Problem 27 - Algebra 1

Question

Write the equation in standard form with integer coefficients. $$3 x+9=\frac{7}{2} y$$

Step-by-Step Solution

Answer

The equation in standard form with integer coefficients is \(6x -7y = -18\).

Step 1: Multiply the entire equation by 2

To eliminate the fraction on the right side, it's important to multiply the entire equation by 2. Doing so gives: \(6x + 18 = 7y\).

Step 2: Rearrange the equation

The next step is to write the equation in the form of Ax + By = C. This could be achieved by transposing 6x to the right side of the equation so it becomes: \(18 = 7y-6x\).

Step 3: Write the variables in order x, y

Finally, the equation needs to be written in the proper order, which is the variables x and y coming before the constant. This yields the equation: \(6x -7y = -18\).

Key Concepts

Integer Coefficients in Equations

When we talk about integer coefficients in equations, we refer to the numbers that multiply the variables being whole numbers. Integers include all positive and negative whole numbers, as well as zero. For an equation to have integer coefficients, every number that multiplies a variable must be an integer. They do not allow fractions or decimals.

• This makes equations simpler and easier to handle when solving or rearranging them.
• In school mathematics, integer coefficients are preferred because they help to avoid errors related to fraction operations.

In practice, converting equations to have integer coefficients often involves multiplying through by a common multiple to clear out any fractions. This step ensures that all terms in the equation are whole numbers.

Understanding Linear Equations

Linear equations are a fundamental concept in algebra. They are equations of the first degree, meaning each variable is raised to the power of one. A linear equation will graph as a straight line.These equations can be represented in several forms, with the standard form being one of them. The standard form of a linear equation is written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) and \(B\) are not both zero.

• In our example, \(6x - 7y = -18\) is the final standard form.
• Standard form is particularly useful for quickly identifying certain properties of the line, such as x and y-intercepts.

Linear equations are straightforward to work with because their solutions are often simple intersections or straight lines.

Rearranging Equations

Rearranging equations is a common task in algebra designed to change the structure of an equation for easier understanding or solving. The goal is often to isolate a specific variable or to write the equation in a standard form like \(Ax + By = C\).

• This involves operations such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number.
• Equations need to be treated with balance, meaning any operation you do to one side must be done to the other side.

In the given exercise, rearranging meant shifting terms around until the equation fit the form \(6x - 7y = -18\). Notice how careful operations allow us to maintain equality while transforming the equation into a desired format.