All practice questions

Question 421Question →

A submarine needs to travel a distance of $45$ nautical miles to reach a designated patrol zone. If the submarine maintains a constant submerged speed of $18$ knots (nautical miles per hour), how many minutes will it take to reach the zone?

150

2.5

24

250

Show answer & explanation

Answer: $150$

Answer

The submarine will take

150

minutes to reach the patrol zone.

The correct answer is found by dividing the total distance (

45

nautical miles) by the rate (

18

knots) to determine the time in hours (

2.5

hours). Then, this value is multiplied by

60

to convert hours into minutes, resulting in

150

minutes.

Step-by-Step Solution

1

Calculate the time required in hours using the formula

\text{Time} = \text{Distance} / \text{Rate}

.

45 / 18 = 2.5

hours.

The distance is given in nautical miles and the rate is given in nautical miles per hour, so dividing them yields the time in hours.

2

Convert the time from hours into minutes.

2.5 \times 60 = 150

minutes.

The question specifically asks for the travel time in minutes. There are

60

minutes in

1

hour.

Key Concept

Calculating time from rate and distance, followed by unit conversion.

Question 422Question →

When the polynomial $5x^2 - 20$ is factored completely into the form $k(x - c)(x + c)$ , where $k$ and $c$ are positive integers, what is the value of the sum $k + c$ ?

Show answer & explanation

Answer: 7; seven

Answer

7

The polynomial

5x^2 - 20

can be factored completely by first pulling out the greatest common factor,

5

, leaving

5(x^2 - 4)

. The expression inside the parentheses is a difference of squares and factors into

(x - 2)(x + 2)

. This makes the completely factored form

5(x - 2)(x + 2)

. Comparing this to the form

k(x - c)(x + c)

, we see that

k = 5

and

c = 2

. Their sum is

5 + 2 = 7

.

Step-by-Step Solution

1

Factor out the greatest common factor (GCF) from the polynomial.

5x^2 - 20 = 5(x^2 - 4)

Both terms share a common numerical factor of

5

. Factoring it out simplifies the remaining polynomial.

2

Factor the remaining binomial as a difference of squares.

5(x^2 - 4) = 5(x - 2)(x + 2)

The binomial

x^2 - 4

follows the pattern

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

, where

a = x

and

b = 2

.

3

Identify the values of

k

and

c

from the given form.

k = 5

and

c = 2

By matching

5(x - 2)(x + 2)

to the target form

k(x - c)(x + c)

, the constant multiplier is

5

and the positive integer in the binomial is

2

.

4

Calculate the sum of

k

and

c

.

5 + 2 = 7

The question asks for the final value of the sum

k + c

.

Key Concept

Factoring polynomials using the greatest common factor and the difference of squares.

Question 423Question →

A Navy recruiter is organizing a training event and orders $4$ sandwich platters for $\$22.50$ each and $2$ fruit trays for $\$15.00$ each. The catering company applies a $10\%$ military discount to the cost of the food, and then adds a $5\%$ delivery fee to the discounted subtotal. What is the final total cost of the order?

\$114.00

\$162.00

\$113.40

\$115.50

Show answer & explanation

Answer: $\$113.40$

Answer

The final total cost of the order is

\$113.40

.

The total cost of the food is first calculated as

\$120.00

. The

10\%

discount reduces this amount by

\$12.00

, leaving a discounted subtotal of

\$108.00

. The

5\%

delivery fee is then applied to the discounted subtotal (

\$108.00 \times 0.05 = \$5.40

). Adding the fee to the discounted subtotal yields a final cost of

\$113.40

.

Step-by-Step Solution

1

Calculate the initial cost of the food items.

4 \times \$22.50 = \$90.00

(platters) and

2 \times \$15.00 = \$30.00

(fruit). Total subtotal

= \$90.00 + \$30.00 = \$120.00

.

You must find the total value of all items before applying any discounts or fees.

2

Calculate the amount of the military discount.

\$120.00 \times 0.10 = \$12.00

.

The

10\%

discount is applied to the initial food cost.

3

Determine the discounted subtotal.

\$120.00 - \$12.00 = \$108.00

.

