Can you solve and verify a linear equation every time, even with fractions and signs in the way? This lesson test puts that to work across 15 original ASVAB-style questions drawn from the Linear Equations lesson. Items include two-step signed equations, distribution, cleared denominators, decimal coefficients, identities and contradictions, formula rearrangement and short geometry applications.
Distribute carefully, combine like terms and isolate the variable. Which value of x satisfies
2(3x−5)−(4x−7)=3(x+2)−11
Show the answer
Answer: A
Correct answer: x = 2. Distribute on both sides, flipping every sign after the minus:
6x − 10 − 4x + 7 = 3x + 6 − 11
Combine like terms:
2x − 3 = 3x − 5
Subtract 2x and add 5 to get x = 2.
Check: the left side is
2(1) − (1) = 1
and the right side is
3(4) − 11 = 1
A (correct): Distributing gives 6x − 10 − 4x + 7 = 3x + 6 − 11, which simplifies to 2x − 3 = 3x − 5, so x = 2.
B: This comes from writing −(4x − 7) as −4x − 7, flipping only the first sign inside the parentheses.
C: This comes from writing 3(x + 2) as 3x + 2, multiplying only the first term by 3.
D: This comes from writing 2(3x − 5) as 6x − 5, multiplying only the first term by 2.
Practice 2 of 24 · medium
If
11x−9=5x+15
then x equals which of the following?
Show the answer
Answer: C
Correct answer: x = 4. Move the smaller x term: subtract 5x from both sides to get
6x − 9 = 15
Add 9 to get 6x = 24, then divide by 6: x = 4. Check:
44 − 9 = 35
and
20 + 15 = 35
A: This comes from subtracting 9 from 15 instead of adding 9 to both sides, giving 6x = 6.
B: This comes from adding 5x to both sides instead of subtracting it, giving 16x = 24.
C (correct): Subtracting 5x and adding 9 gives 6x = 24, so x = 4.
D: This is the right size but the wrong sign; 11(−4) − 9 = −53, while 5(−4) + 15 = −5.
Practice 3 of 24 · medium
Clearing the decimals in
0.15x+2.4=0.6x−0.3
and solving gives which value of x?
Show the answer
Answer: B
Correct answer: x = 6. Because 0.15 has two decimal places, multiply every term by 100:
15x + 240 = 60x − 30
Subtract 15x and add 30: 270 = 45x, so x = 6.
Check:
0.9 + 2.4 = 3.3
and
3.6 − 0.3 = 3.3
A: This comes from multiplying the x terms by 100 but the constants only by 10, which unbalances the equation.
B (correct): Multiplying every term by 100 gives 15x + 240 = 60x − 30, so 270 = 45x and x = 6.
C: This is a decimal slip: 270 was divided by 4.5 instead of 45.
D: This comes from a sign slip when moving the constants, such as writing 240 + 30 as −270.
Practice 4 of 24 · easy
What value of x makes
4x+3=8
true?
Show the answer
Answer: B
Correct answer: 20. Undo the operations in reverse order. Subtract 3 from both sides:
x/4 = 5
Multiply both sides by 4: x = 20. Check:
20/4 + 3 = 5 + 3 = 8
A: This comes from adding 3 to 8 instead of subtracting it, then multiplying 11 by 4.
B (correct): Subtract 3 to get x/4 = 5, then multiply by 4 to get 20.
C: This is x/4, not x; the last step of multiplying by 4 was skipped.
D: This comes from dividing 5 by 4 instead of multiplying by 4 to undo the division.
Practice 5 of 24 · hard
For what value of k does the equation
3(2x−k)=6x+12
have infinitely many solutions?
Show the answer
Answer: D
Correct answer:
k = −4
Distribute:
6x − 3k = 6x + 12
The x terms already match, so the equation is an identity only when the constants also match:
−3k = 12
so
k = −4
Then both sides read 6x + 12, which is true for every x.
A: With k = 4 the left side is 6x − 12, and 6x − 12 = 6x + 12 reduces to −12 = 12, a contradiction with no solution.
