A student provided the steps for solving an equation. Which statement describes the error in the solution?

Question

Equation: 8 - a/2 = 3 (4 - 5a)

Solution: 16 - a = 6 (4 - 5a) (step 1)
16 - a = 24 - 5a (step 2)
16 + 4a = 24 (step 3)
4a = 8 (step 4)
a = 2 (step 5)

Answer 1

The error in the student's solution occurs in step 1. The original equation is \( 8 - \frac{a}{2} = 3(4 - 5a) \).

To clarify the error, let's analyze the student's first step:

Original equation: \[ 8 - \frac{a}{2} = 3(4 - 5a) \]

The student incorrectly multiplied both sides by 2 without maintaining the correct structure of the equation. Instead of obtaining \( 16 - a = 6(4 - 5a) \), they should have first expanded the right side correctly.

The correct approach would involve applying the multiplication correctly:

1. Multiply both sides to eliminate the fraction if they choose to do so, or distribute the 3 on the right.
2. After correctly distributing, the equation from the original should be formed correctly.

Correctly applying the right side gives: \[ 8 - \frac{a}{2} = 12 - 15a \]

Now, to eliminate \( \frac{a}{2} \), it's usually simpler to multiply the entire equation by 2:

• Which yields: \[ 16 - a = 24 - 30a \]

Thus, the main error lies in how the transition was made between step 1 and step 2. The correct first transformation should have been:

1. Step 1: Correctly expand or clear fractions to maintain algebraic equality.

The solution remains flawed from the beginning, leading to incorrect subsequent steps, resulting in an incorrect final answer.

Answer 2

The table gives points on a line. What is the equation of the line

x y
1 3
2 1
3 -1

Answer 3

To find the equation of the line that passes through the given points, we first need to determine the slope of the line using the \(x\) and \(y\) coordinates.

Given Points

1. \((1, 3)\)
2. \((2, 1)\)
3. \((3, -1)\)

Step 1: Calculate the Slope \(m\)

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

We'll use the first two points \((1, 3)\) and \((2, 1)\):

\[ m = \frac{1 - 3}{2 - 1} = \frac{-2}{1} = -2 \]

Step 2: Use the Point-Slope Form

Now that we have the slope \(m = -2\), we can use the point-slope form of a line equation, which is:

\[ y - y_1 = m(x - x_1) \]

Using point \((1, 3)\):

\[ y - 3 = -2(x - 1) \]

Step 3: Simplify to Slope-Intercept Form \(y = mx + b\)

Now, we'll simplify this equation:

\[ y - 3 = -2x + 2 \]

Adding 3 to both sides:

\[ y = -2x + 5 \]

Step 4: Verify with the Remaining Point

We should verify that this equation holds true for the other points:

• For \(x = 2\): \[ y = -2(2) + 5 = -4 + 5 = 1 \] This matches point \((2, 1)\).

• For \(x = 3\): \[ y = -2(3) + 5 = -6 + 5 = -1 \] This matches point \((3, -1)\).

Final Equation

Thus, the equation of the line that passes through the points is:

\[ \boxed{y = -2x + 5} \]

Answer 4

Ramon teaches the cha- cha a lively and playful dance of Cuban origin, at a dance studio. Ramon earns $50 for each lesson and pays the studio $75 per day to rent space. The equation 50x - 75 = 175 represents Ramon's profit on Saturday. What does the solution x = 5 mean in his situation?

Answer 5

In Ramon's situation, the equation \(50x - 75 = 175\) represents his profit from teaching cha-cha dance lessons on Saturday. Let's break down the components of the equation:

• \(50x\): This term represents the total earnings from dance lessons that Ramon gives on Saturday. Here, \(x\) is the number of lessons he teaches, and each lesson earns him $50.
• \(-75\): This term represents the fixed cost of renting the studio space for the day, which is $75.
• \(= 175\): This is the amount of profit he wants to achieve for that day, which is $175.

