All practice questions

Pair the given polynomial expressions on the left with their fully factored equivalents on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings are:

4x^2 - 25

with

(2x - 5)(2x + 5)

;

x^2 - 8x + 15

with

(x - 3)(x - 5)

;

3x^2 + 12x - 15

with

3(x + 5)(x - 1)

; and

2x^3 - 8x

with

2x(x - 2)(x + 2)

.

Each polynomial expression was broken down into its fundamental algebraic components by identifying patterns such as the difference of squares, standard trinomial factoring, and extracting the greatest common factor.

Step-by-Step Solution

1

Analyze the first expression

4x^2 - 25

.

It factors to

(2x - 5)(2x + 5)

.

It follows the difference of squares pattern

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

with

a = 2x

and

b = 5

.

2

Analyze the second expression

x^2 - 8x + 15

.

It factors to

(x - 3)(x - 5)

.

The constant term is positive and the middle coefficient is negative, so both factors must contain negative constants that multiply to

15

and add to

-8

.

3

Analyze the third expression

3x^2 + 12x - 15

.

It factors to

3(x + 5)(x - 1)

.

Factoring out the GCF of

3

gives

3(x^2 + 4x - 5)

. The resulting quadratic factors into binomials with constants

5

and

-1

.

4

Analyze the fourth expression

2x^3 - 8x

.

It factors to

2x(x - 2)(x + 2)

.

Factoring out the GCF of

2x

gives

2x(x^2 - 4)

. The expression

x^2 - 4

is a difference of squares that further factors into

(x - 2)(x + 2)

.

Key Concept

Factoring Polynomials

Estimated Time:2m 0s

Question 2102Question →

Which composite figure correctly corresponds to each set of geometric components? Match each set of transparent geometric components (Sets 1-3) to its correct superimposed composite figure (Figures X-Z). Assume the base square remains stationary while the overlay square undergoes the indicated rotation before being placed exactly on top of the base square.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Set 1 matches Figure X, Set 2 matches Figure Z, and Set 3 matches Figure Y.

Each correct pairing strictly follows spatial transformation rules. By systematically tracking the starting quadrant of the circle in the overlay, applying the stated degree and direction of rotation, and laying it over the specified diagonal base, the correct composite image can be logically deduced.

Step-by-Step Solution

1

Analyze Set 1 rotation and layering.

The top-left circle moves to the top-right. Placed over a top-left to bottom-right diagonal, this matches Figure X.

A 90-degree clockwise rotation shifts elements to the next adjacent quadrant in a clockwise direction.

2

Analyze Set 2 rotation and layering.

The top-left circle moves to the bottom-right. Placed over a bottom-left to top-right diagonal, this matches Figure Z.

A 180-degree rotation shifts elements diagonally to the completely opposite quadrant.

3

Analyze Set 3 rotation and layering.

The bottom-left circle moves to the top-left. Placed over a bottom-left to top-right diagonal, this matches Figure Y.

A 90-degree clockwise rotation shifts the bottom-left quadrant directly up to the top-left quadrant.

Key Concept

Superimposition of Shapes

Question 2103Question →

A drafter is reviewing a set of architectural plans. Connect each described geometric condition with its correct mathematical term.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Two angles summing to

180^\circ

match with Supplementary angles; two angles summing to

90^\circ

match with Complementary angles; opposite equal angles match with Vertical angles; lines forming a right angle match with Perpendicular lines.

Each definition precisely matches the fundamental geometric axioms for angles and lines. Recognizing these definitions is essential for solving more complex geometry problems.

Step-by-Step Solution

1

Identify the geometric term for two angles whose sum is exactly

180^\circ

.

Supplementary angles

By geometric definition, a pair of angles that add up to

180^\circ

(forming a straight line if adjacent) are supplementary.

2

Identify the geometric term for two angles whose sum is exactly

90^\circ

.

Complementary angles

By geometric definition, a pair of angles that add up to

90^\circ

(forming a right angle if adjacent) are complementary.

3

Identify the term for the opposite angles created when two lines intersect.

Vertical angles

Intersecting lines create two pairs of non-adjacent, opposite angles known as vertical angles, which are congruent.

4

Identify the term for straight lines that intersect to form a right angle.

Perpendicular lines

When lines intersect at exactly

90^\circ

, they are defined mathematically as perpendicular.

Key Concept

Basic terminology for angle relationships and line intersections

Estimated Time:45s

Question 2104Question →

An isosceles triangle has a perimeter of $36$ centimeters. The base of the triangle measures $10$ centimeters. What is the area of the triangle in square centimeters?

Show answer & explanation

Answer: 60

Answer

60

To find the area, you must first know the perpendicular height. By splitting the isosceles triangle in half via its altitude, a right triangle is formed with a hypotenuse of

13

(the side length) and a base of

5

(half of the full base). The Pythagorean theorem (

5^2 + h^2 = 13^2

) reveals the height is

12

. The full area is then calculated as

\frac{1}{2} \times \text{base} \times \text{height}

, which is

\frac{1}{2} \times 10 \times 12 = 60

.

Step-by-Step Solution

1

Calculate the length of the two equal sides.

The two equal sides each measure

13

centimeters.

The perimeter is

36

cm and the base is

10

cm. The remaining

26

cm must be divided equally between the two identical legs:

26 \div 2 = 13

cm.

2

Determine the base of the right triangle formed by the altitude.

The base of the right triangle is

5

centimeters.

The altitude of an isosceles triangle bisects the base, splitting the

10

-centimeter base into two equal

5

-centimeter segments.

3

Calculate the height using the Pythagorean theorem.

The height is

12

centimeters.

In the right triangle formed, the hypotenuse is

13

and one leg is

5

. Let

h

be the height.

5^2 + h^2 = 13^2 \Rightarrow 25 + h^2 = 169 \Rightarrow h^2 = 144 \Rightarrow h = 12

.

4

Calculate the total area of the triangle.

The area is

60

square centimeters.

The area formula is

\frac{1}{2} \times \text{base} \times \text{height}

. Thus,

\frac{1}{2} \times 10 \times 12 = 60

.

Key Concept

Using the Pythagorean theorem to find missing dimensions in isosceles triangles to calculate area.

Question 2105Question →

An engineer battalion is transporting heavy equipment in a convoy to a forward operating base $210$ miles away. To arrive on schedule, the convoy must maintain an overall average speed of $42$ mph for the entire journey.

During the first leg, the convoy travels $120$ miles in $2$ hours. They then halt for $30$ minutes to inspect the equipment.

What average speed must the convoy maintain for the remainder of the trip to achieve the target overall average speed?

24 mph

30 mph

36 mph

60 mph

Show answer & explanation

Answer: 36 mph

Answer

The convoy must maintain an average speed of 36 mph for the remainder of the trip.

To achieve an overall average speed of

42

mph over

210

miles, the total journey must take exactly

5

hours. The convoy has already used

2.5

hours (

2

hours driving +

0.5

hours resting) and covered

120

miles. This leaves

90

miles to be covered in the remaining

2.5

hours. Dividing the remaining distance by the remaining time (

90 / 2.5

) gives the required speed of

36

mph.

Step-by-Step Solution

1

Calculate the total time required for the entire journey.

Total Time =

210 \text{ miles} / 42 \text{ mph} = 5 \text{ hours}

.

To find the time available for the second leg, we first need to know the total time allowed for the mission.

2

Calculate the time spent on the first leg and the rest stop.

Time Spent =

2 \text{ hours} + 30 \text{ minutes} = 2.5 \text{ hours}

.

The rest stop must be included in the total elapsed time because the overall average speed accounts for the entire duration of the journey.

3

Determine the remaining distance and remaining time for the final leg.

Remaining Distance =

210 - 120 = 90 \text{ miles}

.
Remaining Time =

5 - 2.5 = 2.5 \text{ hours}

.

Subtracting the distance already covered and the time already spent gives the parameters for the final segment.

