HSPT Math Quiz: Solve Linear Equations

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Solve Linear EquationsQuestion 1 of 20

Solve the inequality $-4(2y-3)\le 8-6y.$

y\ge 2

y\le 2

y< -2

y> -2

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HSPT Math Quiz

HSPT Math Quiz: Solve Linear Equations

Practice Solve Linear Equations in HSPT Math 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 Linear Equations, giving you a quick way to practice the rules, question types, and explanations that matter most for HSPT Math.

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

Solve the inequality $-4(2y-3)\le 8-6y.$

$y\ge 2$ (correct answer)
$y\le 2$
$y< -2$
$y> -2$

Explanation: When solving linear inequalities, you follow the same steps as solving equations, with one crucial exception: flipping the inequality sign when multiplying or dividing by a negative number. Start by distributing the

-4

on the left side:

-4(2y-3) = -8y + 12

. The inequality becomes

-8y + 12 \leq 8 - 6y

. Next, collect like terms by adding

8y

to both sides:

12 \leq 8 - 6y + 8y

, which simplifies to

12 \leq 8 + 2y

. Subtract

8

from both sides:

4 \leq 2y

. Finally, divide both sides by

2

2 \leq y

, or equivalently,

y \geq 2

. Now let's examine why the other answers are incorrect. Choice B (

y \leq 2

) flips the inequality sign incorrectly—this would happen if you mistakenly thought you needed to flip the sign when dividing by positive 2. Choice C (

y < -2

) and Choice D (

y > -2

) both have the wrong number entirely. These errors likely stem from sign mistakes during the distribution step or when moving terms across the inequality. The correct answer is A:

y \geq 2

. Study tip: When solving inequalities, double-check your work by testing a value. For

y \geq 2

, try

y = 3

-4(6-3) = -12

and

8-18 = -10

. Since

-12 \leq -10

is true, your solution checks out. Always remember: only flip the inequality sign when multiplying or dividing by negative numbers.

Question 2

Solve for $x$ : $3(2x-5)+4 = 2(x+7)-2.$

$x=\dfrac{23}{4}$ (correct answer)
$x=\dfrac{21}{4}$
$x=\dfrac{25}{4}$
$x=\dfrac{23}{5}$

Explanation: This equation tests your ability to solve linear equations with parentheses and variables on both sides. When you encounter equations like this, your goal is to systematically eliminate parentheses, combine like terms, and isolate the variable. Start by distributing on both sides:

3(2x-5)+4 = 2(x+7)-2

becomes

6x-15+4 = 2x+14-2

. Simplifying the constants gives you

6x-11 = 2x+12

. Next, collect all terms with

x

on one side and constants on the other. Subtract

2x

from both sides:

4x-11 = 12

. Add 11 to both sides:

4x = 23

. Finally, divide by 4:

x = \frac{23}{4}

. Choice A,

x=\frac{23}{4}

, is correct. Choice B,

x=\frac{21}{4}

, likely results from an arithmetic error when combining constants—perhaps getting

4x = 21

instead of

4x = 23

. Choice C,

x=\frac{25}{4}

, could come from incorrectly handling the signs when distributing, leading to

4x = 25

. Choice D,

x=\frac{23}{5}

, suggests an error in collecting like terms, possibly getting

5x = 23

instead of

4x = 23

. Remember to work methodically: distribute first, combine like terms on each side, then move all variable terms to one side and constants to the other. Double-check your arithmetic at each step, especially when dealing with negative numbers and distribution. These multi-step equations reward careful, systematic work over speed.

Question 3

Solve the inequality $7-2x>3x+12.$ Which of the following is equivalent?

