Problem 32 - Algebra 1

Question

Write the equation in standard form with integer coefficients. $$y=-\frac{1}{3} x-4$$

Step-by-Step Solution

Answer

The standard form of the given equation with integer coefficients is \(x + 3y = -12\).

Step 1: Remove Fraction

In the equation \(y=-\frac{1}{3}x-4\), multiply all terms by -3 to get rid of the fraction: -3y = x + 12

Step 2: Rearrange the equation

The standard form is Ax + By = C, so we need to rearrange the equation, moving x terms on the left side and constant terms on the right: x + 3y = -12

Key Concepts

Standard Form

Standard Form in algebra typically refers to the way a linear equation is written. This is an important format since it offers a uniform way to present lines, making them easier to analyze and compare. For linear equations, the standard form is denoted as \(Ax + By = C\). The key aspects of this format are:

• The terms on one side are arranged so that the variables \(x\) and \(y\) are on the left and the constant is on the right.
• The coefficients \(A\), \(B\), and \(C\) should be integers for simplicity.
• Usually, \(A\) is a non-negative integer.

Understanding the importance of presenting equations in this manner helps in quickly identifying slopes, intercepts, and solutions of systems of equations when paired together. This form is widely preferred in professional settings and when solving systems of equations due to its standardized format.

Linear Equations

Linear equations are equations of the first order, which means they're the simplest forms of equations that involve two variables. They are called 'linear' because they graph as straight lines. The general linear equation can be expressed in various forms including slope-intercept (\(y = mx + b\)) and standard form (\(Ax + By = C\)). Here are some key characteristics of linear equations:

• They have no exponents higher than 1.
• The graph of a linear equation will always be a straight line.
• The solution of a linear equation will provide a specific 'line' or 'path' on the graph plane.

Being able to identify and convert between different forms of linear equations is an essential algebraic skill. This competency aids in solving various algebra problems, including simultaneous equations, inequalities, and more.

Integer Coefficients

An important step in working with linear equations, especially in the context of transferring them to the standard form, is ensuring that the coefficients are integers. Integer coefficients make calculations simpler and the equations cleaner. Steps to convert coefficients to integers include:

• Multiply all terms by a number that clears out fractions or decimals in the equation.
• Aim to have a positive integer leading coefficient for \(x\).
• Reassess the equation to ensure it's in the correct format: \(Ax + By = C\).

By committing to integer coefficients, we facilitate better readability and handiness in further computations. This practice is crucial when solving systems of equations or when precise mathematical relationships need to be maintained, as it helps avoid rounding errors and fractional interpretations.