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SAT Study Guides
3.5 SAT Math - Simultaneous Equations
There is another instance in which you will want to line two equations on top of each other to solve: simultaneous equations. Simultaneous equations look a lot like elimination, but don’t be fooled! With simultaneous equations, you will NOT be solving for one variable, you will be solving for a whole expression.
Example Simultaneous Equation Problems
6. If 7x + 3y = 12and x - y = 10, what is 6x + 4y?
The key to these is to simply decide whether you will be adding or subtracting. Here, we will subtract:
| 7x + 3y = 12 |
| - (x - y = 10) |
| 6x + 4y =2 |
Notice anything? We put the bottom equation in parentheses. Why? To reinforce that if you are subtracting, you MUST distribute that negative! Which not only means 7x + (-)x or 7x - x, but also 3y - (-)y or 3y + y.
Simultaneous equations, like elimination, may need to be multiplied through to make it work.
-Hint: You shouldn’t need to multiply it through by anything much more difficult than 2, 3, -4 or the like. If it’s not obvious right away which way you should go, experiment. If you put equation 1 on top and add and that doesn’t work, try subtracting, then try subtracting with equation 2 on top. Simultaneous equations generally work themselves out to the final expression you want pretty quickly.
Example Problem 2
10. If a + 5b = 22 and 4a + 3b = 18, what is 9a + 11b?
Let’s see what possibilities we have. Adding will only give us 5a; subtracting definitely won’t help. Multiplying the first equation doesn’t seem to help either, but multiplying the second equation by 2 will give us 9a once we get to the adding stage. Let’s see if that works all the way through:
2(4a + 3b) = 18(2)
8a + 6b = 36
Looks like it should work, so let’s try it:
| a + 5b =22 |
| + 8a + 6b = 26 |
| 9a + 11b = 48 |
Yup, it worked!
-SAT math tip: If you’re not sure whether or not to use elimination/substitution or simultaneous equations to solve a problem, look at what they’re asking for. If they’re asking for more than one variable (6x + 4y, a/b, xw, etc.) it will ALWAYS be easier to solve for that expression than it will be to solve for each variable separately!
Example Problem 3
20. If j and k are positive and
| = 8-1, then |
| = |
It is certainly possible to solve for j and solve for k, but doing so is far more difficult than simply solving for the expression asked for. Do you know the answer?
(A) 1/18
(B) 1/6
(C) 2
(D) 8
(E) 18
The answer is (C) 2.
Giving you an expression to solve for instead of a single variable is actually one of the few little gifts the testmakers give you – they’re giving you a big hint on how they want you to solve it, and therefore what concept they’re testing. The example above is testing the definition of negative exponents, and exponents with fractions. But it’s also testing whether you can spot shortcuts and cut right to the information required.
Example Problem 4
In a word problem, simply use a variable for any unknowns in the problem.
Tom is 2 years younger than twice Jane's age. If Jane will be 14 in two years, how old is Tom?
The unknowns here are Tom and Jane. Pick easy variables to remember like t and j instead of always using x and y. You would set this problem up by reading word for word, left to right, using your variables for the unknowns:
‘Tom’: t
‘is’: =
‘2 years younger than’: [something] minus 2
‘twice’: 2 times
‘Jane’s age’: j
Where does 2 times j go? In the [something] spot. So your equation now reads:
t = 2j -2
Then use the next sentence to figure out what to do next. If Jane’s age now is j, her age in two years will be 2 + j, correct? So the next equation is 2 + j = 14. Then solve for those variables just like in the problems above.
You can also use one variable for the one thing you are solving for, and put everything else in those terms.
Example Problem 5
Amy is 18 months older than her brother Sean and three years older than her sister Katie. If the sum of their ages is 20 years, how old is Amy?
For which person would you choose to use the variable? You could pick any of them, but it’s probably wisest to use Amy as your variable, since that’s who the question is asking for. So let’s make Amy a.
What would the other two people be in terms of that variable? Amy is 18 months older than her brother Sean and three years older than her sister Katie. See how they’re trying to trick you? You need to either put all of them in months, or all of them in years before you can do anything else. Let’s put everyone in years. Why? Because the information given is in years. (If you’re not sure how to put 18 months into years, you need to set up a ratio – how many months in one year? Put the number of pieces you’re dividing the main unit into, in this case years, in the denominator, and the number you’re given in the numerator. You now have the fraction 18/12 which reduces to the decimal 1.5.)
So Amy is 1.5 years older than Sean and 3 years older than Katie. What’s Sean’s age in terms of a? What’s Katie’s?
Sean should be a - 1.5 and Katie should be a - 3.
Now write the equation adding them all together:
a + (a - 1.5) + (a - 3) = 20
And solve for your variable:
a + a - 1.5 + a - 3 = 2 -
3a - 4.5 = 20
3a = 24.5
a = 24.5/3
Once you get to this stage with a, you can plug 24.5/3 into your calculator to get 8.16 - a very messy answer when asking for someone’s age. You have to remember what they were talking about, and why they gave someone’s age in months. Keep in mind (based on what you know about remainders) that a remainder of .16 means something. What does it mean? That Amy is 8 years old with a remainder of a few months. So we need to find out how many. You could do the remainder trick to get the whole number that .16 represents, but unfortunately, that’s going to represent a different number than the number of months extra that Amy has because the denominator was not 12 – the number of months in a year. So what do you do? Try putting it into a mixed number. 8.16 as a mixed number is
| 8 |
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