4

Calculate the required average speed for the final leg.

Speed =

90 \text{ miles} / 2.5 \text{ hours} = 36 \text{ mph}

.

Average speed is always calculated as total distance divided by total time for that specific segment.

Key Concept

To find a missing speed in a multi-leg journey, use the total distance and target average speed to find the total allowed time. Subtract the time and distance already used (including rest stops) to find the parameters for the remaining leg.

Alternative Method

You can work entirely in minutes to avoid decimals. Total time is

300

minutes. Time spent is

150

minutes. Time remaining is

150

minutes. Speed needed is

90 \text{ miles} / 150 \text{ minutes} = 0.6 \text{ miles per minute}

. Multiply by

60

to convert to

36

mph.

Estimated Time:1m 30s

Question 2106Question →

If $(x, y)$ is the solution to the system of equations below, what is the value of the expression $x + 3y$ ?

3x - 2y = 14

x = y + 5

21

-9

1

41

Show answer & explanation

Answer: 1

Answer

The value of the expression is 1.

The system can be efficiently solved using substitution because the second equation already isolates

x

. Substituting

x = y + 5

into the first equation yields

3(y + 5) - 2y = 14

. Distributing the

3

gives

3y + 15 - 2y = 14

, which simplifies to

y = -1

. Substituting this back into the second equation gives

x = -1 + 5 = 4

. Finally, the question asks for the value of

x + 3y

, which evaluates to

4 + 3(-1) = 1

.

Step-by-Step Solution

1

Substitute the expression for

x

from the second equation into the first equation.

3(y + 5) - 2y = 14

This isolates a single variable, allowing us to solve for

y

.

2

Distribute the 3 and solve the resulting equation for

y

.

3y + 15 - 2y = 14 \Rightarrow y + 15 = 14 \Rightarrow y = -1

Expanding the terms allows combining like terms to find the value of

y

.

3

Substitute

y = -1

back into the second equation to determine

x

.

x = -1 + 5 \Rightarrow x = 4

Both values are needed for the final expression.

4

Evaluate the target expression

x + 3y

using the values found for

x

and

y

.

4 + 3(-1) = 4 - 3 = 1

The question specifically asks for the evaluated value of this expression, not just the ordered pair.

Key Concept

Solving a system of linear equations using the substitution method and evaluating an algebraic expression.

Question 2107Question →

An electronics store on a military base sells a laptop originally priced at $840$ . The store offers a $25\%$ discount for service members. If a $6\%$ sales tax is applied to the discounted price, what is the final cost of the laptop?

680.40

1008.00

667.80

863.90

Show answer & explanation

Answer: $667.80$

Answer

The final cost of the laptop is $667.80.

The correct answer is found by completing a two-step percentage calculation. First, the

25\%

discount is applied to the

\$840

original price (

840 \times 0.25 = 210

), yielding a sale price of

\$630

. Then, the

6\%

sales tax must be calculated on this new sale price (

630 \times 0.06 = 37.80

) and added to it. Adding the tax to the sale price gives a final total of

\$667.80

.

Step-by-Step Solution

1

Calculate the dollar amount of the military discount.

840 \times 0.25 = 210

To find the discount, multiply the original price by the decimal equivalent of the percentage (

25\% = 0.25

).

2

Subtract the discount from the original price to find the sale price.

840 - 210 = 630

The discount must be subtracted from the original price to determine the amount that will be taxed.

3

Calculate the amount of sales tax on the discounted price.

630 \times 0.06 = 37.80

Sales tax is calculated as a percentage of the final sale price, not the original price.

4

Add the sales tax to the discounted price to find the total final cost.

630 + 37.80 = 667.80

The final amount paid is the sum of the sale price and the calculated tax.

Key Concept

Calculating Percentages of Quantities

Alternative Method

Instead of calculating the discount amount and subtracting, you can determine what percentage of the price you will pay. If there is a

25\%

discount, you pay

75\%

of the original price (

0.75

). To include a

6\%

tax, you multiply by

1.06

. You can calculate the final price directly as:

\$840 \times 0.75 \times 1.06 = \$667.80

.

