SATII数学考试快递——Systems of Equations
Sometimes, a question will have a lone equation containing two variables, and using the methods we’ve discussed up until now will not be enough to solve for the variables. Additional information is needed, and it must come in the form of another equation.
Say, for example, that a single equation uses the two variables x and y. Try as you might, you won’t be able to solve for x or y. But given another equation with the same two variables x and y, then the values of both variables can be found.
These multiple equations containing the same variables are called systems of equations. For the Math IC, there are essentially two types of systems of equations that you will need to be able to solve. The first, easier type involves substitution, and the second involves manipulating equations simultaneously.
Substitution
Simply put, substitution is when the value of one variable is found and then substituted into the other equation to solve for the other variable. It can be as easy as this example:
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In this case, we have two equations. The first equation contains x and y. The second contains only y. To solve for x, you must solve for y in the second equation and substitute that value for y in the first equation. If 2y = 6, then y = 3, and then x = y – 3 + 4 = 3 – 3 + 4 = 4.
Here is a slightly more complicated example.
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Again, you cannot solve for x in terms of k using just the first equation. Instead, you must solve for y in terms of k in the second equation, and then substitute that value in the first equation to solve for x.
Then substitute y = 6k + 1 into the equation 3x = y + 5.
Simultaneous Equations
Simultaneous equations refer to equations that can be added or subtracted from each other in order to find a solution. Consider the following example:
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In this particular problem, you can find the value of x by adding the two equations together:
