Air Force Officer Qualifying Test (AFOQT) Quiz: Solve Algebraic Equations

20 questions · exam conditions

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Solve Algebraic EquationsQuestion 1 of 20

Fuel purchase: $x$ gallons at base, $y$ gallons en route; $x+y=500$ , $3x+2y=1300$ ; factor $x^2-15x+56$ .

x=300,\ y=200;\ (x-7)(x-8)

x=200,\ y=300;\ (x-7)(x-8)

x=300,\ y=200;\ (x-4)(x-14)

x=250,\ y=250;\ (x-7)(x-8)

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Air Force Officer Qualifying Test (AFOQT) Quiz

Air Force Officer Qualifying Test (AFOQT) Quiz: Solve Algebraic Equations

Practice Solve Algebraic Equations in Air Force Officer Qualifying Test (AFOQT) with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Solve Algebraic Equations, giving you a quick way to practice the rules, question types, and explanations that matter most for Air Force Officer Qualifying Test (AFOQT).

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Fuel purchase: $x$ gallons at base, $y$ gallons en route; $x+y=500$ , $3x+2y=1300$ ; factor $x^2-15x+56$ .

$x=300,\ y=200;\ (x-7)(x-8)$ (correct answer)
$x=200,\ y=300;\ (x-7)(x-8)$
$x=300,\ y=200;\ (x-4)(x-14)$
$x=250,\ y=250;\ (x-7)(x-8)$

Explanation: This question tests solving simultaneous equations and factoring expressions, critical for AFOQT Math Knowledge. Solving simultaneous equations involves finding common solutions where equations intersect, while factoring expressions simplifies them for easier manipulation. In this passage, equations represent fuel gallons at base and en route, requiring precise calculations to ensure accuracy. Choice A is correct because it accurately solves the equations with x=300, y=200 and factors the expression as (x-7)(x-8), demonstrating understanding of algebraic manipulation in context. Choice B is incorrect due to a common miscalculation in arithmetic, often seen when mis-substituting. To teach this skill, practice identifying key variables and constants in word problems and using systematic approaches to solve equations. Emphasize understanding the role of each expression component in contextual scenarios.

Question 2

If $3x - 2y = 13$ and $x + y = 6$ , what is the value of $x^2 - y^2$ ?

19
22
24 (correct answer)
26
35

Explanation: This problem tests your ability to solve systems of equations and recognize algebraic patterns. When you see an expression like

x^2 - y^2

, immediately think about the difference of squares factorization:

x^2 - y^2 = (x+y)(x-y)

. You're given two equations:

3x - 2y = 13

and

x + y = 6

. Notice that you already have

(x+y) = 6

, so you just need to find

(x-y)

. To find

x-y

, solve the system. From the second equation,

x = 6 - y

. Substitute into the first equation:

3(6-y) - 2y = 13

, which gives

18 - 3y - 2y = 13

, so

-5y = -5

and

y = 1

. Therefore,

x = 6 - 1 = 5

. Now

x - y = 5 - 1 = 4

. Using the factorization:

x^2 - y^2 = (x+y)(x-y) = (6)(4) = 24

. Choice A (19) might result from incorrectly adding

x^2 + y^2 = 25 + 1 = 26

then subtracting something randomly. Choice C (26) is exactly

x^2 + y^2

, a common sign error. Choice D (35) could come from miscalculating during the substitution process or arithmetic errors. The correct answer is B. Strategy tip: When you see

x^2 - y^2

in a system of equations problem, always check if you can use the difference of squares formula

(x+y)(x-y)

rather than solving for individual variables first. This approach is faster and reduces calculation errors.

Question 3

The sum of two numbers is 28. Three times the smaller number is 6 more than the larger number. What is the product of the two numbers?

28
123.75
150
165.75 (correct answer)
192

Explanation: When you encounter word problems involving two unknown numbers with given relationships, you need to set up a system of equations to solve systematically. Let's call the smaller number

x

and the larger number

y

. From the problem, you can write two equations:

The sum equals 28: $x + y = 28$
Three times the smaller is 6 more than the larger: $3x = y + 6$

From the second equation, rearrange to get

y = 3x - 6

. Substitute this into the first equation:

x + (3x - 6) = 28

. Simplifying:

4x - 6 = 28

, so

4x = 34

, and

x = 8.5

. Therefore

y = 28 - 8.5 = 19.5

. The product is

8.5 \times 19.5 = 165.75

. Let's verify:

8.5 + 19.5 = 28

✓ and

3(8.5) = 25.5

, which is indeed 6 more than 19.5 ✓ Looking at the wrong answers: A) 28 is simply the sum of the two numbers, not their product—this tests whether you confused the operations. B) 123.75 might result from calculation errors in the system of equations or incorrect substitution. D) 192 could come from assuming the numbers are integers and making algebraic mistakes, perhaps thinking the numbers are 12 and 16. For AFOQT word problems, always define your variables clearly, set up the system methodically, and verify your solution against both original conditions. Double-checking prevents careless errors that lead to trap answers.

Question 4

When $6x^3 - 9x^2 + 4x - 6$ is factored completely, which of the following is one of its factors?

$2x + 3$
$3x^2 - 2$
$3x^2 + 2$ (correct answer)
$6x^2 - 2$
2x - 3

Explanation: When you encounter polynomial factoring questions on the AFOQT, look for opportunities to group terms and factor systematically. This expression has four terms, making it perfect for factoring by grouping. Start by grouping the first two terms and the last two terms:

(6x^3 - 9x^2) + (4x - 6)

. From the first group, factor out the greatest common factor

3x^2

3x^2(2x - 3)

. From the second group, factor out

2

2(2x - 3)

. Notice that both groups now contain the factor

(2x - 3)

. This gives you:

3x^2(2x - 3) + 2(2x - 3) = (3x^2 + 2)(2x - 3)

. You can verify this by expanding:

(3x^2 + 2)(2x - 3) = 6x^3 - 9x^2 + 4x - 6

✓ The complete factorization is

(3x^2 + 2)(2x - 3)

, making choice (C)

3x^2 + 2

correct. Choice (A)

2x + 3

has the wrong sign - the actual linear factor is

(2x - 3)

. Choice (B)

3x^2 - 2

also has an incorrect sign in the constant term. Choice (D)

6x^2 - 2

could tempt you if you mistakenly tried to factor out 2 from the entire expression first, but this doesn't work since not all terms are divisible by 2. For AFOQT factoring problems, always verify your answer by expanding back to the original expression. Factoring by grouping is especially useful when you have four terms that don't share a common factor across all terms.

Question 5

If $2x^2 + 5x - 12 = 0$ , what is the sum of the possible values of x?

-2.5 (correct answer)
2.5
-6
6
12

Explanation: When you encounter a quadratic equation and need to find the sum of its roots, you can solve this efficiently using the relationship between coefficients and roots, rather than solving the entire equation. For any quadratic equation in the form

ax^2 + bx + c = 0

, the sum of the roots equals

-\frac{b}{a}

. In the equation

2x^2 + 5x - 12 = 0

, we have

a = 2

and

b = 5

, so the sum of the roots is

-\frac{5}{2} = -2.5

. To verify this works, you could factor the equation:

2x^2 + 5x - 12 = (2x - 3)(x + 4) = 0

, giving roots

x = \frac{3}{2}

and

x = -4

. Indeed,

\frac{3}{2} + (-4) = \frac{3}{2} - \frac{8}{2} = -\frac{5}{2} = -2.5

. Answer A (-2.5) is correct. Answer B (2.5) represents a sign error—you might get this if you forgot the negative sign in the formula

-\frac{b}{a}

. Answer C (-6) could result from incorrectly using

-\frac{c}{a} = -\frac{(-12)}{2} = 6

and then applying a wrong sign, confusing the sum formula with the product formula. Answer D (6) comes from calculating

-\frac{c}{a}

, which actually gives you the product of the roots, not their sum. Remember: for

ax^2 + bx + c = 0

, sum of roots =

-\frac{b}{a}

and product of roots =

\frac{c}{a}

. This coefficient relationship saves time compared to fully solving the quadratic.

Question 6

Given the system of equations: $x+y+z=6$ , $2x-y+z=3$ , and $x+y-z=0$ , what is the value of y?

0
1
2 (correct answer)
3
6

Explanation: When you encounter a system of three linear equations with three unknowns, you need to use systematic elimination or substitution to solve for each variable. This tests your ability to manipulate equations strategically. Let's solve this step by step using the given equations:

Equation 1: $x + y + z = 6$
Equation 2: $2x - y + z = 3$
Equation 3: $x + y - z = 0$

First, add equations 1 and 3 to eliminate

z

(x + y + z) + (x + y - z) = 6 + 0

2x + 2y = 6

x + y = 3

(Equation 4) Next, subtract equation 1 from equation 2 to eliminate

z

(2x - y + z) - (x + y + z) = 3 - 6

x - 2y = -3

(Equation 5) Now solve the simpler 2×2 system using equations 4 and 5: From equation 4:

x = 3 - y

Substitute into equation 5:

(3 - y) - 2y = -3

3 - 3y = -3

-3y = -6

y = 2

Looking at the wrong answers: A)

y = 1

would give

x = 2

and

z = 2

, but this doesn't satisfy equation 2. C)

y = 3

would make

x = 0

, leading to contradictions when checking all three equations. D)

y = 6

would require

x = -3

, which also fails to satisfy the original system. Strategy tip: Always verify your solution by substituting back into all original equations. Systems of equations on the AFOQT often have answer choices that satisfy only one or two equations, so complete verification is essential.

Question 7

Which of the following represents the complete factorization of $12a^3b^2 - 18a^2b^3 + 6a^2b^2$ ?

$6a^2b^2(2a - 3b + 1)$ (correct answer)
$6a^2b^2(2a - 3b)$
$2a^2b^2(6a - 9b + 3)$
$6ab(2a^2b - 3ab^2 + ab)$
3a^2b^2(4a - 6b + 2)

Explanation: When you encounter polynomial factorization problems, you're looking to pull out the greatest common factor (GCF) first, then check if further factoring is possible. To find the complete factorization of

12a^3b^2 - 18a^2b^3 + 6a^2b^2

, start by identifying the GCF of all terms. Look at the coefficients: 12, 18, and 6 have a GCF of 6. For the variables, the lowest power of

a

a^2

and the lowest power of

b

b^2

. So the GCF is

6a^2b^2

. Factor this out:

6a^2b^2(2a - 3b + 1)

. You can verify this by distributing:

6a^2b^2 \cdot 2a = 12a^3b^2

6a^2b^2 \cdot (-3b) = -18a^2b^3

, and

6a^2b^2 \cdot 1 = 6a^2b^2

. This matches the original expression, confirming that choice A is correct. Choice B,

6a^2b^2(2a - 3b)

, is missing the constant term. When expanded, this only gives you the first two terms of the original expression. Choice C,

2a^2b^2(6a - 9b + 3)

, uses an incorrect GCF for the coefficients—while the factored form is mathematically equivalent, it's not the complete factorization since you could still factor out 3 from the parentheses. Choice D,

6ab(2a^2b - 3ab^2 + ab)

, doesn't pull out the full GCF of the variable terms. Remember: complete factorization means pulling out the greatest possible common factor. Always double-check by expanding your answer to ensure it matches the original expression.

Question 8

Given the equations $\frac{1}{2}x + \frac{1}{3}y = 4$ and $x - y = 2$ , find the value of $x+y$ .

1.2
5.6
6.8
8.4
9.2 (correct answer)

Explanation: This problem tests your ability to solve a system of two linear equations with two unknowns. When you encounter systems like this, you have several solution methods available: substitution, elimination, or graphing. Let's use substitution since the second equation is already solved for one variable in terms of the other. From

x - y = 2

, we get

x = y + 2

. Substituting this into the first equation:

\frac{1}{2}(y + 2) + \frac{1}{3}y = 4

Expanding:

\frac{1}{2}y + 1 + \frac{1}{3}y = 4

Combining the y terms:

\frac{3}{6}y + \frac{2}{6}y = 3

, which gives us

\frac{5}{6}y = 3

Solving for y:

y = 3 \times \frac{6}{5} = \frac{18}{5} = 3.6

Now finding x:

x = y + 2 = 3.6 + 2 = 5.6

Therefore:

x + y = 5.6 + 3.6 = 9.2

Answer choice A (1.2) likely results from calculation errors when combining fractions. Choice B (5.6) is the value of x alone, representing the common mistake of stopping partway through the problem. Choice C (8.4) could come from sign errors or incorrect fraction arithmetic during the solving process. The correct answer is D (9.2). Strategy tip: Always verify your solution by substituting both values back into the original equations. This catches arithmetic mistakes and ensures you've answered what the question actually asks for—here, the sum

x + y

, not individual variable values.

Question 9

Given $x + 2y = 5$ and $3x + y = 5$ , what is the value of $4x+3y$ ?

3
5
8
10 (correct answer)
15

Explanation: When you encounter a system of linear equations asking for the value of a specific expression, you don't always need to solve for individual variables. Instead, look for ways to combine the given equations to directly find what you're asked for. You have

x + 2y = 5

and

3x + y = 5

, and you need

4x + 3y

. Notice that

4x + 3y

can be written as

(x + 2y) + (3x + y)

. Since you know both expressions in parentheses equal 5, you get

4x + 3y = 5 + 5 = 10

. Let's verify by solving the system completely. From the first equation:

x = 5 - 2y

. Substituting into the second:

3(5 - 2y) + y = 5

, which gives

15 - 6y + y = 5

, so

-5y = -10

and

y = 2

. Then

x = 5 - 2(2) = 1

. Therefore

4x + 3y = 4(1) + 3(2) = 10

. Choice A (3) might tempt you if you incorrectly subtract the equations instead of adding them. Choice B (5) could result from thinking the answer equals one of the given constants. Choice D (15) might come from calculation errors when solving the system, such as forgetting to distribute negative signs properly. On the AFOQT, always check whether you can manipulate given equations directly to find the target expression before solving for individual variables. This approach saves time and reduces calculation errors—both crucial for standardized test success.

Question 10

One of the factors of $6x^2 - 11x - 10$ is $(3x+2)$ . What is the other factor?

$2x + 5$
$3x - 5$
$2x - 5$ (correct answer)
$2x - 10$
3x - 10

Explanation: When you encounter a factoring problem where one factor is given, you're dealing with polynomial division. Since you know that

(3x + 2)

is a factor of

6x^2 - 11x - 10

, you can find the other factor by dividing the polynomial by the known factor. Let's call the unknown factor

(ax + b)

. Since

(3x + 2)(ax + b) = 6x^2 - 11x - 10

, we can work backwards. The leading coefficient

6x^2

comes from

3x \cdot ax = 3ax^2

, so

3a = 6

, which means

a = 2

. The constant term

-10

comes from

2 \cdot b = 2b

, so

2b = -10

, which means

b = -5

. Let's verify:

(3x + 2)(2x - 5) = 6x^2 - 15x + 4x - 10 = 6x^2 - 11x - 10

✓ The correct answer is C)

2x - 5

. Looking at the wrong answers: A)

2x + 5

would give you

6x^2 + 21x - 10

when multiplied out—the middle term has the wrong sign. B)

3x - 5

would produce

9x^2 - 9x - 10

—the leading coefficient is wrong. D)

2x - 10

would yield

6x^2 - 26x - 20

—both the middle term and constant are incorrect. Study tip: When one factor is given, use the fact that the leading coefficients and constant terms must multiply correctly. This gives you a quick way to check your work without full polynomial multiplication.

Question 11

Consider the system of equations: 3x - 2y = 7 and 4y - 6x = -14. Which statement best describes the solution to this system?

The system has no solution.
The system has exactly one solution at the origin (0,0).
The system has exactly one solution, but not at the origin.
The system has exactly two distinct solutions.
The system has infinitely many solutions. (correct answer)

Explanation: When you encounter a system of linear equations, you need to determine whether the lines intersect at one point, are parallel (no intersection), or are actually the same line (infinitely many solutions). Let's analyze this system by rewriting both equations in standard form. The first equation is already there:

3x - 2y = 7

. For the second equation

4y - 6x = -14

, rearrange to get

-6x + 4y = -14

, or equivalently

6x - 4y = 14

. Now notice something important: if you multiply the first equation by 2, you get

6x - 4y = 14

. This is exactly the second equation! The two equations are actually the same line written in different forms. When this happens, every point on the line satisfies both equations simultaneously, meaning there are infinitely many solutions. Looking at the wrong choices: Choice A suggests no solution, which occurs when lines are parallel but distinct. Choice B claims the unique solution is at the origin, but substituting (0,0) into

3x - 2y = 7

gives

0 = 7

, which is false. Choice C suggests one non-origin solution, which would mean the lines intersect at exactly one point. Choice D mentions two distinct solutions, but linear equations can't have exactly two solutions. Since the problem lists the correct answer as E (infinitely many solutions), but E isn't shown among the choices, this appears to be a formatting issue with the question. Strategy tip: Always check if one equation is a multiple of another—this immediately tells you the lines are identical, yielding infinitely many solutions.

Question 12

What is the complete factorization of $50x^2 - 32y^2$ ?

$(5x - 4y)(10x + 8y)$
$(5x - 4y)(5x + 4y)$
$2(25x^2 - 16y^2)$
$2(5x - 4y)(5x + 4y)$ (correct answer)
2(5x - 4y)²

Explanation: When you encounter an expression with two squared terms separated by subtraction, you're looking at a difference of squares pattern. However, before applying any factoring techniques, always check for a greatest common factor (GCF) first. Looking at

50x^2 - 32y^2

, notice that both coefficients are even: 50 and 32. The GCF of these numbers is 2, so factor that out first:

2(25x^2 - 16y^2)

. Now you have

25x^2 - 16y^2

inside the parentheses, which is a perfect difference of squares since

25x^2 = (5x)^2

and

16y^2 = (4y)^2

. The difference of squares formula is

a^2 - b^2 = (a-b)(a+b)

, so

25x^2 - 16y^2 = (5x-4y)(5x+4y)

. The complete factorization is

2(5x-4y)(5x+4y)

, which is answer D. Choice A,

(5x-4y)(10x+8y)

, expands to

50x^2 + 40xy - 40xy - 32y^2 = 50x^2 - 32y^2

, which seems correct but isn't fully factored since you can still factor out 2 from the second binomial. Choice B,

(5x-4y)(5x+4y)

, gives you

25x^2 - 16y^2

, missing the factor of 2. Choice C,

2(25x^2 - 16y^2)

, is only partially factored since the difference of squares hasn't been applied. Always factor completely by first finding the GCF, then applying special patterns like difference of squares. This two-step approach prevents missing factors.

Question 13

The expression $9x^2 - 30xy + 25y^2$ is equivalent to which of the following?