Estimated Time:1m 30s

Question 2108Question →

A circular target has a diameter of $10$ inches. What is the area of the target in square inches?

25\pi

100\pi

10\pi

5\pi

Show answer & explanation

Answer: $25\pi$

Answer

The correct answer is

25\pi

square inches.

The correct area is found by first dividing the diameter of

10

inches by

2

to get a radius of

5

inches. Squaring the radius gives

25

, which is then multiplied by

\pi

to yield

25\pi

square inches.

Step-by-Step Solution

1

Find the radius of the circular target.

r = \frac{10}{2} = 5

inches.

The area formula requires the radius, which is exactly half of the given diameter.

2

Apply the formula for the area of a circle.

A = \pi r^2 = \pi (5)^2 = 25\pi

.

Substitute the calculated radius into the standard area formula to find the final square footage.

Key Concept

Area of a Circle

Question 2109Question →

Calculate the missing value based on the provided scenario.

Fill in the blanks below

A military hospital pharmacy currently stocks

850

units of a specific antibiotic. If

38\%

of this stock is distributed to various clinics on the base, there will be units remaining in the main pharmacy.

Show answer & explanation

Answer

527

To find the number of remaining units, you can first subtract the distributed percentage from

100\%

(

100\% - 38\% = 62\%

). Next, convert

62\%

to a decimal (

0.62

) and multiply it by the total stock (

0.62 \times 850 = 527

). Alternatively, you could calculate the number of units distributed (

0.38 \times 850 = 323

) and subtract that amount from the original stock (

850 - 323 = 527

).

Step-by-Step Solution

1

Determine the percentage of the antibiotic stock that will remain.

$100\% - 38\% = 62\%

Since

38\%

of the stock is distributed, the remainder is the difference between

100\%

and

38\%

.

2

Convert the remaining percentage to a decimal.

62\% = 0.62

Percentages must be converted into decimals or fractions before they can be used in multiplication.

3

Calculate

62\%

of the total stock.

0.62 \times 850 = 527

Multiplying the decimal representation of the remaining percentage by the total original stock gives the number of units left.

Key Concept

Calculating the remaining amount by applying the complementary percentage to a whole quantity.

Question 2110Question →

A guy wire is attached to the top of a $12$ -foot vertical pole. The wire is anchored to the ground $5$ feet from the base of the pole. What is the length of the guy wire?

13

feet

\sqrt{119}

feet

17

feet

169

feet

Show answer & explanation

Answer: $13$ feet

Answer

The guy wire is

13

feet long.

The length of the guy wire represents the hypotenuse of a right triangle formed by the pole and the ground. According to the Pythagorean theorem (

a^2 + b^2 = c^2

), the square of the hypotenuse equals the sum of the squares of the two legs. Substituting the given values:

12^2 + 5^2 = 144 + 25 = 169

. Taking the square root of

169

gives

13

feet.

Step-by-Step Solution

1

Identify the known values representing the legs of the right triangle.

Leg

a = 5

feet, Leg

b = 12

feet.

The pole and the ground meet at a right angle, so they form the legs of the triangle.

2

Apply the Pythagorean theorem (

a^2 + b^2 = c^2

).

5^2 + 12^2 = c^2

The theorem connects the lengths of the legs to the hypotenuse (the guy wire).

3

Square the lengths of the legs and sum them.

25 + 144 = 169

This calculates the value of the hypotenuse squared (

c^2

).

4

Take the square root of the sum.

c = \sqrt{169} = 13

This isolates

c

and provides the actual length of the hypotenuse.

Key Concept

The Pythagorean theorem is used to find the hypotenuse of a right triangle when both legs are known.

Question 2111Question →

A cafe owner is purchasing a commercial espresso machine. Supplier A lists the machine for $\$2,000$ with a single $30\%$ discount. Supplier B lists the identical machine for $\$2,400$ with successive discounts of $25\%$ and $20\%$ . How much money will the owner save by purchasing the machine from the less expensive supplier?