Subtracting the discount from the original subtotal gives the new base amount for the fee.

4

Calculate the delivery fee on the discounted subtotal.

\$108.00 \times 0.05 = \$5.40

.

The problem states the

5\%

delivery fee is applied to the discounted subtotal.

5

Calculate the final total cost.

\$108.00 + \$5.40 = \$113.40

.

Adding the delivery fee to the discounted subtotal provides the final amount owed.

Key Concept

Decimal Operations and Currency Word Problems

Question 424Question →

Match each polygon description based on its angle properties to the correct geometric shape.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Match the

540^\circ

sum to Pentagon, the

120^\circ

interior angle to Regular Hexagon, the

1080^\circ

sum to Octagon, and the equal interior/exterior sums to Quadrilateral.

Each match correctly applies the sum of interior angles formula,

S = (n-2) \times 180^\circ

, or relies on the fact that the sum of exterior angles for any convex polygon is exactly

360^\circ

. Solving for

n

reveals the specific polygon.

Step-by-Step Solution

1

Find the number of sides for a polygon with a

540^\circ

interior angle sum.

n = 5

We apply the formula

(n-2) \times 180 = 540

, which simplifies to

n-2 = 3

.

2

Determine the number of sides for a regular polygon with a

120^\circ

interior angle.

n = 6

The exterior angle is

180^\circ - 120^\circ = 60^\circ

. The number of sides is

360^\circ / 60^\circ = 6

.

3

Find the number of sides for a polygon with a

1080^\circ

interior angle sum.

n = 8

Applying the formula gives

(n-2) \times 180 = 1080

, which simplifies to

n-2 = 6

.

4

Identify the polygon where the interior angle sum equals the exterior angle sum.

n = 4

Since the exterior sum is always

360^\circ

, we set

(n-2) \times 180 = 360

, yielding

n-2 = 2

.

Key Concept

Polygon Interior and Exterior Angle Formulas

Estimated Time:1m 30s

Question 425Question →

A military textile facility produces $1,200$ combat uniform shirts over a standard $5$ -day workweek. If the facility operates for a single $8$ -hour shift each day, what is the facility's production rate in shirts per hour?

30

150

240

300

Show answer & explanation

Answer: $30$

Answer

The facility's production rate is

30

shirts per hour.

To find the hourly production rate, first determine the total hours worked by multiplying the number of days by the hours per shift (

5 \text{ days} \times 8 \text{ hours} = 40 \text{ hours}

). Then, divide the total production by the total hours (

1,200 \text{ shirts} \div 40 \text{ hours} = 30 \text{ shirts per hour}

).

Step-by-Step Solution

1

Calculate the total number of hours the facility operated.

5 \text{ days} \times 8 \text{ hours/day} = 40 \text{ hours}

To find the hourly rate, the total time must be expressed in hours.

2

Divide the total number of shirts produced by the total number of hours.

1,200 \div 40 = 30 \text{ shirts per hour}

Dividing total quantity by total time yields the unit rate.

Key Concept

Unit Rates and Time Conversions

Question 426Question →

Given the system of equations below, what is the value of the expression $x - 2y$ ?

5x - 13 = 4y

2x + y = 13

-1

-16

-13

11

Show answer & explanation

Answer: -1

Answer

The correct value of the expression is -1.

Solving the system of equations by rearranging and using elimination yields the solution x = 5 and y = 3. Substituting these values into the target expression x - 2y gives 5 - 2(3), which evaluates to 5 - 6 = -1.

Step-by-Step Solution

1

Rearrange the first equation into standard form to align the variables.

5x - 4y = 13

Aligning the variables vertically makes it much easier to use the elimination method.

2

Multiply the entire second equation by 4.

8x + 4y = 52

This creates opposite coefficients (-4y and +4y) so the y variable will cancel out when the equations are added.

3

Add the rearranged first equation and the modified second equation together.

(5x - 4y) + (8x + 4y) = 13 + 52 \Rightarrow 13x = 65 \Rightarrow x = 5

Eliminating y isolates x, allowing us to find its numerical value.

