2x + 5y = 3y + 8 (solve the literal equation for y)
The correct answer and explanation is:
Let’s solve the equation 2x+5y=3y+82x + 5y = 3y + 8 for yy.
Step-by-step solution:
Given:
2x+5y=3y+82x + 5y = 3y + 8
- Get all terms with yy on one side:
Subtract 3y3y from both sides: 2x+5y−3y=3y+8−3y2x + 5y – 3y = 3y + 8 – 3y 2x+2y=82x + 2y = 8
- Isolate the yy-term:
Subtract 2x2x from both sides: 2y=8−2x2y = 8 – 2x
- Solve for yy:
Divide both sides by 2: y=8−2x2y = \frac{8 – 2x}{2}
Simplify the fraction: y=82−2x2y = \frac{8}{2} – \frac{2x}{2} y=4−xy = 4 – x
Final answer:
y=4−x\boxed{y = 4 – x}
Explanation (about 300 words):
When solving an equation literally for a specific variable, the goal is to isolate that variable on one side of the equation so that it stands alone and is expressed in terms of the other variables or constants.
Here, the original equation was 2x+5y=3y+82x + 5y = 3y + 8. We want to solve for yy, meaning we want to rewrite the equation so that yy is by itself on one side.
First, we rearranged the equation to get all terms containing yy on one side. This is important because yy appears on both sides, so combining them makes it easier to isolate yy. Subtracting 3y3y from both sides reduces the number of yy-terms and leads to 2x+2y=82x + 2y = 8.
Next, to isolate yy, subtract 2x2x from both sides, so 2y2y is alone on the left side: 2y=8−2x2y = 8 – 2x.
Finally, dividing both sides by 2 (the coefficient of yy) isolates yy: y=8−2×2=4−x.y = \frac{8 – 2x}{2} = 4 – x.
This expression now clearly shows yy as a function of xx, which is helpful for understanding the relationship between these variables.
In summary, solving for a variable involves using algebraic operations—adding, subtracting, multiplying, or dividing—to rearrange the equation, so the variable is alone on one side. This approach is fundamental in algebra and is widely used in mathematics, physics, engineering, and many other fields.