$(3x - 5y)^2$ (correct answer)
$(3x + 5y)^2$
$(3x - 5y)(3x + 5y)$
$(9x - 5y)(x - 5y)$
$(9x - 25y)^2$

Explanation: When you encounter a trinomial expression like this, you're looking at a potential perfect square trinomial. The key is recognizing the pattern

a^2 - 2ab + b^2 = (a-b)^2

a^2 + 2ab + b^2 = (a+b)^2

. Let's examine

9x^2 - 30xy + 25y^2

. First, identify what could be squared:

9x^2 = (3x)^2

and

25y^2 = (5y)^2

. So we potentially have

(3x)^2

and

(5y)^2

as our first and last terms. For a perfect square trinomial

(3x - 5y)^2

, the middle term should be

-2(3x)(5y) = -30xy

. Since our expression has exactly

-30xy

as the middle term, this confirms that

9x^2 - 30xy + 25y^2 = (3x - 5y)^2

. Choice A is correct because it matches our factorization perfectly. Choice B gives

(3x + 5y)^2 = 9x^2 + 30xy + 25y^2

, which has a positive middle term instead of negative. Choice C represents a difference of squares formula:

(3x - 5y)(3x + 5y) = 9x^2 - 25y^2

, which eliminates the middle term entirely. Choice D expands to

9x^2 - 45xy - 5xy + 25y^2 = 9x^2 - 50xy + 25y^2

, giving an incorrect middle coefficient. Strategy tip: When factoring trinomials, always check if it's a perfect square by verifying that the middle term equals twice the product of the square roots of the first and last terms. This pattern appears frequently on standardized tests.

Question 14

For what value of k will the system of equations $4x - 6y = 10$ and $kx - 9y = 15$ have an infinite number of solutions?

-6
8/3
4
6 (correct answer)
2

Explanation: When you encounter a system of linear equations asking for infinite solutions, you're dealing with the concept of dependent equations - two equations that are actually the same line expressed differently. For a system to have infinite solutions, the equations must be proportional to each other. This means one equation is simply a multiple of the other. Let's examine the given system:

4x - 6y = 10

and

kx - 9y = 15

. To find the proportionality factor, look at the y-coefficients:

\frac{-9}{-6} = \frac{3}{2}

. This means the second equation should be

\frac{3}{2}

times the first equation. Multiplying the first equation by

\frac{3}{2}

\frac{3}{2}(4x - 6y) = \frac{3}{2}(10)

6x - 9y = 15

For the equations to be identical, we need

k = 6

. Let's verify: when

k = 6

, both equations become equivalent forms of the same line, giving us infinite solutions. Now for the wrong answers: Choice A (-6) would give us

-6x - 9y = 15

, which has the wrong sign on the x-term. Choice B (8/3) gives us

\frac{8}{3}x - 9y = 15

, which doesn't match our required coefficient of 6. Choice C (4) gives us

4x - 9y = 15

, which matches the original first equation's x-coefficient but creates inconsistent equations. Remember this key strategy: for infinite solutions, check if one equation can be transformed into the other by multiplication. Calculate the ratio using any coefficient pair, then verify it works for all terms.

Question 15

Which of the following is a factor of $16x^4 - 81y^8$ ?

$2x + 3y^2$ (correct answer)
$4x - 9y^2$
$2x - 3y^4$
$4x^2 + 3y^4$
8x^2 - 9y^4

Explanation: When you encounter an expression like

16x^4 - 81y^8

, recognize this as a difference of squares pattern:

a^2 - b^2 = (a+b)(a-b)

. Here,

16x^4 = (4x^2)^2

and

81y^8 = (9y^4)^2

, so you have

(4x^2)^2 - (9y^4)^2

. Applying the difference of squares formula:

16x^4 - 81y^8 = (4x^2 + 9y^4)(4x^2 - 9y^4)

. But notice that

4x^2 - 9y^4

is itself a difference of squares since

4x^2 = (2x)^2

and

9y^4 = (3y^2)^2

. So you can factor further:

4x^2 - 9y^4 = (2x + 3y^2)(2x - 3y^2)

. The complete factorization is:

16x^4 - 81y^8 = (4x^2 + 9y^4)(2x + 3y^2)(2x - 3y^2)

. Since

(2x + 3y^2)

appears as a factor, choice A is correct. Looking at the wrong answers: B)

4x - 9y^2

doesn't match any factor from our work—the coefficients and exponents are wrong. C)

2x - 3y^4

has the wrong exponent on

y

; it should be

3y^2

, not

3y^4

. D)

4x^2 + 3y^4

has the wrong coefficient on

y

; it should be

9y^4

, not

3y^4

. For AFOQT success, always look for difference of squares patterns when you see subtraction between perfect squares. Factor completely—sometimes you can apply the pattern twice, as in this problem.

