Select the correct answer.
What is the solution for $x$ in the equation?
$\frac{5}{3} x+4=\frac{2}{3} x$
A. $x=-\frac{12}{7}$
B. $x=\frac{12}{7}$
C. $x=-4$
D. $x=4$
Answer (2)
Subtract 3 2 x from both sides: 3 5 x − 3 2 x + 4 = 0 .
Simplify the equation: x + 4 = 0 .
Subtract 4 from both sides: x = − 4 .
The solution is − 4 .
Explanation
Problem Analysis We are given the equation 3 5 x + 4 = 3 2 x and asked to find the value of x that satisfies this equation.
Isolating x terms To solve for x , we first want to isolate the terms with x on one side of the equation. We can subtract 3 2 x from both sides of the equation: 3 5 x + 4 − 3 2 x = 3 2 x − 3 2 x This simplifies to 3 5 x − 3 2 x + 4 = 0
Combining x terms Now, we combine the x terms: 3 5 x − 3 2 x = 3 5 − 2 x = 3 3 x = x So the equation becomes x + 4 = 0
Isolating x To isolate x , we subtract 4 from both sides of the equation: x + 4 − 4 = 0 − 4 This gives us x = − 4
Final Answer Therefore, the solution for x is − 4 . Looking at the multiple choice options, we see that option C is x = − 4 .
Examples
In real-world scenarios, solving linear equations like this can help determine the break-even point in business. For example, if 3 5 x represents the cost of producing x items and 3 2 x − 4 represents the revenue from selling those items (with a fixed cost of 4), solving for x tells you how many items you need to sell to cover your costs. In this case, producing -4 items doesn't make sense in the real world, but it illustrates the mathematical principle. Another example is balancing chemical equations or calculating mixtures in cooking or chemistry, where you need to find the right amount of each ingredient to achieve a desired outcome. Linear equations are fundamental in many quantitative fields.
The solution for x in the equation 3 5 x + 4 = 3 2 x is x = − 4 . Therefore, the correct answer is option C. x = − 4 .
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