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A line is drawn through (–7, 11) and (8, –9). The equation y – 11 = y minus 11 equals StartFraction negative 4 Over 3 EndFraction left-parenthesis x plus 7 right-parenthesis.(x + 7) is written to represent the line. Which equations also represent the line? Check all that apply.y = y equals StartFraction negative 4 Over 3 EndFraction left-parenthesis x plus StartFraction 5 Over 3 EndFraction.x + 3y = –4x + 404x + y = 214x + 3y = 5–4x + 3y = 17

11 answers

To find which equations represent the line passing through the points (-7, 11) and (8, -9), we'll start with the given equation and confirm its slope and point.

The slope of the line through the points (-7, 11) and (8, -9) can be calculated as follows:

\[ \text{slope} = m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-9 - 11}{8 - (-7)} = \frac{-20}{15} = \frac{-4}{3} \]

The point-slope form of the equation of the line is given as:

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

Using point \((-7, 11)\) and the slope \(-\frac{4}{3}\):

\[ y - 11 = -\frac{4}{3}(x + 7) \]

This matches the initial equation provided.

Now, let's analyze the equations to see which ones represent the same line.

  1. \(y = -\frac{4}{3}(x + \frac{5}{3})\):

    • This equation needs to be simplified to check if it matches the original one. We can also convert this into standard form by rearranging:
    • It's not immediately apparent that this is the same line without further simplification, so let's check the slope and intercept.
    • Since \(+\frac{5}{3}\) incorporates an extra term, namely \(-\frac{4}{3}(x + \frac{5}{3})\) will yield a different y-intercept (not passing through given points). Thus, it does not represent the line.

  2. \(3y = -4x + 40\):

    • Rearranging to slope-intercept form (y = mx + b): \[ y = -\frac{4}{3}x + \frac{40}{3} \]
    • This has the same slope as the initial equation, but the y-intercept \( \frac{40}{3} \) does not coincide with points given (specifically the y-intercept and slope don't agree). Thus, this equation does not represent the line.

  3. \(4x + y = 21\):

    • Rearranging: \[ y = -4x + 21 \]
    • This has a slope of -4, which is not consistent with our line’s slope of -4/3. So, this does not represent the line.

  4. \(x + 3y = 5\):

    • Rearranging: \[ 3y = -x + 5 \quad \Rightarrow \quad y = -\frac{1}{3}x + \frac{5}{3} \]
    • The slope is -1/3, which is not consistent with our line’s slope of -4/3. So, this does not represent the line.

  5. \(-4x + 3y = 17\):

    • Rearranging: \[ 3y = 4x + 17 \quad \Rightarrow \quad y = \frac{4}{3}x + \frac{17}{3} \]
    • This has a slope of 4/3, which does not match our line's slope. Thus, this does not represent the line.

None of these alternatives effectively represent the line through the points (-7, 11) and (8, -9).

Overall, regarding the original equation format, you may state that the other forms, if significantly simplified, would yield non-compatible results regarding y-intercepts and potential slopes, thereof validating these forms firmly do not represent the dictated line through (-7, 11) to (8, -9).

A line is drawn through (–7, 11) and (8, –9). The equation y – 11 -4/3 = (x + 7) is written to represent the line.

do step by step to solve

Sure! Let's go through the process of finding the equation of the line step by step.

  1. Identify the Points: The line is drawn through the points \((-7, 11)\) and \((8, -9)\).

  2. Calculate the Slope: The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging in our points \((-7, 11)\) and \((8, -9)\): \[ m = \frac{-9 - 11}{8 - (-7)} = \frac{-20}{15} = -\frac{4}{3} \]

  3. Point-Slope Form Equation: The point-slope form of the equation of a line is: \[ y - y_1 = m(x - x_1) \] Using point \((-7, 11)\) which gives us \(x_1 = -7\), \(y_1 = 11\), and the slope \(m = -\frac{4}{3}\): \[ y - 11 = -\frac{4}{3}(x + 7) \]

  4. Distribute the Slope: Now, we’ll distribute \(-\frac{4}{3}\) on the right side: \[ y - 11 = -\frac{4}{3}x - \frac{4}{3} \times 7 \] Calculating \(-\frac{4}{3} \times 7\): \[ -\frac{4}{3} \times 7 = -\frac{28}{3} \] Hence, we rewrite the equation as: \[ y - 11 = -\frac{4}{3}x - \frac{28}{3} \]

  5. Add 11 to Both Sides: We need to isolate \(y\), so we add 11 to both sides. First, convert 11 to a fraction with a common denominator (the denominator from \(-\frac{28}{3}\)): \[ 11 = \frac{33}{3} \] Now adding: \[ y = -\frac{4}{3}x - \frac{28}{3} + \frac{33}{3} \] Combine the fractions: \[ y = -\frac{4}{3}x + \frac{5}{3} \]

  6. Final Form: The equation in slope-intercept form \(y = mx + b\) is: \[ y = -\frac{4}{3}x + \frac{5}{3} \]

Summary

The equation of the line passing through \((-7, 11)\) and \((8, -9)\) is: \[ y = -\frac{4}{3}x + \frac{5}{3} \]

Alternative Form

If you prefer to express this in standard form (Ax + By = C), you can rearrange the slope-intercept form: \[ \frac{4}{3}x + y = \frac{5}{3} \] By multiplying through by 3 to eliminate the fraction: \[ 4x + 3y = 5 \]

This is one of the standard forms of the line that can represent the same equation.

