3/5x + 1/4y = -1/2 in standard form
The Correct Answer and Explanation is:
To write the equation35x+14y=−12\frac{3}{5}x + \frac{1}{4}y = -\frac{1}{2}53x+41y=−21
in standard form, we want an equation of the form:Ax+By=CAx + By = CAx+By=C
where:
- A, B, and C are integers
- A should be non-negative (positive or zero)
✅ Step-by-step conversion:
1. Eliminate the fractions
To eliminate the denominators 555, 444, and 222, find the least common denominator (LCD), which is 20.
Multiply every term of the equation by 20:20(35x+14y=−12)20\left(\frac{3}{5}x + \frac{1}{4}y = -\frac{1}{2}\right)20(53x+41y=−21)⇒20⋅35x+20⋅14y=20⋅(−12)\Rightarrow 20 \cdot \frac{3}{5}x + 20 \cdot \frac{1}{4}y = 20 \cdot \left(-\frac{1}{2}\right)⇒20⋅53x+20⋅41y=20⋅(−21)⇒12x+5y=−10\Rightarrow 12x + 5y = -10⇒12x+5y=−10
🎯 Final Answer (Standard Form):
12x+5y=−10\boxed{12x + 5y = -10}12x+5y=−10
📘 Explanation
To convert a linear equation into standard form, it’s essential to remove any fractions and ensure the coefficients of xxx, yyy, and the constant term are integers. The standard form, by convention, is written as:Ax+By=CAx + By = CAx+By=C
In the given equation:35x+14y=−12\frac{3}{5}x + \frac{1}{4}y = -\frac{1}{2}53x+41y=−21
we notice fractional coefficients, which are not ideal for standard form. To eliminate these, we multiply the entire equation by the least common denominator (LCD) of all denominators in the equation: 5, 4, and 2. The LCD of 5, 4, and 2 is 20. Multiplying every term by 20 simplifies the equation by clearing out the fractions:
- 20⋅35x=12×20 \cdot \frac{3}{5}x = 12×20⋅53x=12x
- 20⋅14y=5y20 \cdot \frac{1}{4}y = 5y20⋅41y=5y
- 20⋅−12=−1020 \cdot -\frac{1}{2} = -1020⋅−21=−10
So, the equation becomes:12x+5y=−1012x + 5y = -1012x+5y=−10
This is now in standard form: all coefficients are integers, and the xxx-term comes first. While the constant term is negative, this is acceptable; the only strict convention is that the coefficient of xxx, A, should be positive — which it is in this case (A = 12).
This process demonstrates the importance of understanding how to manipulate algebraic expressions to meet format requirements, especially in algebra where form matters for clarity and further operations like graphing or solving systems of equations.
