Problem 23 - Algebra 1

Question

Write the equation in standard form with integer coefficients. $$y+3=0$$

Step-by-Step Solution

Answer

The standard form of the given equation with integer coefficients is \(0x + y = -3\).

Step 1: Identify the Given Equation

The given equation is \(y+3=0\)

Step 2: Transform into Standard Form

To make this look like standard form, we have to write it as Ax + By = C where A, B and C are integers. By simply rearranging the given equation, we can obtain \(0=y+3\) or \(0 = y + 3\), which can be rewritten as \(0x + 1y = -3\)(Since integer coefficients are required, we denote by 1 the coefficient of y), and then as \(0x + y = -3\). So, we have the equation in standard form with integer coefficients.

Key Concepts

Integer Coefficients

Integer coefficients are whole numbers that are used as the multipliers for variables in an equation. In the context of algebra, using integer coefficients helps in keeping the equation simple and easy to interpret. For instance, in the equation given by the exercise, we needed to have coefficients that are integers for all variables.

• An integer can be positive, negative, or zero.
• The equation becomes easier to solve and understand when the coefficients are integers as opposed to fractions or decimals.
• The equation maintains more general validity, as integer solutions are often applicable to a wide range of discrete scenarios.

In the example given, the term with the variable was already integer (1 for y). Making sure that all other terms also use integers ensures clarity. This is why the rewritten standard form equation becomes \(0x + y = -3\).

Linear Equations

Linear equations are fundamental in algebra, representing relationships where two variables change at a constant rate. The standard form of a linear equation is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. Key features of linear equations:

• They graph as a straight line on a coordinate plane.
• They have variables which are only raised to the power of one, meaning no squares or cubes.
• The equation expresses a constant ratio between the change in y and the change in x.

In our exercise, the equation \(y + 3 = 0\) was simplified into the linear standard form \(0x + y = -3\), indicating there is no x contribution in the equation, while \(y\) has a straightforward relation with \(C\). This simplicity helps in easily identifying solutions or graphing it.

Algebraic Manipulation

Algebraic manipulation involves reformatting equations to reveal their underlying structure or make them easier to work with. This technique is essential when transforming equations into a desirable form, such as the standard form with integer coefficients.Steps involved in algebraic manipulation:

• Identify the terms and variables present in the equation.
• Rearrange terms, if necessary, to align with the desired structure \(Ax + By = C\).
• Ensure coefficients are integers by multiplying or manipulating the equation accordingly.

In the exercise, algebraic manipulation helped us convert \(y + 3 = 0\) into the standard form. By rearranging and simplifying, we clarified the integer coefficients, resulting in \(0x + y = -3\). This process showcased how algebraic rules help create neat and manageable equations for further analysis or graphing.