B: This sets k equal to the constant instead of −3k; with k = −12 the equation becomes 6x + 36 = 6x + 12, which has no solution.
C: This divides 12 by 6 instead of by −3; with k = 2 the equation reduces to −6 = 12, which has no solution.
D (correct): Distributing gives 6x − 3k = 6x + 12, so −3k must equal 12, which makes k = −4 and both sides identical.
Practice 6 of 24 · easy
Which of the following is an equation rather than an expression?
Show the answer
Answer: B
Correct answer:
5y − 8 = 12
An equation states that two expressions are equal, so it must contain an equals sign. Only
5y − 8 = 12
does, and solving it gives y = 4. The others are expressions that can only be simplified or evaluated.
A: This has no equals sign; it can be simplified or evaluated but not solved.
B (correct): It contains an equals sign, so it claims two quantities are equal and can be solved (y = 4).
C: This has no equals sign; it simplifies to 5x but there is nothing to solve.
D: This has no equals sign; it can be expanded to 7n − 7 but not solved.
Practice 7 of 24 · hard
What is the solution of
2x+3−5x−1=5
Show the answer
Answer: D
Correct answer: x = 11. The LCD of 2 and 5 is 10. Multiply every term, including the 5:
5(x + 3) − 2(x − 1) = 50
Distribute:
5x + 15 − 2x + 2 = 50
so
3x + 17 = 50
3x = 33 and x = 11.
Check:
14/2 − 10/5 = 7 − 2 = 5
A: This comes from multiplying the fractions by 10 but leaving the 5 on the right side unchanged, giving 3x + 17 = 5.
B: This comes from writing −2(x − 1) as −2x − 2, forgetting that a negative times −1 gives +2.
C: This comes from multiplying only the x by −2 and writing −2(x − 1) as −2x − 1.
D (correct): Multiplying every term by 10 gives 5(x + 3) − 2(x − 1) = 50, so 3x + 17 = 50 and x = 11.
Practice 8 of 24 · medium
Which number, substituted for x, makes the equation
15−4x=43
a true statement?
Show the answer
Answer: A
Correct answer:
x = −7
Subtract 15 from both sides:
−4x = 28
Divide both sides by −4, keeping the sign:
x = −7
Check:
15 − 4(−7) = 15 + 28 = 43
A (correct): Subtracting 15 gives −4x = 28, and dividing by −4 gives x = −7.
B: This divides 28 by 4 but drops the negative sign of the coefficient −4.
C: This comes from adding 15 to 43 instead of subtracting it, then dividing by 4 without the sign.
D: This comes from adding 15 to 43 instead of subtracting it before dividing by −4.
Practice 9 of 24 · medium
Eight less than four times a number is equal to the number plus 19. What is the number?
Show the answer
Answer: D
Correct answer: 9. Let n be the number. Eight less than four times a number is 4n − 8, so
4n − 8 = n + 19
Subtract n and add 8: 3n = 27, so n = 9. Check:
36 − 8 = 28
and
9 + 19 = 28
A: This is the value of 3n; the final step of dividing by 3 was skipped.
B: This comes from writing 4(n − 8), which means four times the difference, instead of 4n − 8.
C: This comes from writing 4n + 8, treating less than as addition.
D (correct): The equation 4n − 8 = n + 19 gives 3n = 27, so n = 9.
Practice 10 of 24 · medium
The sum of four consecutive even integers is 108. What is the smallest of the four integers?
Show the answer
Answer: D
Correct answer: 24. Consecutive even integers differ by 2: n, n + 2, n + 4, n + 6. Their sum is
4n + 12 = 108
so 4n = 96 and n = 24. The integers are 24, 26, 28 and 30, which add to 108.
A: This is the largest of the four integers (24, 26, 28, 30), not the smallest.
B: This is the average, 108 ÷ 4, which falls between the middle two integers and is not even.
C: This comes from spacing the integers by 1 instead of 2, giving 4n + 6 = 108.