Question 16

If $a+b=8$ and $ab=15$ , what is the value of $a^2+b^2$ ?

23
34 (correct answer)
49
64
94

Explanation: When you encounter a system of equations asking for an expression like

a^2 + b^2

, look for algebraic identities that connect what you're given to what you need to find. You have

a + b = 8

and

ab = 15

, and you need

a^2 + b^2

. The key insight is using the algebraic identity:

(a + b)^2 = a^2 + 2ab + b^2

. Rearranging this gives you

a^2 + b^2 = (a + b)^2 - 2ab

. Substituting your known values:

a^2 + b^2 = (8)^2 - 2(15) = 64 - 30 = 34

. This confirms answer choice A is correct. Let's examine why the other options are wrong. Choice B (49) might tempt you if you mistakenly calculated

(a + b)^2 - ab = 64 - 15 = 49

, forgetting the coefficient 2 in front of

ab

. Choice C (64) is simply

(a + b)^2

, which you'd get if you incorrectly thought

a^2 + b^2 = (a + b)^2

, ignoring the cross term

2ab

entirely. Choice D (94) could result from adding instead of subtracting:

(a + b)^2 + 2ab = 64 + 30 = 94

. For AFOQT success, memorize the core algebraic identities, especially

(a + b)^2 = a^2 + 2ab + b^2

and

(a - b)^2 = a^2 - 2ab + b^2

. When you see problems giving you sums and products of variables, immediately think about how these identities can help you find what's being asked without solving for the individual variables.

Question 17

If $2a - 3b = 7$ and $4a + b = 7$ , what is the value of the expression $a - b$ ?

-1
0
1
2
3 (correct answer)

Explanation: When you encounter a system of linear equations like this, you're being tested on your ability to solve for multiple variables and then use those values to evaluate a different expression. The key insight is that you don't always need to find each variable individually—sometimes you can manipulate the equations to find the target expression directly. Given the system

2a - 3b = 7

and

4a + b = 7

, you need to find

a - b

. Start by solving for the individual variables. From the second equation,

b = 7 - 4a

. Substitute this into the first equation:

2a - 3(7 - 4a) = 7

, which simplifies to

2a - 21 + 12a = 7

, so

14a = 28

and

a = 2

. Then

b = 7 - 4(2) = -1

. Therefore,

a - b = 2 - (-1) = 3

. Answer choice A) -1 is the value of

b

alone, which might tempt students who confuse the target expression. Answer choice B) 1 could result from incorrectly calculating

a - b

2 - 1 = 1

, perhaps from a sign error when finding

b

. Answer choice C) 2 is simply the value of

a

, another trap for students who lose track of what they're solving for. For AFOQT algebra problems involving systems of equations, always double-check your solution by substituting back into both original equations. Also, pay careful attention to what the question is actually asking for—it's common to solve for individual variables but then forget to evaluate the specific expression requested.

Question 18

If $5x = 3y + 16$ and $2x + y = 6$ , what is the value of the ratio $x/y$ ?

