Select the choice that represents this two-variable equation after it has been solved for $y$ in terms of $x$.
$\frac{1}{3} y+x=1$
A. $x=-\frac{1}{3} y+1$
B. $y=-x+\frac{2}{3}$
C. $y=-3 x+3$
D. $y=-3 x+1$
E. $y=3 x+1$
Answer (2)
Subtract x from both sides: 3 1 y = 1 − x .
Multiply both sides by 3: y = 3 ( 1 − x ) .
Distribute the 3: y = 3 − 3 x .
Rewrite the equation: y = − 3 x + 3 , so the answer is y = − 3 x + 3 .
Explanation
Understanding the Problem We are given the equation 3 1 y + x = 1 and we want to solve for y in terms of x . This means we want to isolate y on one side of the equation.
Isolating the y term First, we subtract x from both sides of the equation to get 3 1 y = 1 − x .
Multiplying by 3 Next, we multiply both sides of the equation by 3 to isolate y : 3 ⋅ 3 1 y = 3 ⋅ ( 1 − x ) y = 3 ( 1 − x )
Distributing the 3 Now, we distribute the 3 on the right side of the equation: y = 3 ⋅ 1 − 3 ⋅ x y = 3 − 3 x
Rewriting the Equation Finally, we rewrite the equation to match the form of the answer choices: y = − 3 x + 3 So the correct answer is C.
Examples
In physics, this type of equation can represent a linear relationship between two variables, such as the position of an object over time with constant velocity. Solving for one variable in terms of the other allows us to predict the value of one variable if we know the value of the other. For example, if x represents time and y represents position, the equation y = − 3 x + 3 tells us the object's position at any given time.
The equation 3 1 y + x = 1 solved for y gives y = − 3 x + 3 . This matches option C. Thus, the correct answer is C.
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