D (correct): With n, n + 2, n + 4 and n + 6, the sum is 4n + 12 = 108, so n = 24.
Practice 11 of 24 · medium
A rectangular dog run is fenced on all four sides with 78 feet of fencing. Its length is twice its width. What is the length of the dog run?
Show the answer
Answer: C
Correct answer: 26 feet. Let the width be w, so the length is 2w. The perimeter is
2(2w) + 2w = 6w = 78
which gives w = 13 feet. The length is 2(13) = 26 feet. Check:
2(26) + 2(13) = 52 + 26 = 78
A: This is the width; the question asks for the length, which is twice the width.
B: This is half the perimeter, the sum of one length and one width.
C (correct): With width w and length 2w, the perimeter is 2(2w) + 2w = 6w = 78, so w = 13 and the length is 26 feet.
D: This divides 78 by 4 as if the run were a square with four equal sides.
Practice 12 of 24 · hard
The formula
C=95(F−32)
converts degrees Fahrenheit to degrees Celsius. Which formula gives F in terms of C?
Show the answer
Answer: C
Correct answer:
F = 9/5
·C + 32. Undo the operations on F in reverse order. Multiply both sides by 9/5, the reciprocal of 5/9: (9/5)
C = F − 32
Add 32 to both sides:
F = (9/5)
C + 32.
Check with C = 100: (9/5)
(100) + 32 = 212
the boiling point of water in Fahrenheit.
A: This adds 32 correctly but keeps 5/9 instead of multiplying by its reciprocal 9/5.
B: This uses the reciprocal correctly but subtracts 32 instead of adding it to undo F − 32.
C (correct): Multiplying both sides by 9/5 gives (9/5)C = F − 32, and adding 32 isolates F.
D: This adds 32 before undoing the 5/9 factor, so 32 is wrongly multiplied by 9/5.
Practice 13 of 24 · hard
What value of x solves
52x−1=3x+4
Show the answer
Answer: B
Correct answer: x = 23. Cross-multiply, keeping each numerator in parentheses:
3(2x − 1) = 5(x + 4)
Distribute:
6x − 3 = 5x + 20
Subtract 5x and add 3: x = 23.
Check:
(46 − 1)/5 = 9
and
(23 + 4)/3 = 9
A: This comes from dropping the parentheses, writing 6x − 1 = 5x + 4, so only the x terms were multiplied.
B (correct): Cross-multiplying gives 3(2x − 1) = 5(x + 4), so 6x − 3 = 5x + 20 and x = 23.
C: This multiplies each numerator by its own denominator, 5(2x − 1) = 3(x + 4), instead of by the other denominator.
D: This comes from multiplying the left numerator fully but writing 5(x + 4) as 5x + 4.
Practice 14 of 24 · medium
Using the reciprocal of the coefficient, find x when
−43x+5=−7
What is x?
Show the answer
Answer: D
Correct answer: x = 16. Subtract 5 from both sides: −(3/4)
x = −12
Multiply both sides by the reciprocal −4/3:
x = (−12)
(−4/3) = 16
Check: −(3/4)
(16) + 5 = −12 + 5 = −7
A: This multiplies by 4/3 instead of −4/3, dropping the sign of the coefficient.
B: This multiplies −12 by −3/4, the coefficient itself, instead of by its reciprocal.
C: This multiplies −12 by 3/4 instead of by the reciprocal −4/3.
D (correct): Subtracting 5 gives −(3/4)x = −12, and multiplying by −4/3 gives x = 16.
Practice 15 of 24 · easy
Which single step changes
x−9=14
into
x=23
while keeping the equation balanced?
Show the answer
Answer: A
Correct answer: Add 9 to both sides. Use the inverse operation: addition undoes subtraction. Adding 9 to both sides gives
x − 9 + 9 = 14 + 9
so x = 23. Whatever you do to one side must also be done to the other.
A (correct): Adding 9 undoes the subtraction of 9 and is applied to both sides, giving x = 14 + 9 = 23.
