Problem 65 - Algebra 1: Concepts and Skills

Question

Write the equation in standard form with integer coefficients. $$y=-\frac{3}{16} x+\frac{9}{16}$$

Step-by-Step Solution

Answer

The equation in standard form with integer coefficients is: 3x + 16y = 9.

Step 1: Understand the current form

The given equation is in slope-intercept form (y = mx + b), where the slope, m, is -3/16 and the y-intercept, b, is 9/16.

Step 2: Conversion to Standard Form

To convert this equation into standard form, integers need to be introduced. Therefore, multiply every term by 16. This results in 16y = -3x + 9.

Step 3: Rearranging

To be in standard form, the equation now needs to be arranged so that x and y are on the same side of the equation. Thus, rearrange the equation to 3x + 16y = 9.

Key Concepts

Slope-Intercept Form

When dealing with linear equations, one of the most common forms you'll encounter is the slope-intercept form. Defined by the format y = mx + b, this form neatly separates out two crucial characteristics of a line: the slope (m), and the y-intercept (b).

The slope represents the rate at which the y-value of a point on the line changes per unit increase in the x-value. In simpler terms, it tells you how steep the line is. On the other hand, the y-intercept is the point where the line crosses the y-axis, providing an easy starting point for graphing.

Understanding these components is key, as they form the foundation for converting between different types of linear equations.

Integer Coefficients

Integer coefficients are numbers without any fractions or decimals that multiply the variables in an equation. They're important because they make the equation easier to work with, especially when solving by hand or applying certain algebraic methods.

In our exercise, converting to integer coefficients required multiplying every term by the same number – in this case, 16. This eliminates fractions, leading to a cleaner, more standard presentation of the equation. It's important to keep in mind that you must multiply every term, to maintain the equality of the equation.

Converting Equations

Converting equations from one form to another is a common task in algebra, allowing for greater flexibility depending on the context or the problem you're solving.

In the case of the exercise, we converted from the slope-intercept form to the standard form, which typically appears as Ax + By = C. Here, A, B, and C are integers, and A is usually a positive number. To achieve this, one must perform algebraic operations such as multiplication, division, and rearrangement while maintaining the balance of the equation.

Ultimately, the goal is for x and y to be on the same side, and for the resulting coefficients to be integers, making it easier to identify solutions or to graph the equation.