Solve $\frac{x}{2}+\frac{y}{3}=1$ for $x$.
A. $x=3 y-2$
B. $x=-\frac{2}{3} y+2$
C. $x=\frac{2}{3} y \pm 3$
D. $x=-2 y-3$
Answer (2)
Multiply both sides of the equation by 6 to eliminate fractions: 3 x + 2 y = 6 .
Isolate the term with x : 3 x = 6 − 2 y .
Divide both sides by 3: x = 3 6 − 2 y .
Simplify the expression: x = − 3 2 y + 2 . The final answer is x = − 3 2 y + 2 .
Explanation
Understanding the Problem We are given the equation 2 x + 3 y = 1 and asked to solve for x in terms of y . This means we want to isolate x on one side of the equation.
Eliminating Fractions First, let's eliminate the fractions by multiplying both sides of the equation by the least common multiple of 2 and 3, which is 6: 6 ( 2 x + 3 y ) = 6 ( 1 ) Distribute the 6 on the left side: 6 ⋅ 2 x + 6 ⋅ 3 y = 6 3 x + 2 y = 6
Isolating the x term Now, we want to isolate the term with x . Subtract 2 y from both sides of the equation: 3 x = 6 − 2 y
Solving for x Finally, divide both sides by 3 to solve for x :
x = 3 6 − 2 y We can simplify this expression by dividing each term in the numerator by 3: x = 3 6 − 3 2 y x = 2 − 3 2 y Alternatively, we can write this as: x = − 3 2 y + 2
Examples
In physics, if you have a relationship between two variables, such as distance and time, and you know the equation that relates them, you can solve for one variable in terms of the other. For example, if you know the equation that relates the position of an object to time, you can solve for the time it takes for the object to reach a certain position. This is a fundamental skill in many areas of science and engineering.
The solution for x in the equation 2 x + 3 y = 1 is x = − 3 2 y + 2 . This corresponds to option B. The process involves eliminating fractions, isolating the variable, and solving the equation step-by-step.
;