B: Subtracting 9 gives x − 18 = 5; it repeats the operation instead of undoing it.
C: Changing only one side breaks the balance, so the new equation no longer has the same solution.
D: Multiplying gives 9x − 81 = 126; multiplication does not undo subtraction.
Practice 16 of 24 · medium
After combining like terms on the left side, which value of x satisfies
4x+7−x+2=30
Show the answer
Answer: C
Correct answer: x = 7. Combine like terms on the left first:
4x − x = 3x
and
7 + 2 = 9
so
3x + 9 = 30
Subtract 9: 3x = 21. Divide by 3: x = 7. Check:
28 + 7 − 7 + 2 = 30
A: This comes from treating −x as +x, giving 5x + 9 = 30.
B: This comes from adding 9 to 30 instead of subtracting it, giving 3x = 39.
C (correct): Combining like terms gives 3x + 9 = 30, so 3x = 21 and x = 7.
D: This ignores the constants 7 and 2 and solves 3x = 30.
Practice 17 of 24 · medium
A student solved
3(x−4)+10=x+8
and got x = 1. Which mistake most likely produced that answer?
Show the answer
Answer: B
Correct answer: Multiplying 3 by x but not by −4. The correct work is
3x − 12 + 10 = x + 8
so
3x − 2 = x + 8
2x = 10 and x = 5. If the 3 multiplies only the x, you get
3x − 4 + 10 = x + 8
so 2x = 2 and x = 1. Substituting x = 1 into the original gives
−9 + 10 = 1
on the left but 9 on the right, so the check exposes the error.
A: No step in this equation requires dividing by a negative number, so this error cannot occur here.
B (correct): Writing 3x − 4 + 10 = x + 8 gives 2x + 6 = 8, so x = 1; the correct work gives x = 5.
C: There is no minus sign in front of the parentheses here; the 3 is positive.
D: Changing only the left side gives 3x − 12 = x + 8, which leads to x = 10, not x = 1.
Practice 18 of 24 · hard
Which equation has no solution?
Show the answer
Answer: A
Correct answer:
4(x - 2) + 3 = 4x - 1
Simplify each side. An equation has no solution when the x terms cancel and a false statement remains. For
4(x − 2) + 3 = 4x − 1
the left side is 4x − 5, and subtracting 4x leaves
−5 = −1
which is never true.
A (correct): It simplifies to 4x − 5 = 4x − 1; subtracting 4x leaves −5 = −1, which is false.
B: It simplifies to 6x = 6x, which is always true, so every real number is a solution.
C: It simplifies to 5x − 5 = 4x − 5, which gives x = 0; zero is a valid single solution.
D: It simplifies to 6x − 3 = 5x − 2, which has exactly one solution, x = 1.
Practice 19 of 24 · medium
Dividing first is the quickest route here. If
−4(x−6)=52
what is x?
Show the answer
Answer: A
Correct answer:
x = −7
Since 52 is a multiple of −4, divide first instead of distributing:
x − 6 = 52 ÷ (−4) = −13
Add 6:
x = −7
Check:
−4(−7 − 6) = −4(−13) = 52
A (correct): Dividing both sides by −4 gives x − 6 = −13, so x = −7.
B: This divides by 4 instead of −4, giving x − 6 = 13.
C: After x − 6 = −13, this subtracts 6 instead of adding 6.
D: This is the correct size with the wrong sign; −4(7 − 6) = −4, not 52.
Practice 20 of 24 · medium
The answer to
6x−5=2x+5
is not a whole number. Which value of x makes the equation true?
Show the answer
Answer: C
Correct answer:
x = 5/2
Subtract 2x from both sides:
4x − 5 = 5
Add 5: 4x = 10. Divide by 4:
x = 10/4
which reduces to 5/2, or 2.5. Check:
6(2.5) − 5 = 10
and
2(2.5) + 5 = 10
A: This comes from adding 2x to both sides instead of subtracting it, giving 8x = 10.