$x< -1$ (correct answer)
$x> -1$
$x<1$
$x>1$

Explanation: When you encounter linear inequalities, your goal is to isolate the variable on one side while maintaining the inequality's direction. The key difference from equations is that multiplying or dividing by a negative number flips the inequality sign. Starting with

7-2x>3x+12

, collect all terms with

x

on one side and constants on the other. Subtract

3x

from both sides:

7-2x-3x>12

, which gives

7-5x>12

. Next, subtract 7 from both sides:

-5x>5

. Now divide both sides by

-5

. Since you're dividing by a negative number, the inequality sign flips from "greater than" to "less than":

x<-1

. This matches choice A. Let's examine why the other answers are wrong. Choice B (

x>-1

) results from forgetting to flip the inequality sign when dividing by

-5

— this is the most common error students make. Choice C (

x<1

) comes from sign errors during the algebraic manipulation, possibly treating

-2x

+2x

or making arithmetic mistakes with the constants. Choice D (

x>1

) combines both errors: incorrect algebraic manipulation AND forgetting to flip the inequality sign. You can verify your answer by testing a value: if

x=-2

(which satisfies

x<-1

), then

7-2(-2)=11

and

3(-2)+12=6

. Indeed,

11>6

. Remember: always flip the inequality sign when multiplying or dividing both sides by a negative number. This rule trips up many students, so double-check your work whenever negative coefficients appear.

Question 4

Solve for $x$ : $5x-7 = 3(x+1).$

$x = 3$
$x = 4$
$x = 2$
$x = 5$ (correct answer)

Explanation: This is a linear equation that requires you to isolate the variable

x

by using inverse operations and the distributive property. Start by expanding the right side using the distributive property:

3(x+1) = 3x + 3

. Now your equation becomes

5x - 7 = 3x + 3

. Next, collect all terms with

x

on one side and all constants on the other. Subtract

3x

from both sides:

5x - 3x - 7 = 3x - 3x + 3

, which simplifies to

2x - 7 = 3

. Add 7 to both sides:

2x - 7 + 7 = 3 + 7

, giving you

2x = 10

. Finally, divide both sides by 2:

x = 5

. You can verify this by substituting back into the original equation:

5(5) - 7 = 25 - 7 = 18

, and

3(5 + 1) = 3(6) = 18

. Both sides equal 18, confirming our answer. Choice A (

x = 3

) gives

8 = 12

, which is false. Choice B (

x = 4

) gives

13 = 15

, also incorrect. Choice C (

x = 2

) gives

3 = 9

, which doesn't work either. The correct answer is D. When solving linear equations, always work systematically: distribute first, then collect like terms, isolate the variable term, and finally solve for the variable. Always check your answer by substituting back into the original equation—this catches arithmetic errors and confirms you've solved correctly.

Question 5

The width of a rectangle is 4 feet less than twice its length. If the area of the rectangle is 96 square feet, what is the length of the rectangle?

The length is 8 feet (correct answer)
The length is 10 feet
The length is 12 feet
The length is 14 feet

Explanation: When you encounter a rectangle problem involving relationships between dimensions and area, you need to translate the word relationships into algebraic equations and then solve systematically. Let's call the length

l

feet. The problem states "width is 4 feet less than twice its length," so the width is

2l - 4

feet. Since area equals length times width:

l(2l - 4) = 96

. Expanding this equation:

2l^2 - 4l = 96

, or

2l^2 - 4l - 96 = 0

. Dividing by 2:

l^2 - 2l - 48 = 0

. Factoring this quadratic:

(l - 8)(l + 6) = 0

. This gives us

l = 8

l = -6

. Since length cannot be negative,

l = 8

feet. Let's verify: if length is 8 feet, width is

2(8) - 4 = 12

feet, and area is

8 \times 12 = 96

square feet. ✓ Choice A correctly identifies the length as 8 feet. Choice B (10 feet) would give a width of 16 feet and area of 160 square feet—too large. Choice C (12 feet) would give a width of 20 feet and area of 240 square feet—way too large. Choice D (14 feet) would give a width of 24 feet and area of 336 square feet—extremely too large. Strategy tip: Always set up your variable for what the question asks for (here, the length), then express other quantities in terms of that variable. After solving, verify your answer by checking that it satisfies the original area condition.

Question 6

Maria already has 12 stamps and purchases 3 more each week. The equation $12+3w=45$ shows when she will own 45 stamps. How many weeks, $w$ , will that take?

