Linear Expressions and Equations

Understanding Linear Expressions

Linear expressions are algebraic statements that are composed of variables and constants. The defining feature of a linear expression is that each variable is to the first power, or in other words, the highest exponent on any variable is one. For example, '3x + 5' is linear because the variable 'x' is raised to the power of one. If a term such as 'x^2' were included, the expression would no longer be linear. Linear expressions can take various forms, such as '2x - 7', 'a + b', or '4 - 3y', and they are the building blocks for more complex algebraic equations.

Components of Linear Expressions: Variables, Terms, and Coefficients

Linear expressions consist of variables, terms, and coefficients. Variables are symbols, typically letters, that represent unknown values. Terms are the distinct elements of an expression that are combined using addition or subtraction. Coefficients are the numerical factors that multiply the variables within terms. For instance, in the linear expression '5x - 2', 'x' is the variable, '5' is the coefficient of the term '5x', and '-2' is a constant term. The expression is made up of two terms: '5x' and '-2'. A clear understanding of these components is essential for the manipulation and simplification of linear expressions.

Writing Linear Expressions from Word Problems

Converting word problems into linear expressions is a critical skill that requires identifying language that corresponds to mathematical operations. Words like 'total', 'increased by', and 'sum' indicate addition, while 'decreased by', 'less than', and 'difference' suggest subtraction. Multiplication may be implied by 'times', 'product', or 'of', and division by 'per', 'divided by', or 'quotient'. For example, the statement 'the total of a number x and 7' translates to the linear expression 'x + 7', and 'three times a number y decreased by 5' becomes '3y - 5'. This translation from verbal to mathematical representation is crucial for formulating and solving algebraic problems.

Simplifying Linear Expressions

Simplifying linear expressions means rewriting them in their most reduced form while maintaining their original value. This process often involves distributing multiplication over addition or subtraction, combining like terms, and simplifying constants. For instance, to simplify '4(x + 3) + 2x', one would distribute the '4' across the parentheses to get '4x + 12 + 2x', and then combine the like terms '4x' and '2x' to obtain the simplified expression '6x + 12'. Simplification is a fundamental step in solving linear expressions and equations, as it clarifies the structure and facilitates further operations.

Linear Equations and Their Graphical Representation

Linear equations are algebraic statements that equate two linear expressions and often take the form 'ax + by = c', where 'a' and 'b' are coefficients, 'x' and 'y' are variables, and 'c' is a constant. When graphed on a coordinate plane, linear equations with one variable yield lines that are either vertical or horizontal, depending on whether the variable is 'x' or 'y'. Equations with two variables result in straight lines whose slope and y-intercept can be determined from the equation's standard form or slope-intercept form ('y = mx + b'). For example, to graph '2x + 3y = 6', one could solve for 'y' to get 'y = -2/3x + 2', which shows a slope of '-2/3' and a y-intercept of '2', allowing for the plotting of the line.

Solving Linear Equations and Inequalities

Solving linear equations entails finding the value or set of values for the variables that make the equation true. For single-variable equations, this usually involves isolating the variable on one side of the equation. In the case of two-variable systems, methods such as substitution or elimination are used to find the solutions. For example, to solve the system '2x + 3y = 5' and '4x - y = 11', one might solve the second equation for 'y' and substitute into the first to find 'x', then solve for 'y'. Linear inequalities, which use symbols like '