Problem 20 - Algebra 1

Question

Write the equation in standard form with integer coefficients. $$-4 x+5 y+16=0$$

Step-by-Step Solution

Answer

The standard form of the given equation with integer coefficients is \(4x - 5y - 16 = 0\).

Step 1: Arrange equation

To begin, set all terms on one side of the equation equal to zero. The given equation is already set up this way.

Step 2: Ensure A is positive

In the standard form, the coefficient of the x term (A) should be positive. As we see, in the provided equation, the coefficient for x is -4, which is negative. To fix this, multiply the entire equation by -1.

Step 3: Result

The multiplication by -1 gives us the standard form of the equation: \(4x - 5y - 16 = 0\).

Step 4: Verify

Check that the coefficients A, B, and C are integers and A is not negative. You'll see this is the case with this particular equation, as our coefficients are 4, -5, and -16, and 4 is not negative.

Key Concepts

Integer Coefficients

In algebra, it is common to work with equations that contain integer coefficients. Integer coefficients are numbers that appear in front of the variables in an equation and are whole numbers, not fractions or decimals. This is important because integer coefficients simplify calculations and interpretations. For instance, the equation given in the exercise

• Step 1: Given Equation: -4x + 5y + 16 = 0
• Step 4: Standardized Equation with Integer Coefficients: 4x - 5y - 16 = 0

, is expressed with integer coefficients after adjusting the equation for the standard form. Here, we started with first confirming that all terms of the equation initially have integer coefficients, specifically -4, 5, and 16. Then, the equation was adjusted by multiplying through by -1 to ensure the x-term coefficient is positive, while still keeping integer coefficients. This adjustment shows the importance of using integer coefficients in making equations easier to work with and understand.

Linear Equation

A linear equation is one of the simplest forms in algebra, meaning it is an equation that makes a straight line when plotted on a graph. These equations usually consist of two variables, often denoted as x and y, with each term being either a constant or the product of a constant and a single variable. The standard form of a linear equation is represented as \[ Ax + By + C = 0 \] where A, B, and C are real numbers, and A is a non-zero integer. To understand it better:

• A linear equation will always have variables raised only to the power of one (e.g., x¹, y¹).
• It can take various forms, such as slope-intercept form \( y = mx + b \), but often needs conversion to standard form \( Ax + By = C \)
• Having integer coefficients, as shown in the modified equation \( 4x - 5y - 16 = 0 \), allows simplicity in linear transformations and provides a clearer interpretation in different mathematical contexts.

Algebra 1

Algebra 1 is a foundational math course that introduces students to basic algebraic concepts, including working with linear equations, understanding standard form, and manipulating equations to meet certain criteria, such as having integer coefficients. It sets the groundwork for all future math courses.

• Topics like standard form equations teach students how to rearrange equations logically.
• Students learn to recognize and convert equations into standard form, ensuring the coefficients are integers and the x-term is positive.
• Through exercises, such as rearranging \(-4x + 5y + 16 = 0\) to a clean standard form \(4x - 5y - 16 = 0\), students gain proficiency.

This establishes skills such as graph plotting, understanding solution sets, and applying algebra in real-life scenarios, forming the backbone for understanding mathematical relationships and functions on a more advanced level.