$11$ (correct answer)
$9$
$13$
$7$

Explanation: This is a linear equation problem where you need to solve for the variable

w

. When you see an equation already set up like this, your job is to isolate the variable on one side. Starting with

12 + 3w = 45

, you need to get

w

by itself. First, subtract 12 from both sides to eliminate the constant term on the left:

3w = 45 - 12 = 33

. Then divide both sides by 3 to isolate

w

w = 33 ÷ 3 = 11

. So Maria will have 45 stamps after 11 weeks. Let's check why the other answers are wrong. Choice B (9 weeks) would give Maria

12 + 3(9) = 12 + 27 = 39

stamps—not enough. Choice C (13 weeks) would result in

12 + 3(13) = 12 + 39 = 51

stamps—too many. Choice D (7 weeks) would only give her

12 + 3(7) = 12 + 21 = 33

stamps—far short of her goal. Choice A (11 weeks) is correct because

12 + 3(11) = 12 + 33 = 45

stamps exactly. Remember to always verify your answer by substituting it back into the original equation. This catches arithmetic errors and confirms you solved correctly. On linear equation problems like this, the HSPT often includes answer choices that result from common mistakes like forgetting to subtract the initial amount or making simple calculation errors.

Question 7

Solve for $x$ : $0.6x-1.2 = 0.3(4x-2).$

$x=-1$ (correct answer)
$x=1$
$x=2$
$x=-2$

Explanation: When you encounter a linear equation with decimals and parentheses, your goal is to systematically simplify both sides until you can isolate the variable. Start by distributing the 0.3 on the right side:

0.3(4x-2) = 1.2x - 0.6

. Now your equation becomes

0.6x - 1.2 = 1.2x - 0.6

. Next, collect all terms with

x

on one side and constants on the other. Subtract

0.6x

from both sides:

-1.2 = 0.6x - 0.6

. Then add 0.6 to both sides:

-0.6 = 0.6x

. Finally, divide by 0.6:

x = -1

. Let's verify: substituting

x = -1

into the original equation gives

0.6(-1) - 1.2 = -1.8

on the left, and

0.3(4(-1) - 2) = 0.3(-6) = -1.8

on the right. Both sides equal

-1.8

, confirming our answer. Looking at the wrong choices: Choice B (

x = 1

) would give you

-0.6

on the left but

0.6

on the right. Choice C (

x = 2

) produces

0

on the left but

1.8

on the right. Choice D (

x = -2

) results in

-2.4

on the left but

-3

on the right. The correct answer is A. Study tip: When solving equations with decimals, work methodically through distribution, combining like terms, and isolating the variable. Always verify your answer by substituting back into the original equation—this catches arithmetic errors that are easy to make with decimal calculations.

Question 8

Solve for $z$ : $2(3z+4)=z-8.$

$z=-\dfrac{16}{5}$ (correct answer)
$z=-\dfrac{11}{5}$
$z=\dfrac{16}{5}$
$z=-\dfrac{5}{16}$

Explanation: When you encounter a linear equation with parentheses and variables on both sides, your goal is to isolate the variable through systematic algebraic steps. Start by distributing the 2 on the left side:

2(3z+4) = 6z + 8

. So the equation becomes

6z + 8 = z - 8

. Next, collect all terms with

z

on one side and constants on the other. Subtract

z

from both sides:

6z - z + 8 = -8

, which simplifies to

5z + 8 = -8

. Now subtract 8 from both sides:

5z = -16

. Finally, divide both sides by 5:

z = -\frac{16}{5}

. Let's examine why the other answers are incorrect. Choice B (

z = -\frac{11}{5}

) likely comes from an error in combining constants—perhaps incorrectly calculating

8 + (-8)

-11

instead of

-16

. Choice C (

z = \frac{16}{5}

) has the right magnitude but wrong sign, which could happen if you made a sign error when moving terms across the equals sign. Choice D (

z = -\frac{5}{16}

) flips the numerator and denominator, a common mistake when students confuse the final division step. You can verify your answer by substituting back into the original equation:

2(3(-\frac{16}{5}) + 4) = 2(-\frac{48}{5} + \frac{20}{5}) = 2(-\frac{28}{5}) = -\frac{56}{5}

, and

z - 8 = -\frac{16}{5} - \frac{40}{5} = -\frac{56}{5}

✓ Always check your work by substituting your solution back into the original equation—this catches algebraic errors and builds confidence in your answer.

