Problem 26 - Algebra 2

Question

Write each equation in standard form. Identify A, B, and C. \(5 y=10 x-25\)

Step-by-Step Solution

Answer

Standard form: \(2x - y = 5\); A=2, B=-1, C=5.

Step 1: Rearrange the Equation

The given equation is \(5y = 10x - 25\). To write it in standard form, which is \(Ax + By = C\), first rearrange the terms to get all variable terms on one side and the constant term on the other. This gives us \(10x - 5y = 25\).

Step 2: Simplify the Equation

Now that we have \(10x - 5y = 25\), notice that all the coefficients are divisible by 5. Simplifying by dividing each term by 5 gives us \(2x - y = 5\).

Step 3: Identify A, B, and C

The simplified equation \(2x - y = 5\) is in standard form, where \(A = 2\), \(B = -1\), and \(C = 5\).

Key Concepts

Understanding Linear Equations

A linear equation is an equation of the first degree, which means it involves only the first power of the variables. Linear equations graph as straight lines on a Cartesian plane. A simple form of a linear equation in two variables is expressed as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. In a linear equation:

• \( A \) and \( B \) cannot both be zero simultaneously.
• They define the slope and position of the line, while \( C \) determines its intercept with the axes.

To solve problems involving linear equations, you often aim to express them in this standard form, which makes it easier to analyze and graph.

The Role of Coefficients

Coefficients are numerical or constant factors attached to the variables in an equation. In a linear equation like \( 5y = 10x - 25 \), the coefficients of the variables \( y \) and \( x \) play a crucial role. These values:

• Determine the steepness and direction of the line when graphed.
• Are used to perform operations such as addition, subtraction, or factoring.

For the equation to be in standard form, the variable coefficients are positioned as per the \( Ax + By = C \) structure. Here, the problem begins with rearranging and determining coefficients before simplification.

Rearranging Equations

Rearranging equations is a vital skill in algebra that involves shifting terms from one side of an equation to the other to achieve a desired form. When we rearrange \( 5y = 10x - 25 \) to get \( 10x - 5y = 25 \), you:

• Move terms involving variables to one side.
• Move constant terms to the opposite side.

This step is critical for transforming the equation into the standard form. Proper rearrangement helps to easily identify coefficients of variables, which are essential for graphing and solving equations.

Simplifying Equations

Once an equation is rearranged, simplifying it often involves reducing coefficients to their smallest integer values. This simplification makes the equation neater and more intuitive to work with. In the equation \( 10x - 5y = 25 \), simplification involves:

• Dividing every term by a common factor, in this case, 5.
• Resulting in \( 2x - y = 5 \) where coefficients are more manageable.

By simplifying equations, it becomes easier to interpret and solve them, especially in terms of identifying \( A \), \( B \), and \( C \) for the standard form. Such simplification also aids in matching equations to real-world scenarios or datasets where clearer relationships are needed.