How many solutions are there to the equation below?
[tex]4(x-5)=3 x+7[/tex]
A. 1
B. 0
C. Infinitely many
Answer (1)
Expand the left side of the equation: 4 ( x − 5 ) = 4 x − 20 .
Rewrite the equation: 4 x − 20 = 3 x + 7 .
Isolate x : x = 27 .
The equation has one solution: 1 .
Explanation
Understanding the problem We are given the equation 4 ( x − 5 ) = 3 x + 7 and asked to find the number of solutions.
Expanding the equation First, we expand the left side of the equation by distributing the 4:
4 ( x − 5 ) = 4 × x − 4 × 5 = 4 x − 20
So the equation becomes:
4 x − 20 = 3 x + 7
Isolating x Next, we want to isolate x on one side of the equation. We can subtract 3 x from both sides:
4 x − 20 − 3 x = 3 x + 7 − 3 x
This simplifies to:
x − 20 = 7
Solving for x Now, we add 20 to both sides of the equation to solve for x :
x − 20 + 20 = 7 + 20
This simplifies to:
x = 27
Checking the solution Since we found a unique value for x , there is exactly one solution to the equation. We can check our answer by substituting x = 27 into the original equation:
4 ( 27 − 5 ) = 4 ( 22 ) = 88
3 ( 27 ) + 7 = 81 + 7 = 88
Since both sides are equal, our solution is correct.
Final Answer Therefore, there is only one solution to the equation. The answer is A.
Examples
In real life, solving linear equations like this can help determine the break-even point for a business. For example, if a company's cost is represented by 3 x + 7 and their revenue is 4 ( x − 5 ) , solving for x tells them the number of units they need to sell to make their revenue equal to their cost. This is a fundamental concept in business and economics.
