Algebra combines numbers and their operations with variables that represent unknown quantities. Consider the following:
This is an equation (as indicated by the equal sign) stating that the expression on the left is equivalent to the expression on the right. The expression on the left contains 2 terms with 2 variables and 2 coefficients: 3 and 4 are coefficients (numbers combined with variables), and x and y are variables representing unknown quantities. The 15 in the expression on the right is a constant.
This equation can be solved for x in terms of y, as in:
or for y in terms of x, as in:
For each variable present in an equation, that number of equations are necessary to solve for the value of each variable.
Memorize or be able to derive the following rules of exponents:
And these rules for roots:
Occasionally, it will be required to rationalize a denominator containing a root. To rationalize a denominator, multiply numerator and denominator by the root:
Consider the following example: . To rationalize this denominator, it is necessary to multiply numerator and denominator by the conjugate of the denominator. The conjugate includes the same terms of the denominator but the opposite sign:
A linear equation is an equation of the form: , , or ; these are the standard form, slope-intercept form, and point-slope form of a linear equation.
Linear equations contain an independent and a dependent variable. These can be thought of as input and output variables. By convention, the x variable is the input variable or independent variable and the y variable is the output or dependent variable.
The graph of a linear equation is a line. The slope of the line, m, is defined as the “rise over the run,” or more formally as: where and represent points on the line.
The x-intercept of a line is the point at which the line crosses the x-axis.
The y-intercept of a line is the point at which the line crosses the y-axis.
The x-intercept is found by setting y equal to 0 and solving for x.
The y-intercept is found by setting x equal to 0 and solving for y.
Parallel lines are lines that do not intersect; they share the same slope.
Perpendicular lines are lines that intersect at a right angle; their slopes are negative reciprocals of each other.
A system of linear equations is a collection of 2 more linear equations. Systems of linear equations can be solved using methods of substitution, elimination, or graphing. The solution set of a linear system of equations is the intersection point of the lines involved.
Solving by substitution entails solving for one of the variables in terms of the other variable then substituting the found expression into the other equation and finding the exact value of the variable in question. This value can then replace the variable in the other equation and the remaining variable can be found.
Solving the second equation for x:
and substituting into the first equation:
and replacing the y in the first equation with this value:
So is the solution to the system. This represents the point at which the two lines intersect.
The same system can be solved using elimination. The method of elimination involves combining the equations in such a way that one of the variables is eliminated in the process:
Combining these equations through addition will eliminate the y variable:
By substituting this variable into the original equation, the y variable can be found.
A quadratic equation is an equation of the form: , where A, B, and C are real numbers. The graph of a quadratic equation forms a parabola, which can face up or down.
It is useful to know how to solve a quadratic equation. There are various methods for solving, including: factoring, applying the quadratic formula, and graphing.
To factor a quadratic expression means to decompose the form into a form of , where x is a variable, and a and b are real numbers that are sometimes the same.
To find the values for a and b, look for the factors of C that combine to make the coefficient B.
Take note that: ;
The factors that multiply to 16 and combine to 8 are 4 and 4, so the left side expression can be written as:
And, noticing that either can be set to 0 to make the equation true, the solution is .
Graphically, this solution indicates a double root at meaning that the vertex of the parabola is along the x-axis. And, because a is greater than 0, the parabola points up.
Another method for solving a quadratic equation is the quadratic formula:
This formula yields the x-intercepts of a quadratic function. If the square root portion is greater than 0, two real solutions exist. If the square root portion is equal to 0, there is one root, and it is the vertex. If the square root portion is negative, the graph does not cross the x-axis.
To find the vertex of a quadratic in standard form, find for the x value, and evaluate for the y-value.
For quadratics with an A larger than 1, the same method of factorization can be used, with the addition of finding the factors of A such that ,
A linear inequality involves a linear expression and an inequality- < less than, > greater than, less than or equal to, and greater than or equal to.
Linear inequalities are solved like linear equations with one major difference. When multiplying or dividing both sides of a linear inequality by a negative value, the direction of the inequality switches.
Additionally, when graphing linear inequalities, less than or greater than signs are shown with an open circle and a dashed line. Less than or equal to and greater than or equal to inequalities are graphed using a closed circle and a heavy line. The solution of the inequality is not the line itself, but the area of the plane either below or above the line.
Functions take in an input variable and give out an output variable that is distinct for each input. A function must pass the ‘vertical line test,’ meaning that for every input, there can be only one output value.
Consider the following function:
This function is read as: f of x is equal to x squared plus 2. If the parent function and applied vertical translation are not recognized, the function can be graphed with an x,y table, in which various x values are evaluated to find the general shape of the graph.
Functions have a domain and a range. A domain is all of the possible input, or x, values that will yield a real value, and a range is all of the possible output, or y, values associated with the domain.
Functions that contain possible division by 0 or square roots will potentially have domain restrictions. To find the domain restrictions of a function, set the denominator equal to 0 and solve for x, or set the expression inside the root greater than or equal to 0 and solve for x. The domain will be everything except the values found.
Setting the denominator equal to 0 and solving:
Setting the denominator greater than or equal to 0:
However, because x cannot equal , because of the division by zero, the domain is the set of x values greater than .
Simple interest is found using the following formula:
where I is the total interest, p is the initial amount, r is the interest rate as a decimal, and t is the number of years.
Compound interest is found using:
where A is the future value, p is the initial amount, r is the interest rate as a decimal, n is the number of times the interest is compounded per year, and t is the number of years.