key term - Algebraic Expression

Definition

An algebraic expression is a mathematical phrase that combines variables, numbers, and operations, such as addition, subtraction, multiplication, and division, to represent a quantitative relationship. Algebraic expressions are fundamental in solving linear equations and systems of equations.

5 Must Know Facts For Your Next Test

1. Algebraic expressions can contain one or more terms, where each term is a combination of variables and coefficients.
2. The simplest form of an algebraic expression is a single variable or constant, such as $x$ or $5$.
3. Algebraic expressions can be manipulated using the rules of algebra, such as combining like terms, distributing, and factoring.
4. Solving linear equations involves isolating the variable by performing inverse operations on both sides of the equation.
5. Systems of linear equations can be solved using substitution or elimination methods, which involve manipulating algebraic expressions.

Review Questions

• Explain how the concept of an algebraic expression is used in the process of solving linear equations.
  • When solving linear equations, the goal is to isolate the variable on one side of the equation. This involves performing inverse operations on both sides of the equation, which involves manipulating the algebraic expressions. For example, to solve an equation like $2x + 5 = 17$, you would subtract 5 from both sides to get the algebraic expression $2x = 12$, and then divide both sides by 2 to find the value of $x$. The ability to manipulate algebraic expressions is crucial for successfully solving linear equations.
• Describe how the understanding of algebraic expressions is applied when solving systems of linear equations using the substitution method.
  • In the substitution method for solving systems of linear equations, the goal is to isolate a variable in one equation and then substitute that expression into the other equation. This process involves working with algebraic expressions. For instance, if the system of equations is $2x + 3y = 12$ and $x - y = 5$, you would first solve the second equation for $y$ to get $y = x - 5$. Then, you would substitute this expression for $y$ into the first equation, creating a new algebraic expression $2x + 3(x - 5) = 12$, which you would then solve for $x$. The ability to manipulate these algebraic expressions is essential for successfully applying the substitution method to solve systems of linear equations.
• Evaluate how the understanding of algebraic expressions can be applied to solve more complex systems of linear equations using the elimination method.
  • The elimination method for solving systems of linear equations involves manipulating the algebraic expressions in the equations to eliminate a variable. This is done by adding or subtracting the equations to create a new equation with one fewer variable. For example, if the system is $3x + 2y = 14$ and $2x - y = 8$, you could multiply the second equation by 2 to get $4x - 2y = 16$, and then subtract the first equation from the modified second equation to eliminate the $y$ variable and create the new algebraic expression $x = 2$. The ability to perform these algebraic manipulations is essential for successfully applying the elimination method to solve more complex systems of linear equations.

Related terms

Variable:

A symbol, usually a letter, that represents an unknown or changeable quantity in an algebraic expression.

Coefficient:

The numerical factor that multiplies a variable in an algebraic expression.

Constant:

A fixed value or quantity that does not change within the context of a specific problem or expression.