Select the correct answer.
How many solutions for $x$ does the following equation have?
$\frac{4}{5}(20 x+5)=16 x-2$
A. -2
B. infinite
C. 0
D. 1
Answer (2)
Distribute 5 4 on the left side: 5 4 ( 20 x + 5 ) = 16 x + 4 .
Rewrite the equation: 16 x + 4 = 16 x − 2 .
Simplify the equation: 4 = − 2 .
Since 4 = − 2 is a contradiction, the equation has no solutions: 0 .
Explanation
Analyze the problem We are given the equation 5 4 ( 20 x + 5 ) = 16 x − 2 and asked to find the number of solutions for x . To do this, we will simplify the equation and see what we get.
Distribute First, distribute the 5 4 on the left side of the equation: 5 4 ( 20 x + 5 ) = 5 4 ( 20 x ) + 5 4 ( 5 ) = 16 x + 4
Rewrite the equation Now, rewrite the equation with the simplified left side: 16 x + 4 = 16 x − 2
Simplify Subtract 16 x from both sides of the equation: 16 x + 4 − 16 x = 16 x − 2 − 16 x
4 = − 2
Conclusion The equation simplifies to 4 = − 2 , which is a contradiction. This means that there are no values of x that will satisfy the original equation. Therefore, the equation has no solutions.
Examples
Understanding how to solve equations and determine the number of solutions is crucial in many real-world applications. For example, in engineering, you might need to determine if a particular design will work under certain conditions. If the equation representing the design has no solutions, it means the design is not feasible. Similarly, in economics, you might use equations to model supply and demand. If the equation has no solutions, it indicates a market imbalance where supply and demand cannot be equalized.
The equation 5 4 ( 20 x + 5 ) = 16 x − 2 simplifies to a contradiction, 4 = − 2 , indicating that there are no solutions for x . Thus, the answer is C. 0 .
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