In this lesson we’ll look at how to solve a multivariable equation for a certain variable in terms of the others. When you solve an equation for a variable you’re moving the other terms and coefficients around by using inverse operations to isolate the variable you’re solving for.
Read MoreAll functions are equations, but not all equations are functions. Functions are equations that pass the Vertical Line Test. In other words, in order for a graph to be a function, no perfectly vertical line can cross its graph more than once.
Read MoreThe area of a rectangle is the product of its base and its height. We can also express the area of a rectangle as the product of its length and its width. An area is always given in units of length^2 (“length squared”). In other words, if the dimensions of the rectangle are given in inches, the units for area will be in^2.
Read MoreWe know already how to solve systems of linear equations using substitution, elimination, and graphing. This time, we want to talk about how to solve systems using inverse matrices. To walk through this, let’s use a simple system.
Read MoreIn this lesson we’ll look at composite figures made from rectangles and how to find their areas. A composite figure is made by combining different shapes. We’ll find the area of a composite figure by dividing the composite shape into shapes whose areas we already know how to find.
Read MoreIn math and science you might meet a variable with a subscript. Don’t let them scare you, they’re just a way to keep track of variables that could be related to each other in some way. For instance, if we’re doing a problem involving the distance that two people travel, we might use the variables D_A and D_B to indicate the distances related to person A and person B, respectively.
Read MoreWhen we talk about “even, odd, or neither” we’re talking about the symmetry of a function. It’s easiest to visually see even, odd, or neither when looking at a graph. Sometimes it’s difficult or impossible to graph a function, so there is an algebraic way to check as well.
Read MoreIn this lesson, we're going to focus on particular kinds of fractions. We'll start out by talking about positive fractions, and then we’ll deal with negative fractions at the end of the lesson. Up until now, most of the fractions we’ve dealt with are what we call “proper” fractions, where the numerator is less than the denominator.
Read MoreWe can add and subtract mixed numbers, each of which is the sum of a whole number and a fraction. When we need to add or subtract mixed numbers, we deal with the whole numbers separately from the fractions, and we find a common denominator for the fractions.
Read MoreGiven two points A and B in three-dimensional space, we can calculate the distance between them using the distance formula. It doesn’t matter which point is A and which point is B. The fact that we square the differences inside the square root means that all of our values will be positive, which means we’ll get a positive value for the distance between the points.
Read MoreNow we want to look at what happens when we combine two data sets, either by adding them or subtracting them. When we’re combining multiple linear random variables, we can find the mean and standard deviation of the combination using the means and standard deviations of the individual variables.
Read MoreThe inverse of an invertible linear transformation T is also itself a linear transformation. Which means that the inverse transformation is closed under addition and closed under scalar multiplication. In other words, as long as the original transformation T is a linear transformation itself, and is invertible (its inverse is defined, you can find its inverse), then the inverse of the inverse, T, is also a linear transformation.
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Read MoreIn this lesson we’ll look at how to graph linear inequalities on a coordinate plane. To graph a linear inequality, first graph the boundary line. The boundary line will be dashed if the symbol is < or >. The boundary line will be solid if the symbol is ≤ or ≥.
Read MoreImproper integrals are just like definite integrals, except that the lower and/or upper limit of integration is infinite. Remember that a definite integral is an integral that we evaluate over a certain interval. An improper integral is just a definite integral where one end of the interval is +/-infinity.
Read MoreYou can think about radicals (also called “roots”) as the opposite of exponents. We already know that the expression x^2 with the exponent of 2 means “multiply x by itself two times”. The opposite operation would be “what do we have to multiply by itself two times in order to get x^2?”
Read MoreWe already know how to find critical points of a multivariable function and use the second derivative test to classify those critical points. But sometimes we’re asked to find and classify the critical points of a multivariable function that’s subject to a secondary constraint equation.
Read MoreIn this lesson we’ll look at how to use the properties of perpendicular and angle bisectors to find out more information about geometric figures. An angle bisector goes through the vertex of an angle and divides the angle into two congruent angles that each measure half of the original angle.
Read MoreRemember that for a binomial random variable X, we’re looking for the number of successes in a finite number of trials. For a geometric random variable, most of the conditions we put on the binomial random variable still apply: 1) each trial must be independent, 2) each trial can be called a “success” or “failure,” and 3) the probability of success on each trial is constant.
Read MoreIn the last section we talked about the regression line, and how it was the line that best represented the data in a scatterplot. In this section, we’re going to get technical about different measurements related to the regression line.
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