Problem 62 - Algebra 1: Concepts and Skills

Question

Write the equation in standard form with integer coefficients. $$y=5 x-2$$

Step-by-Step Solution

Answer

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

Step 1: Identify the equation

The provided equation is \( y = 5x -2 \). This is currently in slope-intercept form, which is given by \( y = mx + b \), where m is the slope and b is the y-intercept.

Step 2: Transform the equation

The aim is to transform the equation into standard form, which is \( Ax + By = C \). Begin this process by subtracting both sides by \( 5x \) in order to isolate y on one side and all other terms on the other side. So, \( y = 5x - 2 \) becomes \( -5x + y = -2 \).

Step 3: Final Result

The standard form of a linear equation is not supposed to start with a negative number. Therefore, multiply each term in the equation by -1. The sign changes for each term, thus \( -5x + y = -2 \) transforms to \( 5x - y = 2 \). This is now in standard form.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express linear equations using the formula \( y = mx + b \). Here, \( m \) represents the slope of the line. The slope indicates how steep the line is and the direction in which it slants.
The \( b \) in the equation is the y-intercept, which is the point where the line crosses the y-axis, offering a starting point for graphing the line.
This form is popular because it clearly shows the slope and starting point, making it easy to graph a line quickly by plotting the y-intercept and using the slope.

• If the slope \( m \) is positive, the line slopes upwards.
• If \( m \) is negative, the line slopes downwards.

In our exercise, the original equation \( y = 5x - 2 \) is in this form, where \( m = 5 \) and \( b = -2 \). This means the line has a slope of +5 and crosses the y-axis at -2.

Integer Coefficients

Integer coefficients in linear equations refer to the equation containing numbers that are whole integers—positive or negative—without fractions or decimal parts.
Working with integer coefficients is particularly useful in mathematics since they tend to simplify calculations and are easy to understand visually.

• For example, in transforming an equation into standard form, ensuring all coefficients are integers is crucial.
• This standard format is particularly important in hierarchical systems and algorithms that do not handle fractions well.

In the final step of our problem, we ended up with the equation \( 5x - y = 2 \), where 5, -1, and 2 are all integers. Ensuring integer coefficients is a common requirement when writing equations, especially in standard form. It creates a finish free from decimals that is simpler to interpret.

Linear Equations

Linear equations are mathematical expressions that produce a straight line when graphed. They are called 'linear' because they represent lines forming an even path on a coordinate grid.
These equations are basic tools used to model relationships between variables where one variable depends on the other in a constant way.

• A typical linear equation can be written in several forms, such as slope-intercept form \( y = mx + b \) or standard form \( Ax + By = C \).
• The ability to switch between these forms, as illustrated in our exercise, is a crucial algebra skill.

In our exercise, we started with \( y = 5x - 2 \), and through algebraic manipulation, converted it into the standard form \( 5x - y = 2 \). Linear equations not only help in finding relationships in mathematical problems but are also applied in various real-world scenarios where relationships between two changing quantities need to be determined.