Problem 6 - Algebra 1: Concepts and Skills

Question

Identify the coefficient of each variable term. $$ 5 x-4 x+3=9-x $$

Step-by-Step Solution

Answer

The coefficients of the variable terms in the equation are 5, -4, and -1.

Step 1: Identify the first coefficient

The term containing 'x' on the left-hand side of the equation is '5x'. In this term, '5' is the numerical part that is multiplied by 'x', so the coefficient of '5x' is 5.

Step 2: Identify the second coefficient

The next term containing 'x' on the left-hand side of the equation is '-4x'. In this term, '-4' is the numerical part that is multiplied by 'x', so the coefficient of '-4x' is -4.

Step 3: Identify the third coefficient

The term containing 'x' on the right-hand side of the equation is '-x'. When no numerical part appears with the variable, the coefficient is taken as -1. So the coefficient of '-x' is -1.

Key Concepts

Identifying Coefficients

Understanding how to identify coefficients is foundational in algebra. Coefficients are the numerical parts of terms that are multiplied by a variable. For instance, in the expression \( 5x-4x+3 \), the terms with variables are \( 5x \) and \( -4x \). In these terms, '5' and '-4' are the coefficients. They indicate how many times the variable \( x \) is being counted.

When looking at an equation like \( 9-x \), it is important to recognize that the coefficient for \( x \) is implied to be '-1'. It’s because each \( x \) actually stands for 1 \( x \) and if it’s subtracted, then it’s understood to be \( -1x \) or simply \( -x \) without the need to show the '1'. It is critical to correctly identify coefficients as they play a key role in operations like simplification, solving equations, and understanding the behavior of graphs.

It's also wise to remember that coefficients can be positive, negative, or even fractions and decimals. Each coefficient changes the value of the term it's associated with and contributes to the overall expression or equation in unique ways.

Variable Terms

Variable terms consist of two parts: a variable and its coefficient, as in the term \( 5x \), where 'x' is the variable and '5' is the coefficient. A variable is a symbol, often a letter like \( x \) or \( y \), that represents one or more numbers, while the coefficient is the number that multiplies the variable. In any given algebraic expression or equation, there might be multiple variable terms.

For example, in an expression like \( 2x + 3y - 7z \), there are three terms, \( 2x \) has a coefficient 2, \( 3y \) has a coefficient 3, and \( -7z \) has a coefficient -7. Understanding each variable term's role is helpful in solving and graphing equations as well as in describing relationships in functions. The coefficient gives us important information about the rate of change and the direction (positive or negative) in these contexts.

Linear Equations

Linear equations are a central element of algebra and represent relationships where changes happen at a constant rate. The simplest form of a linear equation is \( y = mx + b \), where \( m \) and \( b \) are constants, representing the slope and the y-intercept respectively.

A linear equation like \( 5x - 4x + 3 = 9 - x \) can be simplified by combining like terms, which means adding or subtracting variable terms based on their coefficients. In this case, \( 5x \) and \( -4x \) can be combined because they both contain the variable \( x \). After simplifying, we would then aim to isolate the variable on one side to solve the equation.

Linear equations form straight lines when graphed on a coordinate plane. The slope or gradient given by the coefficient of \( x \) tells us how steep the line is while the constant term, like '3' in this example, indicates the point where the line will cross the y-axis. Recognizing and understanding these components allows for a better grasp of how to manipulate and solve linear equations.