Linear Equations in One Variable

42 questions

All types42 Multiple choice23 Multiple select11 Numeric8

A digital archive charges an annual subscription fee of $\$120$ , which includes $50$ free document downloads per year. For each additional document downloaded beyond the first $50$ , the archive charges a flat rate of $\$1.50$ . If a research group paid a total of $\$231$ to the archive last year, how many total documents did the research group download?

114

124

204

Show answer & explanation

Answer: 124

Answer

The total number of documents downloaded by the research group last year was 124.

Subtracting the base subscription fee of

120 from the total payment of

231 leaves

111 spent strictly on extra downloads. Dividing

111 by the $1.50 per-document rate yields 74 extra downloads. Adding the 50 included free downloads gives a total of 124 documents downloaded.

Step-by-Step Solution

Define the linear equation representing total cost.

120 + 1.50(x - 50) = 231

, where

x

represents the total number of documents downloaded.

The base subscription fee is

120, and the per-document rate of

1.50 applies only to documents in excess of the initial 50 free documents.

Isolate the variable term by subtracting the base subscription fee from both sides.

1.50(x - 50) = 231 - 120 = 111

This determines the portion of the total cost that resulted strictly from additional downloads.

Solve for the number of additional documents

(x - 50)

x - 50 = \frac{111}{1.50} = 74

Dividing the extra cost of

111 by the per-document rate of

1.50 gives the exact count of extra downloads.

Add the initial 50 free documents to find the total document count

x

x = 74 + 50 = 124

The question asks for the total documents downloaded, which combines the 50 free downloads and the 74 additional paid downloads.

Key Concept

Linear Equations in One Variable

Estimated Time:1m 30s

Question 42Question →

If $x$ is the solution to the linear equation $\frac{4x - 3}{3} - \frac{x + 2}{4} = \frac{5x}{6}$ , which of the following statements about $x$ must be true? Select all such statements.

Select all that apply

x

is a multiple of 3

2x - 5 > 6

x^2 - 5x = 6

x

is a prime number

x < 5

Show answer & explanation

Answer: $x$ is a multiple of 3; $2x - 5 > 6$ ; $x^2 - 5x = 6$

Answer

The statements establishing that

x

is a multiple of 3, that

2x - 5 > 6

, and that

x^2 - 5x = 6

are all true.

Solving the linear equation yields

x = 6

. Substituting

x = 6

into each statement shows that

x

is a multiple of 3 (

6 = 3 \times 2

2x - 5 > 6

evaluates to

7 > 6

, and

x^2 - 5x = 6

evaluates to

36 - 30 = 6

. All three statements are correct.

Step-by-Step Solution

Clear denominators by multiplying both sides by the least common multiple.

12 \cdot \left(\frac{4x - 3}{3} - \frac{x + 2}{4}\right) = 12 \cdot \left(\frac{5x}{6}\right) \implies 4(4x - 3) - 3(x + 2) = 2(5x)

The least common multiple of 3, 4, and 6 is 12.

Distribute factors and combine like terms.

16x - 12 - 3x - 6 = 10x \implies 13x - 18 = 10x

Distributing

-3

into

(x + 2)

yields

-3x - 6

Isolate

x

on one side of the equation.

13x - 10x = 18 \implies 3x = 18 \implies x = 6

Subtract

10x

and add 18 to isolate the variable term.

Evaluate each given statement using

x = 6

Statement 1:

6

is a multiple of 3 (True). Statement 2:

2(6) - 5 = 7 > 6

(True). Statement 3:

6^2 - 5(6) = 6

(True). Statement 4: 6 is prime (False). Statement 5:

6 < 5

(False).

Test each option against the calculated solution

x = 6

Key Concept

Linear Equations in One Variable

Estimated Time:1m 30s

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Linear Equations in One Variable

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Step-by-Step Solution

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