Kenya solved the equation below:
[tex]$-6(x-2)+3 x=-3(x+3)+21$[/tex]
What is the solution to Kenya's equation?
A. -4
B. 12
C. no solution
D. infinitely many solutions
Answer (2)
After expanding and simplifying the equation, we find that both sides are equal for any value of x, indicating there are infinitely many solutions. Thus, the correct answer is D. infinitely many solutions.
;
Expand both sides of the equation: − 6 x + 12 + 3 x = − 3 x − 9 + 21 .
Combine like terms: − 3 x + 12 = − 3 x + 12 .
Add 3 x to both sides: 12 = 12 .
Since the equation simplifies to a true statement, the solution is infinitely many solutions. in f ini t e l y man yso l u t i o n s
Explanation
Problem Analysis We are given the equation − 6 ( x − 2 ) + 3 x = − 3 ( x + 3 ) + 21 and asked to find its solution. We will expand and simplify both sides of the equation to isolate x .
Expanding the Equation First, distribute the − 6 on the left side and the − 3 on the right side of the equation: − 6 x + 12 + 3 x = − 3 x − 9 + 21
Combining Like Terms Next, combine like terms on both sides: − 3 x + 12 = − 3 x + 12
Isolating Constants Add 3 x to both sides of the equation: − 3 x + 3 x + 12 = − 3 x + 3 x + 12
12 = 12
Determining the Solution Since the variables have been eliminated and we are left with a true statement ( 12 = 12 ), the equation has infinitely many solutions.
Examples
Understanding how to solve linear equations is crucial in many real-world applications. For instance, consider a scenario where you are comparing two different cell phone plans. Each plan has a monthly fee and a per-minute charge. By setting up a linear equation, you can determine the number of minutes you need to use each month for the plans to cost the same. This helps you make an informed decision based on your usage habits, ensuring you choose the most cost-effective plan. Linear equations are also used in physics to calculate motion, in economics to model supply and demand, and in engineering to design structures.