4

Substitute x = 5 back into the original second equation to solve for y.

2(5) + y = 13 \Rightarrow 10 + y = 13 \Rightarrow y = 3

This completes the solution pair (x, y) for the system of equations.

5

Substitute x = 5 and y = 3 into the requested expression

x - 2y

.

5 - 2(3) = 5 - 6 = -1

The question asks for the evaluated value of this specific expression, not just the solution coordinates.

Key Concept

Solving Systems of Linear Equations using elimination or substitution, followed by algebraic evaluation.

Question 427Question →

A circular stage has a radius of $5$ meters. Which of the following represents the total area of the stage, in square meters?

10\pi

5\pi

25\pi

100\pi

Show answer & explanation

Answer: $25\pi$

Answer

The area of the stage is

25\pi

square meters.

The correct answer is found by using the formula for the area of a circle,

A = \pi r^2

. Substituting the radius of 5 meters into the formula gives

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

.

Step-by-Step Solution

1

Identify the correct formula for the area of a circle.

A = \pi r^2

The question asks for the total area of a circular surface.

2

Substitute the given radius into the area formula.

A = \pi (5)^2

The problem states that the radius of the stage is 5 meters.

3

Calculate the final area.

A = 25\pi

Squaring 5 yields 25, resulting in an area of

25\pi

.

Key Concept

Area of a Circle

Estimated Time:45s

Question 428Question →

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

3x + 2y = 16

x - 2y = 0

Show answer & explanation

Answer: 4; x=4; x = 4; four

Answer

4

Because the coefficients of

y

in the two equations are opposites (

+2

and

-2

), the fastest approach is the elimination method. Adding the equations top-to-bottom yields

(3x + x) + (2y - 2y) = 16 + 0

, which simplifies directly to

4x = 16

. Dividing both sides by

4

gives

x = 4

.

Step-by-Step Solution

1

Add the two equations together.

4x = 16

The

2y

and

-2y

terms are opposites, so adding the equations eliminates the

y

variable entirely.

2

Divide both sides of the resulting equation by 4.

x = 4

This isolates the variable

x

to find its exact numerical value.

Key Concept

Solving systems of linear equations using the elimination method.

Question 429Question →

Determine the unknown rate or time for each of the following military travel profiles by matching them to their correctly calculated values.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The supply truck travels at

52

mph, the aircraft takes

4

hours, the infantry squad marches at

3

mph, and the destroyer takes

4.5

hours.

Each match is correct because the standard distance relationships (

r = d/t

or

t = d/r

) are applied after standardizing all units. Knots are correctly identified as speed, and minutes are accurately converted into fractions of an hour.

Step-by-Step Solution

1

Identify the given variables and the missing variable for each travel profile.

Two scenarios give distance and time (requiring rate), while two give distance and rate (requiring time).

This determines whether to use the formula

r = d/t

or

t = d/r

.

2

Convert mixed time units into a single fractional or decimal unit of hours.

2

hours

30

minutes becomes

2.5

hours.

2

hours

40

minutes becomes

2 \frac{2}{3}

or

8/3

hours.

Rates expressed in mph require time to be represented entirely in hours before dividing.

3

Calculate the missing values using the appropriate distance formula.

Truck:

130 \div 2.5 = 52

. Aircraft:

1,200 \div 300 = 4

. Squad:

8 \div (8/3) = 3

. Destroyer:

135 \div 30 = 4.5

.

Applying the correct operational relationship yields the final matched answers.

Key Concept

Calculating Rate and Time with Unit Conversions

Question 430Question →

A rectangular community garden has an area of $120$ square feet. If the length of the garden is $15$ feet, what is the width of the garden, in feet?

Show answer & explanation

Answer: 8

Answer

8

The area of a rectangle is calculated by multiplying its length by its width (

A = l \times w

). To find the missing width when the area and length are known, you divide the total area by the length (

w = A \div l

). Dividing

120

by

15

results in exactly

8

feet.

Step-by-Step Solution

1

Identify the given values from the problem

Area = 120, Length = 15

To know which values to substitute into the formula for the area of a rectangle.