\$385

\$40

\$80

\$360

Show answer & explanation

Answer: $\$40$

Answer

The owner will save $40 by choosing Supplier A.

Supplier A's final price is

1,400 (which is

2,000 minus a

600 discount). Supplier B's final price is

1,440 (calculated by first reducing

2,400 by 25% to get

1,800, and then reducing

1,800 by 20% to get

1,440). The difference between the two prices is $40, making Supplier A the less expensive option by that amount.

Step-by-Step Solution

1

Calculate the final price for Supplier A.

\$2,000 \times 0.30 = \$600

discount.

\$2,000 - \$600 = \$1,400

.

A single discount is taken directly from the initial price.

2

Calculate the price after the first discount for Supplier B.

\$2,400 \times 0.25 = \$600

discount.

\$2,400 - \$600 = \$1,800

.

The first of the successive discounts is applied to the original listing price.

3

Calculate the price after the second discount for Supplier B.

\$1,800 \times 0.20 = \$360

discount.

\$1,800 - \$360 = \$1,440

.

The second discount must be applied to the new subtotal, not the original price.

4

Find the difference between the two final prices.

\$1,440 - \$1,400 = \$40

.

Subtracting the lower price from the higher price gives the total savings.

Key Concept

Calculating successive discounts and comparing final sale prices.

Question 2112Question →

A vertical flagpole that is originally $32$ feet tall breaks at a certain height due to strong winds. The top part of the pole folds over without completely detaching, and the tip touches the level ground exactly $16$ feet away from the base of the pole. At what height from the ground did the flagpole break?

8 feet

12 feet

16 feet

20 feet

Show answer & explanation

Answer: 12 feet

Answer

12 feet

The scenario creates a right triangle where the vertical leg is the standing part of the pole, the horizontal leg is the ground, and the hypotenuse is the broken top part. Letting the break height be

x

, the hypotenuse becomes

(32 - x)

. Using the Pythagorean theorem:

x^2 + 16^2 = (32 - x)^2

. Expanding the right side gives

x^2 + 256 = 1024 - 64x + x^2

. Subtracting

x^2

from both sides simplifies the equation to

256 = 1024 - 64x

. Adding

64x

and subtracting

256

gives

64x = 768

. Dividing by

64

yields

x = 12

. Therefore, the pole broke

12

feet above the ground.

Step-by-Step Solution

1

Define the variables for the sides of the right triangle formed by the broken flagpole.

Let the height of the break (standing part) be

x

. The length of the fallen part (hypotenuse) is the remainder of the pole, which is

(32 - x)

. The base distance along the ground is

16

feet.

Establishing these algebraic relationships is necessary to apply the Pythagorean theorem to the scenario.

2

Set up the Pythagorean theorem equation (

a^2 + b^2 = c^2

) using the defined variables.

x^2 + 16^2 = (32 - x)^2

The standing pole and the ground form a right angle, making the standing pole and ground distance the legs, and the fallen part the hypotenuse.

3

Expand both sides of the algebraic equation.

x^2 + 256 = 1024 - 64x + x^2

Expanding the squared binomial

(32 - x)^2

correctly is required to isolate the variable

x

.

4

Solve for

x

by combining like terms.

Subtract

x^2

from both sides to get

256 = 1024 - 64x

. Rearrange to

64x = 768

, and divide by

64

to find

x = 12

.

Isolating

x

provides the exact height from the ground where the flagpole snapped.

Key Concept

Applying the Pythagorean Theorem with algebraic expressions

Question 2113Question →

Match each fundamental geometric term related to a circle with its correct definition.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Radius matches with a line segment from the center to the edge; Diameter matches with a line segment passing through the center; Circumference matches with the total distance around the circle; Chord matches with a line segment connecting any two points on the circle.

Each geometric term possesses a precise mathematical definition that distinguishes its function and location within a circle. Correctly identifying these fundamental components is an essential prerequisite for performing geometric calculations such as finding area or circumference.