Question 9

Solve for $t$ : $9-4(3-t)=5t+1.$

$t=-4$ (correct answer)
$t=4$
$t=-2$
$t=2$

Explanation: When you encounter a linear equation with parentheses and variables on both sides, your goal is to isolate the variable through systematic algebraic manipulation. Start by distributing the

-4

to everything inside the parentheses:

9-4(3-t) = 9-12+4t = -3+4t

. So the equation becomes

-3+4t = 5t+1

. Next, collect all terms with

t

on one side and constants on the other. Subtract

4t

from both sides:

-3 = t+1

. Then subtract

1

from both sides:

-4 = t

, so

t = -4

. Let's verify:

9-4(3-(-4)) = 9-4(7) = 9-28 = -19

and

5(-4)+1 = -20+1 = -19

. ✓ Looking at the wrong answers: Choice B (

t=4

) likely comes from a sign error when distributing the negative or when moving terms across the equals sign. Choice C (

t=-2

) might result from incorrectly combining like terms or making an arithmetic mistake during the solving process. Choice D (

t=2

) could occur if you forgot the negative sign when distributing

-4

through the parentheses, leading to

9-4(3-t) = 9-12-4t

instead of the correct

9-12+4t

. The correct answer is A. Strategy tip: Always distribute carefully through parentheses, especially when there's a negative sign in front. Double-check your work by substituting your answer back into the original equation—both sides should give the same value.

Question 10

Solve the inequality $\dfrac{2x+1}{4}\le 3.$

$x\le \dfrac{11}{2}$ (correct answer)
$x\ge \dfrac{11}{2}$
$x< -\dfrac{11}{2}$
$x> -\dfrac{11}{2}$

Explanation: When you encounter an inequality with a fraction, your goal is to isolate the variable while preserving the inequality's direction. The key principle is that you can perform the same operations on both sides of an inequality as you would with an equation, as long as you don't multiply or divide by a negative number. Starting with

\frac{2x+1}{4} \le 3

, multiply both sides by 4 to eliminate the fraction:

2x + 1 \le 12

. Next, subtract 1 from both sides:

2x \le 11

. Finally, divide both sides by 2:

x \le \frac{11}{2}

. Since we only performed operations with positive numbers, the inequality direction stays the same. Let's examine why the other choices are incorrect. Choice B gives

x \ge \frac{11}{2}

, which flips the inequality sign incorrectly—this would happen if you mistakenly thought dividing by positive 2 required flipping the sign. Choices C and D both show

x < -\frac{11}{2}

and

x > -\frac{11}{2}

respectively. These likely result from sign errors when subtracting 1, perhaps getting

2x \le -11

instead of

2x \le 11

. You can verify the correct answer A by testing a value: if

x = 5

, then

\frac{2(5)+1}{4} = \frac{11}{4} = 2.75

, and indeed

2.75 \le 3

. Study tip: Remember that inequality signs only flip when you multiply or divide by a negative number. With fractions, focus on clearing denominators first by multiplying both sides by the denominator.

Question 11

Solve for $x$ : $5-2(x-4)=3(x+1)+2.$

$x=\dfrac85$ (correct answer)
$x=\dfrac{13}{5}$
$x=\dfrac{5}{8}$
$x=-\dfrac85$

Explanation: When you encounter a linear equation with parentheses and variables on both sides, your goal is to systematically isolate the variable using the order of operations in reverse. Start by distributing through the parentheses. On the left side:

5-2(x-4) = 5-2x+8 = 13-2x

. On the right side:

3(x+1)+2 = 3x+3+2 = 3x+5

. Your equation becomes

13-2x = 3x+5

. Next, collect all variable terms on one side and constants on the other. Add

2x

to both sides:

13 = 5x+5

. Subtract 5 from both sides:

8 = 5x

. Finally, divide by 5:

x = \frac{8}{5}

. Looking at the wrong answers: Choice B gives

x = \frac{13}{5}

, which likely results from incorrectly handling the distribution or making sign errors when moving terms. Choice C gives

x = \frac{5}{8}

, which happens when you accidentally flip the fraction at the end—dividing 5 by 8 instead of 8 by 5. Choice D gives

x = -\frac{8}{5}

, which occurs from sign errors, particularly when distributing the negative through

-2(x-4)

or when moving terms across the equals sign. The correct answer is A:

x = \frac{8}{5}

. Study tip: When solving multi-step equations, work methodically through each step and double-check your distribution of negative signs. Always verify your answer by substituting it back into the original equation—both sides should equal the same value.