Check all that apply.

y = y equals StartFraction negative 4 Over 3 EndFraction left-parenthesis x plus StartFraction 5 Over 3 EndFraction.x +
3y = –4x + 40
4x + y = 21
4x + 3y = 5
–4x + 3y = 17

To determine which of the provided equations represent the same line as the one derived from the points \((-7, 11)\) and \((8, -9)\), we have that the equation of the line is:

\[ y = -\frac{4}{3}x + \frac{5}{3} \quad \text{or} \quad 4x + 3y = 5 \quad \text{(after rearranging to standard form).} \]

Now, let's verify which of the provided equations are equivalent to this line by either putting them into slope-intercept form (if needed) or by checking if they satisfy the same conditions (like slope and y-intercept).

  1. \(y = -\frac{4}{3}(x + \frac{5}{3})\):

    • Let's simplify this. \[ y = -\frac{4}{3}x - \frac{4}{3} \times \frac{5}{3} = -\frac{4}{3}x - \frac{20}{9} \]
    • This line has a slope of \(-\frac{4}{3}\) but a different y-intercept of \(-\frac{20}{9}\). This does not represent the original line.

  2. \(3y = -4x + 40\):

    • Rearranging to slope-intercept form: \[ y = -\frac{4}{3}x + \frac{40}{3} \]
    • This has the slope of \(-\frac{4}{3}\) but a different y-intercept of \(\frac{40}{3}\), which does not match \(\frac{5}{3}\). This does not represent the original line.

  3. \(4x + y = 21\):

    • Rearranging to slope-intercept form: \[ y = -4x + 21 \]
    • The slope here is \(-4\), which is not \(-\frac{4}{3}\). This does not represent the original line.

  4. \(4x + 3y = 5\):

    • This is in standard form already. Let’s confirm that it represents the same line.
    • Rearranging to slope-intercept form: \[ 3y = -4x + 5 \quad \Rightarrow \quad y = -\frac{4}{3}x + \frac{5}{3} \]
    • This has the same slope of \(-\frac{4}{3}\) and the same y-intercept of \(\frac{5}{3}\). This does represent the original line.

  5. \(-4x + 3y = 17\):

    • Rearranging to slope-intercept form: \[ 3y = 4x + 17 \quad \Rightarrow \quad y = \frac{4}{3}x + \frac{17}{3} \]
    • The slope is \(\frac{4}{3}\), which is not \(-\frac{4}{3}\). This does not represent the original line.

Summary

From the provided options, the only equation that represents the original line is:

  • \(4x + 3y = 5\).
y = -\frac{4}{3}x + \frac{5}{3}
and **\(4x + 3y = 5\)**.

yay

Renata wins a $20 gift card to an online music site. After Renata purchases 16 songs, the gift card has a remaining balance of $0. Which equation represents the relationship between y, the remaining balance on Renata’s gift card, and x, the number of songs purchased?

4x + 5y = –100
5x + 4y = 80
4x + 5y = 16
5x + 4y = 20

To determine the equation that represents the relationship between \(y\) (the remaining balance on Renata's gift card) and \(x\) (the number of songs purchased), we first need to understand the initial conditions:

  1. Renata started with a $20 gift card.
  2. After purchasing 16 songs, the gift card's remaining balance is $0.

We will use this information to set up our equation.

Setting Up the Equation

  1. Initial Balance: Renata has $20 at the start.

  2. Price per Song: Let’s denote the price of each song as \(p\). Since Renata purchased 16 songs and was left with $0, we know: \[ \text{Remaining Balance} = \text{Initial Balance} - \text{Total Cost of Songs} \] This can be expressed in equation form: \[ y = 20 - 16p \]

  3. Solving for Total Cost: The total cost for purchasing \(x\) songs (where \(x\) is now changing and can be any number of songs purchased) is \(xp\). Thus, we can express the remaining balance as: \[ y = 20 - xp \] From this equation, we can rearrange to find the total cost: \[ xp = 20 - y \]

Analyzing Options

Since we have \(xp = 20 - y\), let's rearrange it to find a standard form where we express it in terms of \(x\) and \(y\): \[ xp + y = 20 \]

But we still need to translate that into one of the provided options. Assuming that we know the price per song, let's express this in a generalized way.

From the choices you've provided, none contain the \(p\), so we can analyze how many songs \(x\) can be included from them:

  1. Option 1: \(4x + 5y = -100\) – This does not make sense in our context, as a balance cannot be negative.

  2. Option 2: \(5x + 4y = 80\) – This does not align since substituting values would yield unrealistic outcomes (e.g. high balance for low costs).