Step-by-Step Solution

1

Identify the term that represents the perimeter or boundary distance of the circle.

Circumference matches with 'The total distance around the outer boundary of a circle.'

Circumference is the only term listed that measures a 1D distance around a shape rather than a straight interior line segment.

2

Differentiate between the line segments based on their relationship to the circle's center point.

Radius connects the center to the edge. Diameter passes all the way through the center from edge to edge.

This establishes the mathematical relationship where the diameter is exactly twice the length of the radius.

3

Determine the general term for a line segment connecting any two points on the circle's edge.

Chord matches with 'A line segment whose endpoints lie on the circle, but which does not necessarily pass through the center.'

A diameter is a special type of chord, but 'chord' is the broader category that does not require passing through the center.

Key Concept

Fundamental vocabulary and structural properties of circles.

Question 2114Question →

An armored cavalry troop conducts a training movement. They travel $45$ miles over rough terrain at a speed of $15$ miles per hour. After a $90$ -minute halt for vehicle maintenance, they proceed $75$ miles along a highway at $50$ miles per hour. What is their overall average speed, in miles per hour, for the entire operation?

Show answer & explanation

Answer: 20

Answer

The overall average speed is

20

miles per hour.

The correct answer is

20

because average speed is the ratio of total distance to total time. The total distance is

120

miles (

45 + 75

). The total time includes the travel times (

3

hours and

1.5

hours) plus the maintenance halt (

1.5

hours), which equals

6

hours. Dividing

120

miles by

6

hours gives an overall average speed of

20

miles per hour.

Step-by-Step Solution

1

Calculate the time taken for the first travel segment.

\frac{45}{15} = 3

hours

Time is required to determine the total elapsed time of the operation.

2

Convert the maintenance halt time from minutes to hours.

90

minutes

= 1.5

hours

All time units must be consistent (hours) before they can be added together.

3

Calculate the time taken for the second travel segment.

\frac{75}{50} = 1.5

hours

This gives the travel time for the final leg of the route.

4

Calculate the total distance traveled.

45 + 75 = 120

miles

Total distance is the numerator in the average speed formula.

5

Calculate the total elapsed time for the entire operation.

3 + 1.5 + 1.5 = 6

hours

Total time is the denominator in the average speed formula, and it must include the maintenance halt.

6

Calculate the overall average speed.

\frac{120}{6} = 20

miles per hour

Average speed is defined as the total distance divided by the total time.

Key Concept

Average speed is calculated by dividing the total distance traveled by the total time elapsed, including any rest stops or delays.

Question 2115Question →

A quartermaster depot receives a shipment of $2,500$ emergency field rations. Upon inspection, $36\%$ of the rations are classified as vegetarian. Of these vegetarian rations, $15\%$ are also certified as gluten-free. How many gluten-free vegetarian rations are in the shipment?

375

525

135

1,350

Show answer & explanation

Answer: 135

Answer

135

The correct answer is found by completing two sequential percentage calculations. First, find

36\%

of the

2,500

total rations (

2,500 \times 0.36 = 900

). Then, find

15\%

of those

900

vegetarian rations (

900 \times 0.15 = 135

).

Step-by-Step Solution

1

Calculate the number of vegetarian rations by finding

36\%

of the total

2,500

rations.

2,500 \times 0.36 = 900

vegetarian rations.

The first percentage applies to the entire shipment of rations.

2

Calculate the number of gluten-free vegetarian rations by finding

15\%

of the result from step 1.

900 \times 0.15 = 135

gluten-free vegetarian rations.

The problem asks for

15\%

'Of these vegetarian rations', meaning the base for the second percentage is

900

, not

2,500

.

Key Concept

Calculating percentages of a specified subset in a multi-step word problem.

Question 2116Question →

A landscaper is placing a decorative border around a circular flower bed. The radius of the flower bed is $12$ feet. What is the total length of the border needed in terms of $\pi$ ?