Analyzing the Equation

The equation itself can be interpreted step by step:

1. Calculate Total Earnings: The total earnings from \(x\) lessons is \(50x\).
2. Subtract Costs: From those earnings, he subtracts the studio rental cost of $75.
3. Set Profit Target: The equation sets this profit (earnings minus costs) to equal $175.

What Does \(x = 5\) Mean?

To understand what the solution \(x = 5\) means:

• Interpretation: The solution \(x = 5\) indicates that Ramon teaches 5 lessons on Saturday.

• Profit Calculation: We can verify this by substituting \(x = 5\) back into the equation:

  \[ 50(5) - 75 = 175 \]

  This simplifies to:

  \[ 250 - 75 = 175 \]

  Which confirms that:

  \[ 175 = 175 \]

Conclusion

Thus, \(x = 5\) means that Ramon needs to teach 5 dance lessons on Saturday to achieve a profit of $175 after paying the studio rental fee of $75.

Answer 6

The table shows the monthly cost of an online music subscription based on the number of songs downloaded. How can you find the cost, C, of downloading n songs, for n > 1?

Number of Songs (n) Cost (C)
1 $5.00
2 $5.35
3 $5.70
4 $6.05

Answer 7

To find the cost, \( C \), of downloading \( n \) songs based on the table provided, we can analyze the values given to determine a relationship or pattern.

Given Data:

• For \( n = 1 \), \( C = 5.00 \)
• For \( n = 2 \), \( C = 5.35 \)
• For \( n = 3 \), \( C = 5.70 \)
• For \( n = 4 \), \( C = 6.05 \)

Step 1: Calculate the Increase in Cost

Let’s calculate the increase in cost for each additional song:

• From 1 to 2 songs: \[ 5.35 - 5.00 = 0.35 \]

• From 2 to 3 songs: \[ 5.70 - 5.35 = 0.35 \]

• From 3 to 4 songs: \[ 6.05 - 5.70 = 0.35 \]

Step 2: Determine a Pattern

From the calculations, we see that the cost increases by $0.35 for each additional song downloaded after the first song.

Step 3: Formulate the Cost Function

With the base cost of $5.00 for the first song and an increase of $0.35 for each additional song, we can write a formula to express the cost \( C \) as a function of the number of songs \( n \):

• When \( n = 1 \), \( C = 5.00 \)
• For \( n > 1 \), the cost can be expressed as:

\[ C(n) = 5.00 + 0.35(n - 1) \]

Step 4: Simplify the Formula

We can simplify this equation:

\[ C(n) = 5.00 + 0.35n - 0.35 \] \[ C(n) = 4.65 + 0.35n \]

Conclusion

Thus, the cost \( C \) of downloading \( n \) songs for \( n > 1 \) can be expressed by the equation:

\[ C(n) = 0.35n + 4.65 \]

This formula will allow you to calculate the cost for any \( n > 1 \).

Answer 8

Ashley can ride her bicycle 15 miles in 2 hours. There are 60 minutes in 1 hour and there are 1,760 yards in 1 mile/ how many yards does ashley travel each minute

Answer 9

To find out how many yards Ashley travels each minute, we can follow these steps:

Step 1: Calculate Total Miles Traveled per Minute

Ashley rides 15 miles in 2 hours. First, we need to find out how many miles she travels in 1 hour, and then in 1 minute.

1. Miles per hour: \[ \text{Miles per hour} = \frac{15 \text{ miles}}{2 \text{ hours}} = 7.5 \text{ miles per hour} \]

2. Miles per minute: Since there are 60 minutes in 1 hour: \[ \text{Miles per minute} = \frac{7.5 \text{ miles}}{60 \text{ minutes}} = 0.125 \text{ miles per minute} \]

Step 2: Convert Miles per Minute to Yards per Minute

We know that there are 1,760 yards in 1 mile. So, to find the number of yards per minute, we multiply the number of miles per minute by the number of yards per mile:

\[ \text{Yards per minute} = 0.125 \text{ miles per minute} \times 1,760 \text{ yards per mile} \]

Calculating this:

\[ \text{Yards per minute} = 0.125 \times 1,760 = 220 \text{ yards per minute} \]

Conclusion

Ashley travels 220 yards each minute.