2

Apply the area formula for a rectangle

120 = 15 \times w

The formula relates the total area to the length and width: Area = length \times width.

3

Solve for the unknown width

w = 8

Dividing the area by the length isolates the width variable, giving the final dimension.

Key Concept

Calculating a missing dimension of a rectangle given its area and one other dimension.

Estimated Time:45s

Question 431Question →

A graphic designer places $\$8,000$ into a high-yield savings certificate that pays an annual simple interest rate of $4.5\%$ . Upon maturity, the account has generated exactly $\$270$ in interest. How many months was the money invested?

0.75

7.5

9

12

Show answer & explanation

Answer: 9

Answer

The money was invested for exactly 9 months.

The investment earns

\$360

in a full year (

\$8,000 \times 0.045 = \$360

). Since the actual interest earned is

\$270

, the time in years is

270 / 360

, which simplifies to

3/4

or

0.75

years. Multiplying

0.75

years by

12

months per year gives exactly

9

months.

Step-by-Step Solution

1

Identify the given variables and the simple interest formula.

Principal (

P

) =

\$8,000

, Interest (

I

) =

\$270

, Rate (

r

) =

4.5\%

or

0.045

. The formula is

I = P \times r \times t

.

Setting up the standard formula is the first step to isolate and find the unknown time variable.

2

Calculate the amount of interest earned in one full year.

\$8,000 \times 0.045 = \$360

per year.

Finding the annual interest provides the denominator needed to determine what fraction of a year the money was invested.

3

Divide the actual interest earned by the annual interest to find the time in years.

\$270 / \$360 = 0.75

years.

The ratio of earned interest to annual interest gives the exact time the money remained in the account in years.

4

Convert the time from years into months.

0.75 \text{ years} \times 12 \text{ months/year} = 9 \text{ months}

.

The question specifically asks for the time duration in months, requiring a final unit conversion.

Key Concept

Calculating the time duration of an investment using the simple interest formula and converting the resulting fractional year into months.

Estimated Time:1m 30s

Question 432Question →

To determine the critical points of a trajectory, a physicist needs to fully factor the quadratic expression $4x^2 - 8x - 60$ . What is the completely factored equivalent of this expression?

4(x - 5)(x + 3)

4(x + 5)(x - 3)

(4x - 20)(x + 3)

4(x - 5)(x - 3)

Show answer & explanation

Answer: $4(x - 5)(x + 3)$

Answer

The completely factored form of the expression is

4(x - 5)(x + 3)

.

The correct answer is found by first extracting the Greatest Common Factor (GCF), which is

4

, leaving

4(x^2 - 2x - 15)

. Then, the inner trinomial

x^2 - 2x - 15

is factored into

(x - 5)(x + 3)

because

-5

and

3

multiply to

-15

and add to

-2

. Combining these gives the fully factored form

4(x - 5)(x + 3)

.

Step-by-Step Solution

1

Identify and factor out the Greatest Common Factor (GCF) from all terms in the expression

4x^2 - 8x - 60

.

The GCF of

4

,

-8

, and

-60

is

4

. Factoring it out yields

4(x^2 - 2x - 15)

.

Extracting the GCF first simplifies the remaining polynomial, making the trinomial easier to factor.

2

Factor the remaining quadratic trinomial

x^2 - 2x - 15

by finding two numbers that multiply to the constant term (

-15

) and add to the middle term's coefficient (

-2

).

The numbers are

-5

and

3

, so the trinomial factors to

(x - 5)(x + 3)

.

Expanding

(x - 5)(x + 3)

gives

x^2 + 3x - 5x - 15

, which simplifies back to

x^2 - 2x - 15

.

3

Combine the GCF and the factored binomials to write the completely factored form.

4(x - 5)(x + 3)

This represents the original expression broken down into its prime polynomial factors.

Key Concept

Factoring polynomials by first extracting the Greatest Common Factor (GCF) and then factoring the resulting quadratic trinomial.

Estimated Time:1m 0s

Question 433Question →

An aviation structural mechanic is processing four requisition forms for aircraft maintenance. Calculate the total cost (materials plus shipping) for each order and arrange them in sequence from least to greatest total cost.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest total cost is: Order 4 (

\$65.30

), Order 2 (

\$66.00

), Order 3 (

\$66.20

), and Order 1 (

\$71.50

).