24\pi

feet

12\pi

feet

144\pi

feet

576\pi

feet

Show answer & explanation

Answer: $24\pi$ feet

Answer

The total length of the border is

24\pi

feet, which is found by calculating the circumference of the circle.

To find the total length of the border, you must calculate the circumference of the circular flower bed. The formula for circumference is

C = 2\pi r

. Substituting the given radius of

12

feet into the formula gives

C = 2 \cdot \pi \cdot 12 = 24\pi

feet.

Step-by-Step Solution

1

Identify the mathematical concept required to find the length of a border around a circle.

The length of the border corresponds to the circumference of the circle.

Perimeter (circumference) measures the distance around the outside of a 2D shape.

2

Select the correct formula for the circumference of a circle.

Formula:

C = 2\pi r

The problem provides the radius (

r

), so this is the most direct formula to use.

3

Substitute the given radius value into the formula and solve.

C = 2 \cdot \pi \cdot 12 = 24\pi

feet.

Replacing

r

with

12

computes the exact circumference in terms of

\pi

.

Key Concept

Circumference of a circle

Question 2117Question →

Given the system of linear equations below, what is the value of $y$ ?

5x + 2y = 12

3x - 4y = 2

2

11

1

-2

Show answer & explanation

Answer: 1

Answer

The correct value for y is 1.

By multiplying the first equation by 2, we get 10x + 4y = 24. Adding this to the second equation (3x - 4y = 2) eliminates y, resulting in 13x = 26, which simplifies to x = 2. Substituting x = 2 back into the first equation gives 5(2) + 2y = 12, or 10 + 2y = 12. Subtracting 10 from both sides leaves 2y = 2, yielding the final correct answer of y = 1.

Step-by-Step Solution

1

Multiply the first equation by 2 to align the coefficients of y for elimination.

10x + 4y = 24

Creating a +4y term in the first equation allows y to be eliminated when added to the -4y term in the second equation.

2

Add the modified first equation to the second equation.

(10x + 3x) + (4y - 4y) = 24 + 2 \Rightarrow 13x = 26

This eliminates the variable y, leaving a single-variable equation that can be easily solved for x.

3

Divide both sides by 13 to solve for x.

x = 2

Isolating x provides the first part of the coordinate solution.

4

Substitute x = 2 back into the first original equation to solve for y.

5(2) + 2y = 12 \Rightarrow 10 + 2y = 12 \Rightarrow 2y = 2 \Rightarrow y = 1

Since the question asks specifically for the value of y, this final substitution step is required.

Key Concept

Solving Systems of Linear Equations using the Elimination Method

Question 2118Question →

A hiker walks $8$ miles due north, then turns and walks $15$ miles due east. What is the straight-line distance, in miles, from the hiker's starting point to their current location?

Show answer & explanation

Answer: 17

Answer

17

Because the hiker's path consists of two perpendicular segments (north and east), they form the legs of a right triangle measuring

8

miles and

15

miles. The straight-line distance from the start to the end is the hypotenuse of this triangle. Applying the Pythagorean theorem (

a^2 + b^2 = c^2

), we get

8^2 + 15^2 = c^2

, which simplifies to

64 + 225 = 289

. The square root of

289

is

17

. Therefore, the straight-line distance is

17

miles.

Step-by-Step Solution

1

Visualize the geometrical shape formed by the hiker's path.

A right triangle is formed with legs of

8

miles and

15

miles.

The directions 'due north' and 'due east' are perpendicular, creating a

90^\circ

angle.

2

Set up the Pythagorean theorem equation.

8^2 + 15^2 = c^2

The straight-line distance corresponds to the hypotenuse (

c

) of the right triangle.

3

Square the values of the legs and add them together.

64 + 225 = 289

To evaluate the left side of the equation.

4

Calculate the square root of the sum.

c = \sqrt{289} = 17

To solve for

c

and find the final straight-line distance.

Key Concept

Calculating distance using the Pythagorean theorem

Question 2119Question →

Read the geometric scenario below and calculate the missing perimeter. Enter only the numerical value.

