Problem 18 - Algebra 1

Question

Write the equation in standard form with integer coefficients. $$4 x-y-7=0$$

Step-by-Step Solution

Answer

Standard form equation: \(4x - y = 7\).

Step 1 - Identify the Given Equation

The given equation is \(4x - y - 7 = 0\). This equation is almost in standard form, but we have to modify it a little for perfect standard form.

Step 2 - Rearrange the Equation in Standard Form

Rearrange the given equation so it fits into the standard form (Ax + By = C). Simply re-order it to put the y term after the x term to get the equation into standard form. \(4x - y = 7\)

Step 3 - Final check

Check the final equation to make sure it fits the standard form (Ax + By = C), where A, B, and C are integers. As we see, the equation \(4x - y = 7\) fits this form: A=4, B=-1, and C=7 are all integer coefficients.

Key Concepts

What is Standard Form?

When working with linear equations in algebra, **standard form** is a specific way of writing them. In mathematics, a linear equation is in standard form when it fits the structure \(Ax + By = C\). Here, \(A\), \(B\), and \(C\) are constants, and \(x\) and \(y\) are variables. This form is straightforward and quite popular because it standardizes equations, making them easier to work with in certain situations.

Key features of the standard form include:

• A tidy arrangement with the \(x\)-term followed by the \(y\)-term.
• It clearly separates variables from constants.
• It often makes it easier to visualize or interpret the equation's application and meaning.

Understanding standard form helps you in sketching graphs and solving systems of equations.

Understanding Integer Coefficients

In any linear equation, the numbers multiplying the variables are known as **coefficients**. When we say something has **integer coefficients**, it simply means these numbers are whole numbers, which can be positive, negative, or zero.

Working with integer coefficients is important because:

• They simplify the process of mathematical computation and interpretation.
• Using integers helps ensure that equations are easy to work with, especially for solving or graphing.
• Integer coefficients are more relevant in various practical applications compared to fractions or decimals.

For example, in the equation \(4x - y = 7\), the coefficients are \(4\) for \(x\) and \(-1\) for \(y\), demonstrating that both are integers. This integer factor makes the equation cleaner and often more practical for various applications.

Rearranging Equations into Standard Form

Sometimes, equations need a little adjusting to meet a specific format, such as the standard form. **Rearranging equations** is a common technique to convert them into a desired shape.

Here’s a simple method to rearrange an equation:

• Identify the existing structure of the equation. For instance, take \(4x - y - 7 = 0\).
• Your goal is to isolate the constants on one side and the variables \(x\) and \(y\) on the other. Move terms around as needed.
• In our example, adding \(7\) to both sides gives \(4x - y = 7\), which fits the standard form \(Ax + By = C\).

After rearranging, always verify that the coefficients are integers and the formats fit \(Ax + By = C\). This ensures that the equation is not only in standard form but also tidy, with well-organized parts and integer coefficients.