to count solutions
Look for intersection points of the two lines. Each intersection corresponds to a solution of the equation. The number of intersection points tells you how many solutions the equation has.
Step-by-step Explanation
Simplify the equation
Start with the given equation:
Since both sides are multiplied by 5 (and 5 is not 0), divide both sides by 5 to simplify:
Eliminate the variable term
Now remove from both sides by subtracting from each side of the equation:
This simplifies to:
Interpret the result and answer the question
The equation is never true, no matter what is. That means there is no value of that makes the original equation true.
So, the equation has zero solutions, and the correct answer is D) Zero.