Problem 35 - Algebra 2

Question

Write each equation in standard form. Identify A, B, and C. \(\frac{1}{2} x+\frac{1}{2} y=6\)

Step-by-Step Solution

Answer

The equation in standard form is \(x + y = 12\) where \(A = 1\), \(B = 1\), and \(C = 12\).

Step 1: Understand the Equation

The given equation is \( \frac{1}{2} x + \frac{1}{2} y = 6 \). Our goal is to write this in standard form, which is \( Ax + By = C \), where A, B, and C are integers.

Step 2: Eliminate Fractions

Multiply each term in the equation \( \frac{1}{2} x + \frac{1}{2} y = 6 \) by 2 to eliminate the fractions. This gives us: \( x + y = 12 \).

Step 3: Identify A, B, and C

In the equation \( x + y = 12 \), the coefficients correspond to: \( A = 1 \), \( B = 1 \), and \( C = 12 \). These values satisfy the standard form criteria.

Key Concepts

Linear Equation

A linear equation is a type of equation where the highest power of the variable is one. Unlike quadratic or cubic equations, linear equations do not have squared (x^2) or cubic (x^3) terms. They often take the form of a straight line when graphed.
In our exercise, the linear equation given is \( \frac{1}{2} x + \frac{1}{2} y = 6 \). This is a straightforward example where both \( x \) and \( y \) appear with coefficients.

This type of equation can be written in different ways, but one common form is the standard form \( Ax + By = C \). Here \( A \), \( B \), and \( C \) are constants, and both \( A \) and \( B \) should not be zero at the same time. Understanding how to manipulate equations into this form is crucial in algebra.

Eliminate Fractions

Fractions can make equations messy and complicated, especially when trying to write them in standard form. To simplify the process, we aim to eliminate fractions as a first step.

For instance, in the original equation \( \frac{1}{2} x + \frac{1}{2} y = 6 \), we have fractions that complicate reaching the standard form. To eliminate these fractions, we multiply every term by 2, the least common denominator.

• Multiplying \( \frac{1}{2} x \) by 2 gives \( x \)
• Multiplying \( \frac{1}{2} y \) by 2 gives \( y \)
• Multiplying 6 by 2 simplifies to 12

As a result, the equation becomes \( x + y = 12 \). This step helps to clear the fractions and makes the equation easier to handle.

Integer Coefficients

After eliminating fractions, the next crucial step is to ensure all coefficients in the equation are integers. Coefficients are numbers that multiply the variables in an equation. Using integer coefficients makes the equation clearer and more consistent with standard form.

In our example, once the fractions are eliminated, we have \( x + y = 12 \). Here, the coefficients are both \( 1 \), which are integers, and \( 12 \) is also an integer.

• \( A = 1 \)
• \( B = 1 \)
• \( C = 12 \)

Ensuring integer coefficients is fundamental when converting to standard form so that the equation is easier to interpret and graph.

Identify Coefficients

Identifying coefficients is the final step to completing the standard form. A coefficient is a number that multiplies a variable within any given term of an equation.

In the equation \( x + y = 12 \), the coefficients need to be identified as part of expressing the equation in standard form, \( Ax + By = C \).

By inspection:

• The coefficient of \( x \) is 1, so \( A = 1 \).
• The coefficient of \( y \) is 1, so \( B = 1 \).
• The constant term on the right side of the equation is 12, so \( C = 12 \).

This identification process confirms that the equation meets the criteria to be in standard form, making it clearer and ready for further analysis or graphing.