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Writing Equations

Algebra: Writing Equations

Read the following SAT test question and then select the correct answer. 

Always read the question carefully and identify the bottom line.  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that your answer matches the bottom line; the specific question the problem asked you to solve.

The c cars in a car service use a total of g gallons of gasoline per week. If each of the cars uses the same amount of gasoline, then, at this rate, which of the following represents the number of gallons used by 5 of the cars in 2 weeks?

Bottom line: gal in 2 wks = ?

Assess your Options:  You could try to work backwards from the answer choices by plugging in a number for each variable, but you want to avoid working from the answer choices when you do not have to.  Instead, write an equation using the information that you are given in the problem.

Attack the Problem:  Start with the most basic information that you are given and logically translate the words into a math problem.  You know that c stands for cars and g stands for gallons of gasoline.  If all of the cars use the same amount of gasoline, then the total number of gallons must be divided evenly among each of the cars:



Now you know that there are 5 cars.  You might be tempted to put the 5 with the c, but think about it this way: that would mean that the same number of gallons was divided among more cars, so each car was using less gasoline, which is impossible!   If there are more cars, the total amount of gasoline must increase:



Now all you have to do is turn 1 week into 2 weeks by multiplying both sides of your equation by 2:



Loop Back: You found the gallons for 2 weeks, so look down at your answer choices.

(A) ten times c times g
(B) (2 times g) over (5 times c)
(C) (5 times g) over (2 times c)
(D) g over (10 times c)
(E)(10 times g) over c

The correct answer is (E).

Alternative method using Knowsys strategies:  If you struggle with writing equations, choose a number to represent the variable you are given in the problem.  You know you have 5 cars, but pick a number to represent the gallons that these cars use.  Any number that is not already in the problem will work; avoid  0 or 1 because multiple equations may work with these choices. Let’s say that g = 10.  In one week, those 5 cars will use 10 gallons.  How many gallons will they use in 2 weeks?  20 gallons!

Notice that the problem is a “which of the following” question.  Start with answer choice E instead of A.

Plug in the 10 for g and the 5 for c.  10 times 10 is 100, and then if you divide 100 by 5, you get 20.  That matches the answer that you found, so E must be correct.  None of the other answer choices will equal 20.  Strategies are tools to help you – remember that you get the same number of points for the correct answer no matter how you work the problem!


On sat.collegeboard.org, 31% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Functions

Algebra: Functions

Read the following SAT test question and then select the correct answer.

Use the same method for every math question on the SAT.  Start by reading the question carefully and identifying the bottom line; what do you need to find?  Then assess your options and choose the most efficient method to attack the problem.  When you have an answer, loop back to make sure that it matches the bottom line.

In the xy-plane, the graph of the line with equation y = a intersects the graph of the quadratic function f(x) = x² - 6x + 8 in exactly one point. What is the value of a?

Bottom Line: a = ?

Assess your Options:  You could just try plugging this into your calculator, but if you do not think carefully about what you are doing, you are likely to answer a question that was not asked.  Instead, think through every piece of information that you were given in this problem.

Attack the Problem:  What kind of graph is the function that you are given?  A parabola!  You know this because it has an x².  Picture a parabola in your mind (you know that this is a normal, upward-facing parabola because there is no negative before the x²).  Draw a u-shaped parabola on the xy-axis as part of your scratch work.

Now think about the fact that when y equals a certain number, it creates a vertical line. No matter what y equals, that vertical line will only ever intercept the graph at one point. That's not very useful! However, try flipping the given equation on its head: consider a = y. Remember that a =  is just like x =  and will create a horizontal line. Depending on what x equals, the horizontal line might cross the graph at two points, at no point at all, or at exactly one point--the vertex. You know that you must find the vertex of the parabola, so solve your function for x by setting your polynomial equal to zero and finding the roots of the equation:

x² - 6x + 8 = 0
(x – 2)(x – 4) = 0
(x – 2) = 0 and (x – 4) = 0
x = 2 and x = 4

You just found the two places where the parabola crosses the x-axis: 2 and 4.  All parabolas are symmetrical.  That means that the vertex must be halfway between these two numbers at x = 3.  You found the x value of the vertex, but you need the y value. 

Plug in 3 for the x in your original equation:

f(x) = x² - 6x + 8
f(3) = (3)² - 6(3) +8
f(3) = 9 – 18 + 8
f(3) = -1

Loop Back:  When you solve a function for the f(x), you solve for y.  In this problem, you are told that y = a.  You have solved for a, so you are ready to look down at your answer choices.

(A) -3
(B) -1
(C) 1
(D) 3
(E) 4

The correct answer is (B).


On sat.collegeboard.org, 31% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

Graphing Functions

Algebra: Graphing Functions:

Read the following SAT test question and then select the correct answer.

Always use the Knowsys Method on all math questions. This will help you think systematically and avoid careless mistakes. First, read the entire question carefully. Identify the bottom line and note it at the top of your scratch work. Next assess your options: What could I do? What should I do? Choose the most efficient method to attack the problem, and loop back to make sure that your answer matches the bottom line you were looking for.



In the xy-plane, the graph of the equation above assumes its maximum value at x = 2. What is the value of b?

First, consider your bottom line. "What is the value of b?" At the top of your scratch work, write b = ?

Next, start assessing your options. What does it mean that the chart "assumes its maximum" at x = 2? Look at all the parts of the function. The highest power is 2, so you know that this is a quadratic function and that the chart will have a parabola. Since the coefficient of that variable is -2, you also know that the parabola will open downward. If the graph's maximum value is located at x = 2, you know that the vertex of the parabola will be somewhere to the right of the origin, on the vertical line two spaces to the right of the y-axis.

What can you do with that knowledge? Think about how you can move a parabola to the right of the origin. You might remember the formula . If you've forgotten, the point (h, k) represents the vertex of the parabola. You need to combine this with the function you were originally given.



Next, use FOIL and the Distributive Property to square the binomial (x - 2) and multiply in the coefficient.



At this point, you should stop and double-check your bottom line. You don't need to worry about solving for k, x, or y because you have already solved what your bottom line was asking: the value of b. Always keep your bottom line in mind so you remember to loop back and so you can be sure you answer what was asked.

Now that you know that b = 8, look at the answer choices:

(A) -8
(B) -4
(C) 4
(D) 8
(E) 10

The correct answer is D.


On sat.collegeboard.org, 35% of responses were correct.


For more help with math, visit www.myknowsys.com!