B: This inverts the fraction, dividing 4 by 10 instead of 10 by 4.
C (correct): Subtracting 2x and adding 5 gives 4x = 10, so x = 10/4 = 5/2.
D: This comes from subtracting 5 on both sides so the constants cancel, instead of adding 5 to undo −5.
Practice 21 of 24 · easy
A number plus 5, all multiplied by 3, gives 27, as in
3(x+5)=27
What is x?
Show the answer
Answer: D
Correct answer: x = 4. Distribute:
3x + 15 = 27
Subtract 15: 3x = 12. Divide by 3: x = 4. You can also divide both sides by 3 first:
x + 5 = 9
so x = 4.
A: This comes from writing 3(x + 5) as 3x + 5, multiplying only the x by 3.
B: This comes from dividing by 3 to get x + 5 = 9 and then adding 5 instead of subtracting it.
C: This subtracts 3 from 27 and then divides by 3, mixing up which number multiplies the parentheses.
D (correct): Distributing gives 3x + 15 = 27, so 3x = 12 and x = 4.
Practice 22 of 24 · easy
A plumber charges a $65 service fee plus $48 per hour of work. A customer's bill came to $281. How many hours did the plumber work?
Show the answer
Answer: A
Correct answer: 4.5 hours. Let h be the hours worked:
65 + 48h = 281
Subtract 65 from both sides: 48h = 216. Divide by 48: h = 4.5 hours. Check:
65 + 48(4.5) = 65 + 216 = 281
A (correct): The equation 65 + 48h = 281 gives 48h = 216, so h = 4.5.
B: This divides the whole bill, $281, by $48 without first removing the $65 fee.
C: This adds the $65 fee to the bill instead of subtracting it, then divides $346 by $48.
D: This removes the fee correctly but divides $216 by the fee, $65, instead of the hourly rate.
Practice 23 of 24 · easy
Which value of x is a solution of
9−2x=3
Show the answer
Answer: C
Correct answer: 3. Subtract 9 from both sides:
−2x = −6
Divide by −2: x = 3. Check by substitution:
9 − 2(3) = 3
so both sides match.
A: Substituting gives 9 − 2(−3) = 15, not 3; the sign was lost when dividing by −2.
B: This is 9 − 3, the value of 2x, before dividing by 2; substituting gives 9 − 12 = −3.
C (correct): Substituting gives 9 − 2(3) = 9 − 6 = 3, which matches the right side.
D: This adds 3 to 9 instead of subtracting; substituting gives 9 − 24 = −15.
Practice 24 of 24 · medium
Simple interest is given by
I=Prt
where P is the principal, r is the annual rate and t is the time in years. Which formula correctly solves for r?
Show the answer
Answer: B
Correct answer:
r = I/(Pt)
In I = Prt, the letter r is multiplied by P and by t. Treat P and t like numbers and divide both sides by Pt:
r = I/(Pt)
For example, if I = 90, P = 1,500 and t = 2, then
r = 90/3,000 = 0.03
or 3%.
A: This is the reciprocal of the correct expression; it divides Pt by I instead of I by Pt.
B (correct): Since r is multiplied by both P and t, dividing both sides by Pt isolates r.
C: This subtracts Pt, but r is multiplied by Pt, so division is the inverse operation needed.
D: This multiplies by Pt instead of dividing, which repeats the operation rather than undoing it.
About this test
Each question is paired with an explanation that solves it in clear steps and checks the result by substitution, the same habit the lesson recommends. Reading these explanations is one of the quickest ways to tighten your process.
Use the outcome to decide what to revisit. Misses on fraction problems call for another look at Clearing Several Denominators; confusion about no-solution cases points to that section and Worked Example 5. When you want more reps, 24 extra practice questions are linked to this lesson before you move on to inequalities.
The same 124 lessons, 6,516 questions and 82 timed practice tests on your phone, plus flashcards with spaced review and progress tracking. Free to download, and it works offline.
ASVAB Camp
English ▾