Problem 40 - Algebra 2

Question

Write each equation in standard form. Identify A, B, and C. \(0.25 x=0.1+0.2 y\)

Step-by-Step Solution

Answer

Standard form: \(25x - 20y = 10\). A = 25, B = -20, C = 10.

Step 1: Identify the Given Equation

The given equation is in the form of fractions or decimals: \(0.25x = 0.1 + 0.2y\). Our goal is to convert it into the standard form of a linear equation: \(Ax + By = C\).

Step 2: Eliminate Decimals

To eliminate decimals, multiply the entire equation by 100 to turn the decimals into whole numbers. This is because the highest place value occupied by the decimals is hundredths. \[100(0.25x) = 100(0.1) + 100(0.2y)\]Simplifying gives: \[25x = 10 + 20y\]

Step 3: Rearrange to Standard Form

Move all terms involving variables to the left side of the equation and the constant to the right side so it fits the standard form \(Ax + By = C\).\[25x - 20y = 10\]

Step 4: Identify A, B, and C

Now, the equation \(25x - 20y = 10\) is in standard form. Identify the coefficients: \(A = 25\), \(B = -20\), and \(C = 10\).

Key Concepts

Standard Form

The standard form of a linear equation is crucial for consistency and simplicity in algebra. It is usually written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers. This form is preferred because it presents the equation in a clear and organized manner. Having equations in standard form helps when comparing multiple equations or graphing on a coordinate plane.
To achieve the standard form, it’s important that:

• \(A\), \(B\), and \(C\) must be integers (whole numbers).
• \(A\) should be positive.
• The greatest common divisor of \(A\), \(B\), and \(C\) should be 1, meaning they are simplified to the smallest possible integer terms.

In practice, converting equations to standard form may involve removing decimals or fractions, rearranging terms, and sometimes simplifying the equation further by dividing all terms by their greatest common divisor.

Coefficient Identification

Identifying coefficients in an equation is an essential skill when working with standard form equations. In the equation \(Ax + By = C\), coefficients \(A\) and \(B\) are the multipliers of the variables \(x\) and \(y\) respectively, while \(C\) is the constant term on the other side of the equal sign.
This process involves:

• Negative signs: Pay attention to negative signs as they are part of the coefficient, e.g., in \(-20y\), \(B\) is \(-20\).
• Coefficient simplification: If the equation has been simplified to the smallest integer terms, \(A\), \(B\), and \(C\) should not have any common divisors.

Having a clear understanding of each coefficient makes it easier to graph lines, write equations of lines parallel or perpendicular to a given line, and solve systems of equations. It sets the stage for deeper exploration into linear equations and their properties.

Equation Manipulation

Equation manipulation involves a series of steps to adjust and transform an equation into a desired form, like the standard form in this case. The process typically includes:

• Clearing Fractions and Decimals: Multiply every term by the least common denominator or a power of ten to eliminate decimals or fractions, converting them into integers.
• Term Rearrangement: Move variables to one side and constants to the other to fit the form \(Ax + By = C\). This may require switching signs and combining like terms.
• Simplification: Sometimes, you can simplify the equation by dividing all terms by their greatest common factor. This helps keep the equation in its most simplified form.

Each step of manipulating equations seeks to maintain equality while altering the equation's appearance, making it more useful or comparable in its new form. Practice helps to develop accuracy and efficiency in this skill, which is invaluable for algebra and precalculus. Always ensure to check your final equation for accuracy and compliance with the standard form criteria.