Problem 16 - Algebra 1: Concepts and Skills

Question

Write the equation in standard form with integer coefficients. \(y=3 x-8\)

Step-by-Step Solution

Answer

The standard form of the given equation with integer coefficients is \(3x - y = 8\).

Step 1: Write down the original equation

The original equation is given by \(y=3x-8\).

Step 2: Rearrange the equation into standard form

To transform the given equation into standard form, subtract \(3x\) from both sides of the equation. This gives \(-3x + y = -8\). This equation is in standard form, but traditionally, the coefficient of \(x\) is positive.

Step 3: Make the coefficient of x positive

Multiply every term by \(-1\) to make the coefficient of \(x\) positive. This gives \(3x - y = 8\), which is the equation in standard form with integer coefficients.

Key Concepts

Integer Coefficients

In the world of algebra, an equation written with integer coefficients means all the numbers used as coefficients are whole numbers. These coefficients should be integers, like \(1\), \(-3\), \(5\), rather than fractions or decimals. This is especially important in the context of writing an equation in standard form. The standard form of a linear equation is typically written as \(Ax + By = C\). Here, \(A\), \(B\), and \(C\) are integer coefficients.

• All terms must be integers.
• No fractions or decimal numbers.
• The equation should be free of any denominators.

In practice, making sure that the coefficients are integers might require multiplying each term by a common factor. For example, if you have a fraction, like \(\frac{1}{2}x - y = \frac{2}{3}\), multiply every term by \(6\) to eliminate the fractions and express the equation with integer coefficients. This concept is crucial for clear communication in mathematics, ensuring that everyone understands the values used are complete numbers.

Linear Equation Transformation

Transforming a linear equation involves manipulating it to fit a desired form. For linear equations, the standard form is \(Ax + By = C\), where both \(A\) and \(B\) are not zero. Transformations might include:

• Rearranging terms to get all variables on one side and constants on the other.
• Adjusting the signs to conform to standard notation, such as making \(A\) positive.
• Ensuring integer coefficients by adjusting terms appropriately.

One of the most common transformations is moving terms from one side of the equation to the other, maintaining balance by performing the same operation on both sides. Additionally, to maintain integer coefficients, you might need to multiply or divide all terms appropriately. In this process, each move should be deliberate, slowly evolving your equation into the desired format without altering its original solution set.

Rearranging Equations

Rearranging equations is a critical skill in algebra, allowing you to rewrite an equation in a more useful form. When you rearrange equations, the goal can be to make one variable the subject or to express the equation in standard form.

• Subtract or add terms to both sides to isolate certain variables.
• Move terms around to align with the desired form, such as \(Ax + By = C\).
• Always keep the equation balanced by performing operations on both sides equally.

For instance, if you begin with an equation like \(y = 3x - 8\), to rearrange it into standard form, move \(3x\) to the left side by subtracting \(3x\) from both sides, leading to \(-3x + y = -8\). In a subsequent step, make adjustments like multiplying by \(-1\) to ensure the coefficient of \(x\) is positive, resulting in \(3x - y = 8\). Rearranging is a foundational skill, ensuring flexibility in mathematical problem-solving.