Problem 29 - Algebra 1

Question

Write the equation in standard form with integer coefficients. $$y=\frac{5}{2} x+9$$

Step-by-Step Solution

Answer

The standard form of the equation with integer coefficients is \(5x - 2y = -18\).

Step 1: Identify the present form

First, identify the given equation \(y = \frac{5}{2}x + 9\) as being in slope-intercept form.

Step 2: Convert to standard form

Next, convert this equation to standard form. To do this, multiply every term in the equation by 2 to get rid of the fraction, producing \(2y = 5x + 18\).

Step 3: Finalize the standard form with integer coefficients

Finally, rearrange the terms to match the standard form \(Ax + By = C\). This gives the final equation as \(5x - 2y = -18\).

Key Concepts

Standard Form

The standard form of a linear equation is represented as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) should be non-negative. This format is popular because it provides a clear structure where both variables, \(x\) and \(y\), are on the same side. It is particularly useful in applications like graphing, comparing equations, or solving systems of equations, as it allows for easy manipulation and comparison.

When converting equations to standard form from other forms, you must ensure that all coefficients are integers and rearranged accordingly. Additionally, simplify fractions and ensure all terms are on one side of the equation. This process ensures the equation is easy to work with and meets the criteria for completeness and clarity. Converting from slope-intercept form, for example, requires moving terms around and sometimes multiplying through to eliminate fractions.

Integer Coefficients

Working with integer coefficients involves making sure the coefficients (numbers taking place of \(A\), \(B\), and \(C\) in an equation) are whole numbers. This means there should be no fractions or decimals in the equation. Having integer coefficients can make the equation easier to read, use, and interpret, especially in graphing and algebraic calculations.

To change fractions to integers, multiply every term in the equation by the denominator of the fraction. For example, in the equation \(y = \frac{5}{2}x + 9\), multiply every term by 2 to clear the fraction: \(2y = 5x + 18\), which ensures all coefficients become whole numbers. This step is crucial for maintaining the equation's integrity while rendering it more practical for further algebraic procedures.

Slope-Intercept Form

The slope-intercept form of a linear equation is \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept. This form is useful when you want to quickly identify these two key characteristics of a line, making it a common choice for initial graph plotting and analysis.

Slope-intercept form allows you to see how alterations to \(m\) and \(b\) will affect the line's position and orientation. For instance, increasing \(m\) makes the line steeper, while changing \(b\) shifts the line up or down on the graph. This form is direct and relates directly to the graph, making it intuitive for solving problems involving direct comparisons or identifying parallel and perpendicular relationships between lines.

However, to convert this form to others, such as standard form, it involves rearranging terms and sometimes dealing with fractions or decimals to ensure proper integer coefficients.