Fill in the blanks below

In

\triangle PQR

, an altitude

PS

is drawn to the base

QR

, creating two right triangles,

\triangle PQS

and

\triangle PSR

. If the length of side

PQ

is

13

units, the length of side

PR

is

15

units, and the altitude

PS

is

12

units, the perimeter of the entire

\triangle PQR

is units.

Show answer & explanation

Answer

42

To find the perimeter of

\triangle PQR

, we need the lengths of its three outer sides:

PQ

,

PR

, and

QR

. We are given

PQ = 13

and

PR = 15

. The altitude

PS

divides

QR

into two segments,

QS

and

SR

, forming two right triangles. Using the Pythagorean theorem in

\triangle PQS

, we find

QS = \sqrt{13^2 - 12^2} = \sqrt{25} = 5

. Using the Pythagorean theorem in

\triangle PSR

, we find

SR = \sqrt{15^2 - 12^2} = \sqrt{81} = 9

. The total base is

QR = QS + SR = 5 + 9 = 14

. The perimeter is the sum of the outer sides:

13 + 15 + 14 = 42

.

Step-by-Step Solution

1

Use the Pythagorean theorem in right

\triangle PQS

to find the length of base segment

QS

.

QS = \sqrt{13^2 - 12^2} = \sqrt{169 - 144} = \sqrt{25} = 5

The altitude

PS

forms a right angle with the base, making

PQ

the hypotenuse. The missing leg is found by subtracting the square of the altitude from the square of the hypotenuse.

2

Use the Pythagorean theorem in right

\triangle PSR

to find the length of base segment

SR

.

SR = \sqrt{15^2 - 12^2} = \sqrt{225 - 144} = \sqrt{81} = 9

Similarly,

PR

is the hypotenuse of the second right triangle, so the missing leg is found by subtracting the square of the altitude from the square of this hypotenuse.

3

Calculate the total length of the base

QR

.

QR = QS + SR = 5 + 9 = 14

The entire base

QR

is the sum of the two segments created by the altitude.

4

Calculate the perimeter of the full

\triangle PQR

.

Perimeter =

PQ + PR + QR = 13 + 15 + 14 = 42

The perimeter is the total length of the outer boundary of the triangle, which is the sum of its three main sides.

Key Concept

Applying the Pythagorean theorem in multi-step geometric figures to find missing side lengths and perimeter.

Estimated Time:2m 0s

Question 2120Question →

An auto repair shop owner orders a bulk supply of replacement brake pads with an original total cost of $\$800$ . The automotive parts distributor applies a $15\%$ bulk discount to the order. In addition, the shop receives an extra $5\%$ discount on the reduced price for paying the invoice in full upon delivery.

Determine the shop's final purchase price for the brake pads and the total amount of money saved compared to the original cost.

Fill in the blanks below

The final purchase price is $\$

and the total amount saved is

\$$.

Show answer & explanation

Answer

The final purchase price is

646 and the total amount saved is

154.

The correct final price is found by taking 15% off

800 to get

680, then taking 5% off

680 to arrive at

646. The total savings is the sum of both discounts (

120 +

34 = $154).

Step-by-Step Solution

1

Calculate the amount of the 15% bulk discount.

\$800 \times 0.15 = \$120

To find the savings provided by the initial distributor discount.

2

Calculate the intermediate price after the first discount.

\$800 - \$120 = \$680

The subsequent 5% discount must be applied to this reduced price, not the original price.

3

Calculate the amount of the 5% payment discount.

\$680 \times 0.05 = \$34

To determine the additional savings from paying the invoice in full upon delivery.

4

Calculate the final purchase price.

\$680 - \$34 = \$646

This is the final amount the shop must pay after all successive discounts have been applied.

5

Calculate the total amount saved.

\$120 + \$34 = \$154

To find the complete difference between the original cost and the final price.

Key Concept

Successive discounts must be calculated sequentially on the new reduced price at each step, rather than adding the percentages together.

Answer

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