Question 12

Tickets to a concert cost $8 each plus a one-time $4 service fee. If Shawn has at most $60 to spend, which inequality gives the possible number of tickets, $n$ , he can buy, and what is the greatest whole number of tickets he can afford?

$8n+4\le60,\; n=7$ (correct answer)
$8n+4\ge60,\; n=6$
$8n-4\le60,\; n=8$
$8n+4<60,\; n=6$

Explanation: When you encounter word problems involving costs and spending limits, you're dealing with linear inequalities. The key is translating the written constraints into mathematical expressions that capture both the cost structure and the spending limitation. Here, each ticket costs $8, so $

n

tickets cost

8n

dollars. There's also a one-time

4 service fee regardless of how many tickets Shawn buys. This means his total cost is

8n + 4

. Since he has "at most

60" to spend, his total cost must be less than or equal to

60, giving us

8n + 4 \leq 60

. To find the maximum number of tickets, solve:

8n + 4 \leq 60

, so

8n \leq 56

, which means

n \leq 7

. The greatest whole number satisfying this is

n = 7

. Looking at the wrong answers: Choice B uses

\geq

instead of

\leq

, which would mean Shawn must spend at least

60 rather than at most

60. Choice C subtracts the service fee (

8n - 4

) instead of adding it, misrepresenting the cost structure. Choice D uses a strict inequality (

<

) rather than "less than or equal to" (

\leq

), which incorrectly excludes the possibility of spending exactly

60, leading to

n = 6

$ instead of the correct maximum of 7. Remember that "at most" translates to

\leq

, "at least" means

\geq

, and fixed fees are always added to variable costs. Double-check your inequality direction by asking whether the constraint is an upper limit or lower limit on spending.

Question 13

If $1.5 - 0.5(4x - 6) \ge 2(x + 3.25)$ , which of the following best describes the solution set for $x$ ?

$x \le -0.5$ (correct answer)
$x \ge -0.5$
$x \le 0.5$
$x \le -2$

Explanation: First, distribute the constants on both sides of the inequality:

1.5 - 2x + 3 \ge 2x + 6.5

. Combine like terms on the left side:

4.5 - 2x \ge 2x + 6.5

. To gather the

x

terms, add

2x

to both sides:

4.5 \ge 4x + 6.5

. To isolate the

x

term, subtract 6.5 from both sides:

-2 \ge 4x

. Finally, divide by 4:

-0.5 \ge x

. This is equivalent to

x \le -0.5

. Distractor B results from incorrectly flipping the inequality sign. Distractor D results from a sign error when distributing -0.5 to -6.

Question 14

Two cars leave the same point at the same time, traveling in opposite directions. One car travels at a constant speed of 45 mph. After 3 hours, they are 285 miles apart. If $s$ is the speed of the second car in mph, which equation represents this situation?

$3(s + 45) = 285$ (correct answer)
$3s + 45 = 285$
$3(s - 45) = 285$
$s + 45 = 285/3$

Explanation: The distance formula is distance = rate × time. The first car travels for 3 hours at 45 mph, covering a distance of

45 \times 3

miles. The second car travels for 3 hours at

s

mph, covering a distance of

s \times 3

miles. Since they are traveling in opposite directions, their distances from the starting point add up to the total distance between them. Therefore,

3 \times 45 + 3s = 285

. Factoring out the common factor of 3 gives

3(45 + s) = 285

, which is equivalent to

3(s + 45) = 285

. Choice D is the same equation, but it represents the next step in solving the problem, not the initial setup based on the combined rate. Choice A is the best representation of the initial physical situation, where the combined speed of separation is

(s+45)

mph.