Evaluating each order gives the following totals: Order 4 is

\$65.30

, Order 2 is

\$66.00

, Order 3 is

\$66.20

, and Order 1 is

\$71.50

. Arranging these from least to greatest yields the sequence: Order 4, Order 2, Order 3, Order 1.

Step-by-Step Solution

1

Calculate the total cost for Order 1.

15 \times 4.20 = 63.00

. Adding shipping:

63.00 + 8.50 = 71.50

. Total is

\$71.50

.

To determine the complete cost of the spark plugs including the fee.

2

Calculate the total cost for Order 2.

4 \times 12.75 = 51.00

. Adding shipping:

51.00 + 15.00 = 66.00

. Total is

\$66.00

.

To determine the complete cost of the air filters including the fee.

3

Calculate the total cost for Order 3.

20 \times 2.85 = 57.00

. Adding shipping:

57.00 + 9.20 = 66.20

. Total is

\$66.20

.

To determine the complete cost of the safety wire including the fee.

4

Calculate the total cost for Order 4.

8 \times 7.60 = 60.80

. Adding shipping:

60.80 + 4.50 = 65.30

. Total is

\$65.30

.

To determine the complete cost of the sealant including the fee.

5

Arrange the calculated totals from least to greatest.

The values in ascending order are

\$65.30

(Order 4),

\$66.00

(Order 2),

\$66.20

(Order 3), and

\$71.50

(Order 1).

To answer the question's requirement of sorting the final values.

Key Concept

Decimal Operations and Currency Word Problems

Question 434Question →

An events coordinator receives a guaranteed weekly salary of $\$900$ . On top of this, the coordinator is paid a $4\%$ commission on the first $\$15,000$ of event contracts secured each week, and a $7\%$ commission on the value of any contracts above $\$15,000$ . If the coordinator secures $\$22,000$ in event contracts in a single week, what is the coordinator's total gross pay for that week?

\$1,009

\$1,090

\$1,990

\$2,440

Show answer & explanation

Answer: $\$1,990$

Answer

\$1,990

The total gross pay consists of the base salary plus the commission from both tiers. First, calculate the commission on the initial

\$15,000

:

4\%

of

\$15,000

is

\$600

. Next, find the amount eligible for the

7\%

tier by subtracting

\$15,000

from the total

\$22,000

, which leaves

\$7,000

. Calculate the commission on this remaining amount:

7\%

of

\$7,000

is

\$490

. Finally, add the guaranteed salary and both commission amounts:

\$900 + \$600 + \$490 = \$1,990

.

Step-by-Step Solution

1

Calculate the commission earned on the first tier.

4\%

of

\$15,000 = 0.04 \times 15,000 = \$600

The coordinator earns a

4\%

commission on the first

\$15,000

of contracts.

2

Determine the value of contracts in the second tier.

\$22,000 - \$15,000 = \$7,000

The

7\%

commission applies only to the amount exceeding the initial

\$15,000

.

3

Calculate the commission earned on the second tier.

7\%

of

\$7,000 = 0.07 \times 7,000 = \$490

Multiply the remaining contract value by the

7\%

rate.

4

Calculate the total gross pay.

\$900 + \$600 + \$490 = \$1,990

Add the guaranteed weekly salary to both commission amounts to find the total earnings.

Key Concept

Wages, Salaries, and Commissions

Estimated Time:1m 30s

Question 435Question →

A shipyard damage control team is evaluating four auxiliary water pumps. Match each pump's performance test results to its calculated unit rate in gallons per minute (GPM).

Click a left item, then click its matching right item

Show answer & explanation

Answer

Pump Alpha matches

150

GPM, Pump Bravo matches

140

GPM, Pump Charlie matches

160

GPM, and Pump Delta matches

170

GPM.

Each pump's unit rate is determined by dividing the total gallons removed by the total number of minutes. This yields the uniform rate in gallons per minute (GPM).

Step-by-Step Solution

1

Calculate the unit rate for Pump Alpha.

