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:

　　If x – 4 = y – 3 and 2y = 6, what is x?

　　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.

　　Suppose 3x = y + 5 and 2y – 2= 12k. Solve for x in terms of k.

　　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:

　　Suppose 2x + 3y = 5 and –1x – 3y = –7. What is x?

　　In this particular problem, you can find the value of x by adding the two equations together:　　Here is another example:

　　6x + 2y = 11 and 5x + y = 10. What is x + y?

　　By subtracting the second equation from the first:　　Some test-takers might have seen this problem and been tempted to immediately start trying to solve for x and y individually. The better test-taker notices that by subtracting the second equation from the first, the answer is given.　　Give this last example a try:

　　2x + 3y = –6 and –4x + 16y = 13. What is the value of y?

　　The question asks you to solve for y, which means that you should find a way to eliminate one of the variables by adding or subtracting the two equations. 4x is simply twice 2x, so by multiplying the first equation by 2, you can then add the equations together to find y.

　　2　(2x + 3y = –6) = 4x + 6y = –12

　　Now add the equations and solve for y.　　When you solve for one variable, like we have in this last example, you can solve for the second variable using either of the original equations. If the last question had asked you to calculate the value of xy, for example, you could solve for y, as above, and then solve for x by substitution into either equation. Once you know the independent values of x and y, you can multiply them together.　　Simultaneous equations on the Math IC will all be this simple. They will have solutions that can be found easily by adding or subtracting the equations given. Only as a last resort should you solve for one variable in terms of the other and then plug that value into the other equation to solve for the second variable.