Rewrite the equation $2x + 4y + 2 = 3y + 5$ in general form.
A) $2x + y - 3 = 0$
B) $2x + y = 3$
C) $y = -2x + 3$
D) $2x + 7y + 7 = 0
Answer (1)
Subtract 3 y from both sides of the equation.
Subtract 5 from both sides of the equation.
Simplify the equation to the general form.
The final answer is 2 x + y − 3 = 0 .
Explanation
Understanding the Problem We are given the equation 2 x + 4 y + 2 = 3 y + 5 and asked to rewrite it in general form, which is A x + B y + C = 0 .
Subtracting 3y from both sides To rewrite the equation in general form, we need to move all terms to one side of the equation and simplify. We start by subtracting 3 y from both sides: 2 x + 4 y + 2 − 3 y = 3 y + 5 − 3 y 2 x + y + 2 = 5
Subtracting 5 from both sides Next, we subtract 5 from both sides: 2 x + y + 2 − 5 = 5 − 5 2 x + y − 3 = 0
Identifying the General Form The equation is now in the general form A x + B y + C = 0 , where A = 2 , B = 1 , and C = − 3 .
Final Answer Comparing our result 2 x + y − 3 = 0 with the given options, we see that it matches option A.
Examples
In architecture, the general form of a linear equation can help define the constraints of a building's design, such as the relationship between the height and width of a room to meet specific aesthetic or structural requirements. For instance, an architect might use the equation 2 x + y − 3 = 0 to ensure that for every 2 units of increase in the width ( x ), the height ( y ) decreases by 3 units to maintain a certain proportion. This ensures that the design adheres to predefined spatial relationships, contributing to the overall harmony and functionality of the structure.