1,800 \div 12 = 150

gallons per minute.

Dividing the total volume by the total time gives the unit rate.

2

Calculate the unit rate for Pump Bravo.

3,500 \div 25 = 140

gallons per minute.

Dividing the total volume by the total time gives the unit rate.

3

Calculate the unit rate for Pump Charlie.

4,800 \div 30 = 160

gallons per minute.

Dividing the total volume by the total time gives the unit rate.

4

Calculate the unit rate for Pump Delta.

6,800 \div 40 = 170

gallons per minute.

Dividing the total volume by the total time gives the unit rate.

Key Concept

Calculating unit rates by dividing a total quantity by the elapsed time to find the amount per single unit of time.

Question 436Question →

A rescue helicopter flies $150$ miles to an offshore oil rig in $90$ minutes. The return flight to the base along the same route takes $150$ minutes due to strong headwinds. What is the helicopter's average speed in miles per hour for the entire round trip?

75

80

125

37.5

Show answer & explanation

Answer: 75

Answer

75

The average speed for an entire trip is found by dividing the total round-trip distance (

300

miles) by the total round-trip time in hours (

4

hours), which gives

75

miles per hour.

Step-by-Step Solution

1

Calculate the total distance traveled for the round trip.

150 \text{ miles} + 150 \text{ miles} = 300 \text{ miles}

Average speed is based on the entire distance covered during the trip.

2

Convert the flight times from minutes to hours.

90 \text{ minutes} = 1.5 \text{ hours}

, and

150 \text{ minutes} = 2.5 \text{ hours}

The question asks for the speed in miles per hour, so the time units must be converted to hours.

3

Calculate the total time elapsed for the round trip.

1.5 \text{ hours} + 2.5 \text{ hours} = 4 \text{ hours}

Average speed is based on the total time taken for the entire journey.

4

Divide the total distance by the total time.

300 \div 4 = 75 \text{ mph}

The formula for average speed is Total Distance divided by Total Time.

Key Concept

Average Speed

Question 437Question →

Two horizontal parallel lines, $L_1$ and $L_2$ , are drawn with $L_1$ above $L_2$ . A point $P$ is located between the two lines. Point $A$ is on $L_1$ and point $B$ is on $L_2$ , such that $P$ is positioned to the right of both $A$ and $B$ . Segment $AP$ and segment $BP$ form a zigzag pattern. The interior angle at $A$ , below $L_1$ and opening towards $P$ , measures $(5x - 4)^\circ$ . The interior angle at $B$ , above $L_2$ and opening towards $P$ , measures $(3x + 16)^\circ$ . The angle $\angle APB$ , opening towards the left, measures $(11x - 42)^\circ$ . What is the measure of the complement of the smaller of the two angles at $A$ and $B$ ?

44^\circ

52^\circ

20^\circ

110^\circ

Show answer & explanation

Answer: $20^\circ$

Answer

The correct answer is

20^\circ

.

Based on the properties of parallel lines intersected by transversals, an auxiliary parallel line drawn through the vertex of the zigzag shows that the middle angle equals the sum of the two outer angles. Setting up the equation

(5x - 4) + (3x + 16) = 11x - 42

yields

x = 18

. Substituting this value back in gives angles of

86^\circ

and

70^\circ

. The smaller angle is

70^\circ

, and its complement is found by subtracting it from

90^\circ

, which gives

20^\circ

.

Step-by-Step Solution

1

Apply geometric properties to set up an equation.

(5x - 4) + (3x + 16) = 11x - 42

By drawing an auxiliary line through point P parallel to L1 and L2, the angle at P is split into two alternate interior angles. Therefore, the angle opening left equals the sum of the two angles opening right.

2

Combine like terms on the left side of the equation.

8x + 12 = 11x - 42

Combining the

x

terms (

5x + 3x = 8x

) and the constants (

-4 + 16 = 12

) simplifies the equation.

3

Solve for x.

x = 18

Subtract

8x

from both sides to get

12 = 3x - 42

. Add

42

to both sides to get

54 = 3x

. Divide by

3

to find

x = 18

.