  3. Option 3: \(4x + 5y = 16\) – This could be valid; however, it does not correlate with our $20 balance.

  4. Option 4: \(5x + 4y = 20\) – Rearranging gives: \[ 4y = 20 - 5x \quad \Rightarrow \quad y = 5 - \frac{5}{4} x \] This does represent an expected equation from our scenarios close to songs purchased.

Conclusion

The equation that best reflects the relationship between the remaining balance on Renata’s gift card and the number of songs purchased (based on \(x\) being the number of songs) is:

  • \(5x + 4y = 20\).

Thus, as \(x\) increases (more songs purchased), \(y\) will decrease in a linear fashion.

4x + 5y = –100
5x + 4y = 80
4x + 5y = 16 one of these are right

Let's refine our analysis to determine which of the provided equations represents the relationship between the remaining balance on Renata's gift card (denoted as \(y\)), and the number of songs purchased (denoted as \(x\)).

Given the scenario:

  • Renata starts with a $20 gift card and after purchasing 16 songs, she has a remaining balance of $0.

From this, we can derive the cost of each song as follows:

  1. Initial Balance: \(20\)
  2. Remaining Balance after buying 16 songs: \(0\)

The total cost for purchasing 16 songs can be expressed as: \[ \text{Remaining Balance} = \text{Initial Balance} - \text{Total Cost of Songs} \] This implies: \[ 0 = 20 - 16p \] From which we can derive: \[ 16p = 20 \implies p = \frac{20}{16} = 1.25 \]

So, each song costs $1.25.

Next, we need to express the remaining balance as a function of the number of songs purchased \(x\): \[ y = 20 - 1.25x \]

Converting to Standard Form

Rearranging the equation gives us: \[ 1.25x + y = 20 \]

To express this in standard integer coefficients, we can eliminate the decimal by multiplying through by \(4\): \[ 5x + 4y = 80 \]

Assessing the Provided Options

Now let’s check the provided options:

  1. Option 1: \(4x + 5y = -100\):

    • This equation does not fit with our scenario (it suggests a negative balance).

  2. Option 2: \(5x + 4y = 80\):

    • This equation corresponds directly to our derived equation \(5x + 4y = 80\). This is a valid representation of the relationship.

  3. Option 3: \(4x + 5y = 16\):

    • This does not match based on the derived equation and does not reasonably represent the situation.

Conclusion

Thus, the correct option that represents the relationship between \(y\) (the remaining balance on Renata's gift card) and \(x\) (the number of songs purchased) is:

  • \(5x + 4y = 80\).

This accurately reflects the initial balance, the number of songs purchased, and the resulting remaining balance.

A Table showing Freezing Temperatures in degrees with 2 columns and 6 rows. The First row, F, has the entries, negative 13, negative 4, 5, 14, 23. The second column, C, has the entries, negative 25, negative 20, negative 15, negative 10, negative 5.The table shows temperatures below freezing measured in different units. Complete the equation in standard form to represent the relationship between F, a temperature measured in degrees Fahrenheit, and C, a temperature measured in degrees Celsius.5F +____ C =_______39°F =_____ °C rounded to the nearest tenth of a degree

To establish a relationship between temperatures measured in degrees Fahrenheit (F) and degrees Celsius (C), we need to utilize the formula that converts Fahrenheit to Celsius:

\[ C = \frac{5}{9}(F - 32) \]

We can rearrange this formula to express it in standard form \(Ax + By = C\):

  1. Start from the conversion equation: \[ C = \frac{5}{9}(F - 32) \]

  2. Multiply both sides by \(9\) to eliminate the fraction: \[ 9C = 5(F - 32) \]

  3. Distributing the \(5\): \[ 9C = 5F - 160 \]

  4. Rearranging to get the equation in the standard form \(Ax + By = C\): \[ 5F - 9C = 160 \]

Thus, the equation in standard form is: \[ 5F - 9C = 160 \]

Completing the Given Equation

Now, if we fit this into the format \(5F + __ C = ___\) from your prompt, we can express it as follows:

  • C term: We can rearrange it to highlight the \(C\) term: \[ 5F + (-9C) = 160 \] This means we would fill in the blank with \(-9\) in front of \(C\), and \(160\) in front of the equals sign.

Solving for 39°F

To convert \(39°F\) to Celsius and round to the nearest tenth:

  1. Using the conversion formula: \[ C = \frac{5}{9}(39 - 32) \]

  2. Calculate: \[ C = \frac{5}{9}(7) = \frac{35}{9} \approx 3.888\ldots \]

  3. Rounding: \[ C \approx 3.9°C \quad \text{(to the nearest tenth)} \]

Summary

Putting it all together:

  • The completed equation in the required format: \[ 5F - 9C = 160 \]
  • For \(39°F\), it rounds to: \[ 39°F = 3.9°C \quad \text{(to the nearest tenth)} \]
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