Algebra problem

For -2k / (k + 1) = 3/4, find the value of k.

We need to cross-multiply to solve for k. We then have -8k = 3k + 3. Subtracting 3k from both sides, we get -11k = 3, which gives k = -3/11.

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Data Analysis problem

*CALCULATOR REQUIRED

The following table shows points (x, y) on the same line. Which of the following choices shows variable y in terms of variable x for the line?

A. y = -5x/2 - 12.15
B. y = -2x/5 - 2.28
C. y = -2x/5 - 1.48
D. y = -5x/2 - 11.35


Since every point on the table (x, y) shares the same line, we can determine the slope of the line using any pair of three available points. We will use the first two for convenience. The slope is found by the formula (change in y) / (change in x) = [0 - 0.4] / (-3.7 - (-4.7)] = -0.4 / 1 = -2/5. This rules out choices A and D. We now have the equation in the form y = -2x/5 + b, where b is the y-intercept or point on the line where x = 0, that is, (0, b). Using the point-slope formula (and the first point on the table) we have the equation as y - 0.4 = -2/5[x - (-4.7)], which simplifies to y = -2x/5 - 1.88 + 0.4, or y = -2x/5 - 1.48. Be sure to check this equation with all of the given points x above to see if you are getting the corresponding y values. Thus the correct answer choice is (C).

You may also plug in each equation in the answer choices if you forget how to set up your equation, but it may take extra time. You may, however, take a shortcut after you find the correct slope -2/5 to check answers (B) and (C) with the table values to find the correct y-intercept.

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A hybrid Algebra-Geometry Problem

A parabola, whose equation is ax² + bx + 3 = y, has a vertex at (-1, -2). Find the value of a + b.

We know that any parabola can also be expressed in the form of a(x – c)² + d = y, where (c, d) is the point of the vertex of the parabola. In this example c = -1, and d = -2. Thus we have the equation in the vertex form as a(x –  (-1))² + (-2) = y. Setting this expression for y and the given expression for y equal, we have a(x + 1)² – 2 = ax² + bx + 3. Simplifying the left side, we have ax² + 2ax + a – 2 = ax² + bx + 3. Since the two sides are equal, we can compare their coefficients. The second and third terms provide with information we need, so we have 2a = b, and a – 2 = 3, which gives a = 5, and b = 2(5) = 10. Thus a + b = 5 + 10 = 15.

The second way of solving this problem is to remember that the x-coordinate of the vertex of a parabola is given by -b/2a, so that -1 = -b/2a, so that 2a = b. Now, we can plug in the x-coordinate of the given vertex point into the given equation to get a second expression for constants a and b: a(-1)² + b(-1) + 3 = -2, which gives a – b = -5. Substituting 2a = b into this equation, we have a – 2a = -5, so that a = 5, and b = 10 as before. 

Always check the constants you found with the vertex coordinate in the original equation to see if you are correct. If you are not sure how to use either of the two methods, skip this question on the exam and come back to it.  

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The Redesigned Math SAT: should I be afraid?

You do NOT have to be afraid. There are plenty of great books out there. However, being a mathematician myself, I tend to sort out books before I buy them. Now I write my own study guides.

 The new SAT Math exam coming up in 2016 will require the knowledge of trigonometry, geometry, sequences, logic, algebra, word problems, graphical reasoning, data analysis, among other topics.

 Typically, word problems must be solved in the following way: you are given information in words, and you are required to put it in mathematical form.

 Example word problem: Rita is 8 years older than John was 3 years ago. Tom will be twice the current age of John in 4 years time. How old is Tom now if Rita is 25?

 Solution: let r be Rita's age now, j John's age now, and t Tom's age now. Then we have r = j - 3 + 8 (we got this from sentence "Rita is 8 years older than John was 3 years ago."), t + 4 = 2j (from "Tom will be twice the current age of John in 4 years time."). Since r = 25, we have 25 = j - 3 + 8, which means that j = 25 - 5 = 20, and t = 2(20) - 4 = 40 - 4 = 36. Thus Tom is 36 years old.

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Please visit our website RevisedSAT.com to practice for free.

The Redesigned Math SAT: should I be afraid?

You do NOT have to be afraid. There are plenty of great books out there. However, being a mathematician myself, I tend to sort out books before I buy them. Now I write my own study guides.

 The new SAT Math exam coming up in 2016 will require the knowledge of trigonometry, geometry, sequences, logic, algebra, word problems, graphical reasoning, data analysis, among other topics.

 Typically, word problems must be solved in the following way: you are given information in words, and you are required to put it in mathematical form.

 Example word problem: Rita is 8 years older than John was 3 years ago. Tom will be twice the current age of John in 4 years time. How old is Tom now if Rita is 25?

 Solution: let r be Rita's age now, j John's age now, and t Tom's age now. Then we have r = j - 3 + 8 (we got this from sentence "Rita is 8 years older than John was 3 years ago."), t + 4 = 2j (from "Tom will be twice the current age of John in 4 years time."). Since r = 25, we have 25 = j - 3 + 8, which means that j = 25 - 5 = 20, and t = 2(20) - 4 = 40 - 4 = 36. Thus Tom is 36 years old.

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Please visit our website RevisedSAT.com to practice for free.