Systems of equations are multiple equations that all have a common solution. Students encounter these systems of equations when there are multiple ‘unknowns’ - or variables - that have not been given to them yet. When this happens, the goal for students is to use given information in the equations to solve for all variables. To solve systems of equations, it is helpful for students to have a background understanding of simple algebraic equations, variables, and graphing linear equations. How do I solve a system of equations? There are three methods used to solve systems of equations: graphing, substitution, and elimination. To solve a system by graphing, you simply graph the given equations and find the point(s) where they all intersect. The coordinate of this point will give you the values of the variables that you are solving for. This is most efficient when the equations are already written in slope-intercept form. The next method is substitution. Substitution is best used when one of the equations is in terms of one of the variables such as y=2x+4, but the equations can always be manipulated. The first step in this method is to solve one of the equations for one variable. Once an expression for the variable is found, substitute or plug in the expression into the other equation where the original variable was to solve for the number value of the next variable. The final step is to substitute the number value that was found in for its corresponding variable in the original equation. The third method is elimination. Elimination is adding the equations together in order to create an equation with only one variable. This can only be done when the coefficients of one variable in both equations are opposites and will cancel each other out once added together. Elimination is best used when this is already occurring in the equations, but the equations can also be manipulated into creating common coefficients by either multiplying or dividing equations by a certain number. The next step would be to use the equation that we created to find the value of the variable and then plug that value back into an original equation to find the remaining variable. Here’s an example of a problem where solving a system of equations is necessary: Logan has answered 0.8 as many math questions as Spanish questions, and he’s answered 5 more English questions than Spanish questions. If Logan has answered 33 questions in total, how many math questions has Logan answered? How do I solve this problem? The first step is to create equations from the word problem. To do this, we must assign variables to each unknown part of the problem. The variables x, y, and z will represent the amount of math, Spanish, and English questions that Logan has answered respectively. As Logan has answered 0.8 times as many math questions as Spanish ones, the equation to represent this would be 0.8y=x. The second equation would be z=y+5 to represent how Logan has answered five more English questions than Spanish ones. The final equation would be x+y+z=33 to represent how Logan answered 33 questions in total. Looking back at the original question, the goal of this problem is to find how many math questions Logan has answered. Since the first equation we found was 0.8y=x, we can see that we only need the y variable to find the value of x, or the amount of math questions that were answered. Because there are two equations already solved in terms of two variables, 0.8y=x and z=y+5, substitution would be the most efficient method. To do this method, we would substitute these equations for the x and y variables of the third equation which would give us (0.8y)+y+(y+5)=33. The next step would be to solve this equation for the y variable by combining like terms: 2.8y=28 which would give us y=10 or 10 Spanish questions answered. Now that we have found the value of the y variable, we can plug it back into the equation, 0.8y=x, to find the value of x. By substituting y with its value of 10, we would get 0.8(10)=x which would give us a value of 8 for x. What’s the answer? Logan answered 8 math questions. What concepts did we use? To solve that example problem, we used several different math concepts. The first one we used was how to write equations from word problems. Through our understanding of the problem, we were able to assign each unknown aspect of the problem a variable and then create equations based off of their relationships in the problem which we would then recognize as a system of equations. The second concept that we used is solving the system. The equations weren’t written in slope-intercept form, so graphing would not have been an efficient method. There were no variables that were opposite each other, so we also ruled out the elimination method. By recognizing substitution as the best method to use, we were able to efficiently use our math skills to solve for the unknown variables in a system of equations. If this is what you want for your child - It's easy to get started!Call Now (303) 502-8345 and Learn More!
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Updated November 02, 2020 By Daniel Pinzow
Functions are relations that derive one output for each input, or one y-value for any x-value inserted into the equation. For example, the equations: y = x + 3 \text{ and } y = x^3 - 1
are functions because every x-value produces a different y-value. In graphical terms, a function is a relation where the first numbers in the ordered pair have one and only one value as its second number, the other part of the ordered pair.
An ordered pair is a point on an x-y coordinate graph with an x and y-value. For example, (2, −2) is an ordered pair with 2 as the x-value and −2 as the y-value. When given a set of ordered pairs, ensure that no x-value has more than one y-value paired to it. When given the set of ordered pairs [(2, −2), (4, −5), (6, −8), (2, 0)], you know that this is not a function because an x-value – in this case – 2, has more than one y-value. However, this set of ordered pairs [( −2, 4), ( −1, 1), (0, 0), (1, 1), (2, 4)] is a function because a y-value is allowed to have more than one corresponding x-value.
It is relatively easy to determine whether an equation is a function by solving for y. When you are given an equation and a specific value for x, there should only be one corresponding y-value for that x-value. For example y = x + 1
is a function because y will always be one greater than x. Equations with exponents can also be functions. For example: y = x^2 - 1
is a function; although x-values of 1 and −1 give the same y-value (0), that is the only possible y-value for each of those x-values. However: y^2 = x + 5
is not a function; if you assume that x = 4, then y^2 = 4 + 5 = 9 \\ y^2 = 9
has two possible answers (3 and −3).
Determining whether a relation is a function on a graph is relatively easy by using the vertical line test. If a vertical line crosses the relation on the graph only once in all locations, the relation is a function. However, if a vertical line crosses the relation more than once, the relation is not a function. Using the vertical line test, all lines except for vertical lines are functions. Circles, squares and other closed shapes are not functions, but parabolic and exponential curves are functions.
An input-output chart displays the output, or result, for each input, or original value. Any input-output chart where an input has two or more different outputs is not a function. For example, if you see the number 6 in two different input spaces, and the output is 3 in one case and 9 in another, the relation is not a function. However, if two different inputs have the same output, it is still possible that the relation is a function, especially if squared numbers are involved. |