Algebra is a branch of mathematics that deals with the manipulation of symbols and the rules for manipulating those symbols. Algebra has its roots in ancient civilizations, but it was not until the 16th century that it began to take on the form that we recognize today. Algebra is used to solve equations and to find unknown values, which makes it a powerful tool in a wide range of fields, including science, engineering, and finance.
Definitions:
- Variable: A letter or symbol that represents a value or a set of values.
- Equation: A statement that shows that two expressions are equal.
- Inequality: A statement that shows that one expression is greater than or less than another.
- Coefficient: A number that is multiplied by a variable in an algebraic expression.
- Term: A part of an algebraic expression that is separated by + or – signs.
Examples:
- Solving for a variable:
Suppose we are given the equation 2x + 3 = 7. We can solve for the value of x by subtracting 3 from both sides of the equation to get 2x = 4, and then dividing both sides by 2 to get x = 2. So, in this case, the value of x is 2.
- Solving a quadratic equation:
A quadratic equation is an equation in the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. To solve a quadratic equation, we can use the quadratic formula: x = (-b +/- sqrt(b^2 – 4ac)) / (2a). For example, suppose we are given the equation x^2 – 5x + 6 = 0. Using the quadratic formula, we find that x = (5 +/- sqrt(5^2 – 416)) / (2*1) = (5 +/- sqrt(25 – 24)) / 2 = (5 +/- sqrt(1)) / 2 = (5 +/- 1) / 2 = 3 or 2.
- Simplifying algebraic expressions:
We can simplify algebraic expressions by combining like terms and using the properties of operations. For example, the expression 3x + 2x + 4 can be simplified to 5x + 4, and the expression (x + 2)^2 can be simplified to x^2 + 4x + 4.
- Graphing linear equations:
A linear equation is an equation in the form y = mx + b, where m and b are constants and x and y are variables. We can graph a linear equation by plotting a set of points that satisfy the equation and then drawing a straight line through those points. For example, the equation y = 2x + 3 can be graphed by plotting the points (0,3), (1,5), (2,7), and so on. The graph of this equation is a straight line with a slope of 2 and a y-intercept of 3.
- Solving systems of equations:
A system of equations is a set of two or more equations that must be solved simultaneously. We can solve a system of equations by graphing the equations and finding the point at which the graphs intersect, or by using algebraic methods such as substitution or elimination. For example, the system of equations y = 2x + 3 and y = -x + 5 can be solved by graphing the equations and finding the point at which they intersect, or by solving for x in the first equation and substituting that value into the equation.