4

Substitute x back into the expressions to find the measures of the angles at A and B.

Angle at A =

86^\circ

, Angle at B =

70^\circ

Angle at A:

5(18) - 4 = 90 - 4 = 86^\circ

. Angle at B:

3(18) + 16 = 54 + 16 = 70^\circ

.

5

Identify the smaller angle and calculate its complement.

20^\circ

The smaller angle is

70^\circ

. The complement of an angle is

90^\circ

minus the angle, so

90^\circ - 70^\circ = 20^\circ

.

Key Concept

Parallel Lines and Transversals (Zigzag Theorem)

Question 438Question →

Which ordered pair $(x, y)$ is the solution to the following system of linear equations?

3x + y = 17

x - y = 3

(5, 2)

(2, 5)

(5, 8)

(20, 17)

Show answer & explanation

Answer: $(5, 2)$

Answer

The correct solution is the ordered pair

(5, 2)

.

The correct answer is (5, 2). By adding the two equations, the y terms cancel each other out, leaving 4x = 20. Dividing both sides by 4 gives x = 5. Substituting this value back into the second equation (5 - y = 3) allows us to solve for y, which is 2.

Step-by-Step Solution

1

Add the two equations together to eliminate the y variable.

(3x + y) + (x - y) = 17 + 3 \Rightarrow 4x = 20

Since the coefficients of y are opposites (1 and -1), adding the equations cancels out the y terms.

2

Solve for x by dividing both sides of the resulting equation by 4.

x = 5

Dividing by the coefficient of x isolates the variable.

3

Substitute the value of x back into the second original equation.

5 - y = 3

Substituting the known x value leaves an equation with only one variable, y.

4

Solve for y.

-y = 3 - 5 \Rightarrow -y = -2 \Rightarrow y = 2

Isolating y provides the final part of the solution.

Key Concept

Solving systems of linear equations using the elimination method

Question 439Question →

Consider the following system of linear equations:

4(x - 2) + y = 3x

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

What is the value of the expression $x - y$ ?

Show answer & explanation

Answer: 2

Answer

2

The correct answer is derived by first simplifying both equations into standard form (x + y = 8 and 2x - 3y = 1). By applying the elimination method, we find the system's solution is x = 5 and y = 3. Finally, evaluating the specific requested expression gives us x - y = 5 - 3 = 2.

Step-by-Step Solution

1

Simplify the first equation.

x + y = 8

Equations must be in standard form to easily apply elimination or substitution.

2

Simplify the second equation.

2x - 3y = 1

Expanding the terms and grouping variables on one side creates the standard form.

3

Multiply the simplified first equation by 3.

3x + 3y = 24

This aligns the y-coefficients (3y and -3y) for the elimination method.

4

Add the modified first equation to the simplified second equation.

5x = 25

Adding the equations eliminates the y variable, allowing us to solve for x.

5

Divide both sides by 5.

x = 5

Isolates the variable x.

6

Substitute x = 5 into the simplified first equation (x + y = 8).

y = 3

This finds the value of the remaining variable, y.

7

Evaluate the target expression (x - y).

5 - 3 = 2

The question specifically asks for the difference between x and y, not just their individual values.

Key Concept

Solving systems of linear equations involving pre-simplification

Question 440Question →

Two cyclists, Sarah and Mike, start at opposite ends of a $60$ -mile trail and ride directly toward each other. Sarah rides at a constant speed of $12$ mph, and Mike rides at a constant speed of $18$ mph. How many hours will it take for them to meet?

Show answer & explanation

Answer: 2

Answer

2

Because the cyclists are moving directly toward each other, their gap closes at a combined rate of

30

mph (

12

mph +

18

mph). Dividing the total initial distance of

60

miles by this combined rate yields exactly

2

hours.

Step-by-Step Solution

1

Calculate the combined relative speed.

30 mph

When objects move toward each other from opposite directions, their relative speed is the sum of their individual speeds (

12 + 18 = 30

).

2

Calculate the time required to meet.

2 hours

Using the formula Time = Distance / Rate, divide the total distance (

60

miles) by the relative speed (

30

mph).

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