Question 15

If $2|x - 3| + 5 = 17$ , what is the sum of all possible values of $x$ ?

6 (correct answer)
9
12
15

Explanation: When you encounter absolute value equations, remember that the expression inside the absolute value bars can be either positive or negative, leading to multiple solutions. Start by isolating the absolute value term. From

2|x - 3| + 5 = 17

, subtract 5 from both sides:

2|x - 3| = 12

. Then divide by 2:

|x - 3| = 6

. Now solve the two cases. Since

|x - 3| = 6

, either:

Case 1: $x - 3 = 6$ , so $x = 9$
Case 2: $x - 3 = -6$ , so $x = -3$

You can verify both solutions work in the original equation. The question asks for the sum of all possible values:

9 + (-3) = 6

. Looking at the wrong answers: Answer B (9) represents only one of the two solutions—a common error when students forget that absolute value equations typically have two solutions. Answer C (12) might come from adding the absolute values of both solutions (

|9| + |-3| = 12

) instead of their actual sum. Answer D (15) could result from arithmetic errors during the isolation steps or incorrectly calculating

9 + 3 = 12

and then making another error. The key strategy for absolute value equations is to always consider both the positive and negative cases after isolating the absolute value expression. Remember that

|A| = B

means

A = B

A = -B

. Always verify your solutions in the original equation, and read carefully—questions may ask for individual solutions, their sum, or their product.

Question 16

A number decreased by 15 is equal to 3 times the number increased by 9. What is the number?

The number is -15
The number is -12 (correct answer)
The number is -9
The number is -6

Explanation: When you see a word problem asking you to find an unknown number, you're dealing with algebraic translation. The key is converting the English phrases into mathematical expressions, then solving the resulting equation. Let's call the unknown number

x

. "A number decreased by 15" translates to

x - 15

. "3 times the number increased by 9" means

3x + 9

. Since these expressions are equal, you get:

x - 15 = 3x + 9

. To solve, collect like terms. Subtract

x

from both sides:

-15 = 2x + 9

. Then subtract 9 from both sides:

-24 = 2x

. Finally, divide by 2:

x = -12

. You can verify this:

-12 - 15 = -27

and

3(-12) + 9 = -36 + 9 = -27

✓ Looking at the wrong answers: Choice A (-15) likely comes from focusing too much on the "decreased by 15" part without solving the full equation. Choice C (-9) might result from confusing the constants in the problem or making sign errors. Choice D (-6) could come from arithmetic mistakes when combining terms or dividing incorrectly. The correct answer is B. Remember this translation pattern: "decreased by" means subtraction, "increased by" means addition, and "times" indicates multiplication. Always check your answer by substituting back into the original word problem to ensure both sides truly equal each other.

Question 17

Maria has 3 times as many books as Carlos. If Maria gives Carlos 12 books, they will have equal numbers of books. How many books does Maria have originally?

Maria originally has 36 books (correct answer)
Maria originally has 42 books
Maria originally has 48 books
Maria originally has 54 books

Explanation: When you encounter word problems involving relationships between quantities that change, setting up equations with variables is your most reliable approach. These problems test your ability to translate verbal descriptions into mathematical expressions. Let's define variables: Let

C

= Carlos's original number of books and

M

= Maria's original number of books. From the problem, we know two key relationships. First, "Maria has 3 times as many books as Carlos," which gives us

M = 3C

. Second, "If Maria gives Carlos 12 books, they will have equal numbers," meaning Maria will have

M - 12

books and Carlos will have

C + 12

books, so

M - 12 = C + 12

. From the second equation:

M - 12 = C + 12

, so

M = C + 24

. Now substitute the first equation:

3C = C + 24

. Solving:

2C = 24

, so

C = 12

. Therefore,

M = 3(12) = 36

. Let's verify: Maria starts with 36, Carlos with 12. After the transfer, Maria has 24 and Carlos has 24. ✓ Choice A is correct with 36 books. Choice B (42 books) would mean Carlos originally had 14 books, but after the transfer they'd have 30 and 26 respectively—not equal. Choice C (48 books) implies Carlos had 16 originally, leading to 36 and 28 after transfer. Choice D (54 books) means Carlos had 18, resulting in 42 and 30 after transfer. Always verify your solution by checking it against both original conditions. This catches algebraic errors and confirms your setup was correct.