-1/17
34/11
-17 (correct answer)
17/2
11/34

Explanation: When you encounter a system of two equations with two unknowns, you need to solve for both variables first, then calculate the requested ratio. This tests your ability to manipulate linear equations systematically. Starting with the given equations:

$5x = 3y + 16$ ... (1)
$2x + y = 6$ ... (2)

From equation (2), solve for

y

y = 6 - 2x

Substitute this into equation (1):

5x = 3(6 - 2x) + 16

5x = 18 - 6x + 16

5x = 34 - 6x

11x = 34

x = \frac{34}{11}

Now find

y

y = 6 - 2(\frac{34}{11}) = 6 - \frac{68}{11} = \frac{66 - 68}{11} = -\frac{2}{11}

Therefore:

\frac{x}{y} = \frac{\frac{34}{11}}{-\frac{2}{11}} = \frac{34}{11} \times \frac{-11}{2} = -17

Choice A (-1/17) is the reciprocal of the correct answer with a sign error. Choice B (34/11) gives you just the value of

x

, not the ratio

x/y

. This is a common trap when students stop solving too early. Choice D (17/2) has the correct magnitude but the wrong sign, suggesting an error in tracking negative values during substitution. Always solve completely for both variables before calculating ratios, and double-check your arithmetic with negative numbers. Substitution errors are especially common under time pressure on the AFOQT.

Question 19

If the system of equations is defined by 4x + 7y = 15 and 7x + 4y = 18, what is the value of the expression 3x + y?

1
2
3
5
7 (correct answer)

Explanation: When you encounter a system of linear equations, you have several solution methods: substitution, elimination, or addition/subtraction. This problem tests whether you can solve the system efficiently and then evaluate a specific expression. Let's solve using the elimination method. Given:

$4x + 7y = 15$ (Equation 1)
$7x + 4y = 18$ (Equation 2)

Adding these equations together:

(4x + 7y) + (7x + 4y) = 15 + 18

This gives us:

11x + 11y = 33

Factoring:

11(x + y) = 33

Therefore:

x + y = 3

Now we can find individual values. Subtracting Equation 1 from Equation 2:

(7x + 4y) - (4x + 7y) = 18 - 15

3x - 3y = 3

x - y = 1

With

x + y = 3

and

x - y = 1

, we get

x = 2

and

y = 1

. Therefore:

3x + y = 3(2) + 1 = 7

Since 7 isn't among the given options A through D, the answer must be E (which isn't shown but would be the "none of the above" choice). Answer choice A (1) might tempt you if you calculated just

y

. Choice B (2) represents the value of

x

alone. Choice C (3) equals

x + y

, which is a key intermediate step but not what's asked. Choice D (5) could result from arithmetic errors in the solving process. Strategy tip: Always double-check your solution by substituting back into the original equations, and remember that "none of the above" is often correct when your careful work yields an answer not listed.

Question 20

What is one of the factors of the expression 6x² + 21x - 45?

x - 5
2x + 3
x + 3
6x - 9
2x - 3 (correct answer)

Explanation: When you encounter a quadratic expression that needs to be factored, start by looking for common factors among all terms, then use systematic factoring techniques. To factor

6x^2 + 21x - 45

, first notice that all coefficients (6, 21, -45) share a common factor of 3. Factor this out:

3(2x^2 + 7x - 15)

. Now factor the quadratic

2x^2 + 7x - 15

. You need two numbers that multiply to

(2)(-15) = -30

and add to

7

. Those numbers are

10

and

-3

. Rewrite the middle term:

2x^2 + 10x - 3x - 15

. Group and factor:

2x(x + 5) - 3(x + 5) = (2x - 3)(x + 5)

. Therefore:

6x^2 + 21x - 45 = 3(x + 5)(2x - 3)

The factors are

3

(x + 5)

, and

(2x - 3)

. However, none of these appear exactly in the choices given. Let's check each option by substitution or division:

Choice A: $(x - 5)$ has the wrong sign compared to our factor $(x + 5)$
Choice B: $(2x + 3)$ has the wrong sign in the constant term compared to $(2x - 3)$
Choice C: $(x + 3)$ would give us different coefficients when expanded with other factors
Choice D: $(6x - 9) = 3(2x - 3)$ , which includes the common factor 3, but this exact form isn't a simple factor

Since none of the given options A-D are actual factors of the expression, the correct answer must be E (none of the above). Strategy tip: Always factor out common factors first, then systematically factor the remaining quadratic. Double-check by expanding your factors back to the original expression.