Problem 19 - Algebra 1

Question

Write the equation in standard form with integer coefficients. $$x+3 y-4=0$$

Step-by-Step Solution

Answer

The equation in standard form with integer coefficients is \(x + 3y = 4\).

Step 1: Identify the form of the given equation

The given equation is \(x + 3y - 4 = 0\). Rearranging, we get \(x + 3y = 4\). It looks like the equation is already in the standard form.

Step 2: Check the coefficients

Now we need to check if all the coefficients are integers. For \(x + 3y = 4\), the coefficients of x and y, and the constant on the right side of the equation are 1, 3, and 4 respectively, and all are integers.

Key Concepts

Integer Coefficients

In mathematics, an integer coefficient is simply a whole number found as a coefficient in an equation. Coefficients are the numbers that multiply the variables, like the numerical values in front of the variables in a linear equation. When dealing with integer coefficients, it's necessary to ensure that these numbers, including constants, are all integers (whole numbers that can be positive, negative, or zero).

Why is having integer coefficients important? It’s mostly about simplicity and clarity.

• They make equations easier to work with, especially when performing operations like addition, subtraction, or multiplication.
• Integer coefficients can help to ensure that solutions are also integers, making the equation easier to interpret and solve, particularly in practical scenarios where fractions or decimals may be less desirable.

In our initial equation, the coefficients are 1 for the variable \(x\) (implied as we do not actually write the 1), 3 for \(y\), and 4 as the constant. All these numbers are integers, which means our coefficients are exactly how mathematicians like them: neat and tidy whole numbers.

Standard Form

A linear equation in standard form is typically written as:\[Ax + By = C\]Here, \(A\), \(B\), and \(C\) are integers, and it is standard for \(A\) to be positive. This form is particularly useful because:

• It keeps the equation organized and easy to read, which is helpful when analyzing or graphing.
• Ensures that all variables and constants are clearly defined on their respective sides of the equality.

In the given problem, the equation is\[x + 3y - 4 = 0\]To convert it to standard form, we move the constant to the other side:\[x + 3y = 4\]The rearrangement ensures that the equation fits the standard format. This simplicity is often preferred as it allows uniform processing of equations, especially when solving systems of equations or performing more advanced algebraic operations.

Rearranging Equations

Rearranging equations involves changing the form of an equation without altering its inherent equality or the solutions it produces. This transformation is done through basic operations like addition, subtraction, multiplication, or division.

In practice, reorganizing an equation can help in several ways:

• It allows for easier reading and interpretation, especially when matching the equation to a standard form.
• Can assist in isolating variables or constants required for solutions or deeper analysis.

In our exercise, you began with\[x + 3y - 4 = 0\]Through rearranging, we obtained\[x + 3y = 4\]This is achieved by simply adjusting the terms to reflect familiar equation structures. By understanding how to reposition parts of an equation, solving and manipulating equations becomes a more approachable task, allowing you to handle a variety of algebraic expressions effectively.