Question 18

If $\frac{x}{3} + \frac{x}{4} = \frac{x}{2} - 5$ , what is the value of $x$ ?

$x = -50$
$x = -60$ (correct answer)
$x = -70$
$x = -80$

Explanation: When you encounter an equation with fractions containing the same variable, your goal is to combine like terms and isolate the variable. Start by finding a common denominator to simplify the fractions on the left side. To solve

\frac{x}{3} + \frac{x}{4} = \frac{x}{2} - 5

, first combine the left side using the common denominator 12:

\frac{4x}{12} + \frac{3x}{12} = \frac{7x}{12}

Now your equation becomes:

\frac{7x}{12} = \frac{x}{2} - 5

Convert

\frac{x}{2}

to twelfths:

\frac{x}{2} = \frac{6x}{12}

So:

\frac{7x}{12} = \frac{6x}{12} - 5

Subtract

\frac{6x}{12}

from both sides:

\frac{x}{12} = -5

Multiply both sides by 12:

x = -60

Choice B (

x = -60

) is correct. You can verify by substituting back:

\frac{-60}{3} + \frac{-60}{4} = -20 + (-15) = -35

, and

\frac{-60}{2} - 5 = -30 - 5 = -35

✓ Choice A (

x = -50

) gives you

-\frac{65}{6}

on the left and

-30

on the right—not equal. Choice C (

x = -70

) produces

-\frac{175}{12}

versus

-40

—also unequal. Choice D (

x = -80

) yields

-\frac{200}{12}

compared to

-45

—incorrect as well. When solving fraction equations, always find common denominators first, then systematically eliminate fractions by multiplying through. Double-check your answer by substituting back into the original equation—this catches arithmetic errors that are easy to make with fraction calculations.

Question 19

For what value of $a$ does the equation $2(ax + 3) - 4x = 6$ have infinitely many solutions?

$a = -2$
$a = 0$
$a = 2$ (correct answer)
No value of $a$ exists.

Explanation: First, simplify the equation. Distribute the 2:

2ax + 6 - 4x = 6

. Subtract 6 from both sides:

2ax - 4x = 0

. Factor out

x

x(2a - 4) = 0

. For this equation to have infinitely many solutions, it must be true for any value of

x

. This occurs when the coefficient of

x

is zero. So, we set

2a - 4 = 0

. Adding 4 to both sides gives

2a = 4

. Dividing by 2 gives

a = 2

. When

a=2

, the equation becomes

x(0)=0

, or

0=0

, which is always true.

Question 20

Solve for $y$ : $5(2y-3)=10y-15.$

All real numbers (correct answer)
No solution
$y=3$
$y=-3$

Explanation: When you encounter an equation like this, you're dealing with a linear equation that might have one solution, no solution, or infinitely many solutions. The key is to simplify both sides and see what happens. Let's solve

5(2y-3)=10y-15

step by step. First, distribute the 5 on the left side:

5(2y) - 5(3) = 10y - 15

, which gives us

10y - 15 = 10y - 15

. Notice that both sides are now identical. When you subtract

10y

from both sides, you get

-15 = -15

, and when you add 15 to both sides, you get

0 = 0

. This is always true, regardless of what value you substitute for

y

. Let's check the answer choices: Choice A is correct because any real number you substitute for

y

will make the original equation true. Choice B (no solution) would occur if we ended up with something like

0 = 5

, which is never true. Choice C (

y = 3

) suggests there's only one solution - you can verify this isn't the case by substituting other values like

y = 0

y = 1

into the original equation and seeing they work too. Choice D (

y = -3

) has the same problem as choice C - it's just one specific value when infinitely many work. Watch for this pattern on algebra problems: when both sides of an equation simplify to exactly the same expression, you have an identity that's true for all real numbers. Don't assume every equation has just one solution!