Question
Graph the proportional relationship y = 2x by plotting points.
a. (2, 1) & (0, 0)
b. (2, 4) & (1, 2) & (0, 0)
c. (1, 4) & (0, 0)
d. (2, 6) & (1, 3) & (0, 0)
a. (2, 1) & (0, 0)
b. (2, 4) & (1, 2) & (0, 0)
c. (1, 4) & (0, 0)
d. (2, 6) & (1, 3) & (0, 0)
Answers
Answered by
GPT-4o mini
To graph the proportional relationship represented by the equation \( y = 2x \), you can plot points that satisfy the equation. We'll find the coordinates of \( y \) for different values of \( x \):
1. When \( x = 0 \):
- \( y = 2(0) = 0 \)
- Point: \( (0, 0) \)
2. When \( x = 1 \):
- \( y = 2(1) = 2 \)
- Point: \( (1, 2) \)
3. When \( x = 2 \):
- \( y = 2(2) = 4 \)
- Point: \( (2, 4) \)
4. When \( x = 3 \):
- \( y = 2(3) = 6 \)
- Point: \( (3, 6) \)
Now, let's evaluate the options based on the points satisfying the equation \( y = 2x \):
- **Option a**: (2, 1) & (0, 0)
- (2, 1) does not satisfy \( y = 2x \) since \( 1 \neq 4 \).
- **Option b**: (2, 4) & (1, 2) & (0, 0)
- All points are correct. (2, 4), (1, 2), and (0, 0) satisfy \( y = 2x \).
- **Option c**: (1, 4) & (0, 0)
- (1, 4) does not satisfy \( y = 2x \) since \( 4 \neq 2 \).
- **Option d**: (2, 6) & (1, 3) & (0, 0)
- (2, 6) does not satisfy \( y = 2x \) since \( 6 \neq 4 \) and (1, 3) does not satisfy since \( 3 \neq 2 \).
Based on the evaluations, the correct option is:
**b. (2, 4) & (1, 2) & (0, 0)**.
1. When \( x = 0 \):
- \( y = 2(0) = 0 \)
- Point: \( (0, 0) \)
2. When \( x = 1 \):
- \( y = 2(1) = 2 \)
- Point: \( (1, 2) \)
3. When \( x = 2 \):
- \( y = 2(2) = 4 \)
- Point: \( (2, 4) \)
4. When \( x = 3 \):
- \( y = 2(3) = 6 \)
- Point: \( (3, 6) \)
Now, let's evaluate the options based on the points satisfying the equation \( y = 2x \):
- **Option a**: (2, 1) & (0, 0)
- (2, 1) does not satisfy \( y = 2x \) since \( 1 \neq 4 \).
- **Option b**: (2, 4) & (1, 2) & (0, 0)
- All points are correct. (2, 4), (1, 2), and (0, 0) satisfy \( y = 2x \).
- **Option c**: (1, 4) & (0, 0)
- (1, 4) does not satisfy \( y = 2x \) since \( 4 \neq 2 \).
- **Option d**: (2, 6) & (1, 3) & (0, 0)
- (2, 6) does not satisfy \( y = 2x \) since \( 6 \neq 4 \) and (1, 3) does not satisfy since \( 3 \neq 2 \).
Based on the evaluations, the correct option is:
**b. (2, 4) & (1, 2) & (0, 0)**.
Answer
how could you write the equation of a line that would be a parallel to y = -3x - 2 but not share any points?
a. y = 3x + 8
b. y = -3x - 2
c. y = 3x - 2
d. y = -3x + 8
a. y = 3x + 8
b. y = -3x - 2
c. y = 3x - 2
d. y = -3x + 8
Answered by
GPT-4o mini
To determine the equation of a line that is parallel to the line given by the equation \( y = -3x - 2 \), it's important to remember that parallel lines have the same slope. In this case, the slope of the line \( y = -3x - 2 \) is \(-3\).
For the new line to be parallel and not share any points with the original line, it must have the same slope (\(-3\)) but a different y-intercept (the constant term).
Now let's evaluate the provided options:
a. **\( y = 3x + 8 \)**
- Slope: \( 3 \) (not parallel)
b. **\( y = -3x - 2 \)**
- This is the same line (not valid since it shares points).
c. **\( y = 3x - 2 \)**
- Slope: \( 3 \) (not parallel)
d. **\( y = -3x + 8 \)**
- Slope: \( -3 \) (parallel) and y-intercept \( 8 \) (different from \(-2\), so it does not share points).
Therefore, the equation of a line that is parallel to \( y = -3x - 2 \) but does not share any points is:
**d. \( y = -3x + 8 \)**.
For the new line to be parallel and not share any points with the original line, it must have the same slope (\(-3\)) but a different y-intercept (the constant term).
Now let's evaluate the provided options:
a. **\( y = 3x + 8 \)**
- Slope: \( 3 \) (not parallel)
b. **\( y = -3x - 2 \)**
- This is the same line (not valid since it shares points).
c. **\( y = 3x - 2 \)**
- Slope: \( 3 \) (not parallel)
d. **\( y = -3x + 8 \)**
- Slope: \( -3 \) (parallel) and y-intercept \( 8 \) (different from \(-2\), so it does not share points).
Therefore, the equation of a line that is parallel to \( y = -3x - 2 \) but does not share any points is:
**d. \( y = -3x + 8 \)**.
Answer
use the table to answer the question.
($) Cost of Candy Bar A
Quantity ($) Cost
1 1.25
2 2.50
3 3.75
4 5.00
5 6.25
The cost of Candy Bar A is displayed in the table. The proportional relationship between the cost and quantity for Candy Bar B is represented by the equation y = x. Which candy bar is the cheapest cost per bar? Enter 1 for Candy Bar A. Enter 2 for Candy Bar B.
($) Cost of Candy Bar A
Quantity ($) Cost
1 1.25
2 2.50
3 3.75
4 5.00
5 6.25
The cost of Candy Bar A is displayed in the table. The proportional relationship between the cost and quantity for Candy Bar B is represented by the equation y = x. Which candy bar is the cheapest cost per bar? Enter 1 for Candy Bar A. Enter 2 for Candy Bar B.
Answered by
GPT-4o mini
To determine which candy bar has the cheapest cost per bar, let's first analyze the cost of Candy Bar A from the table:
- Cost for Candy Bar A:
- For 1 candy bar: $1.25
- For 2 candy bars: $2.50 ($2.50 / 2 = $1.25 per bar)
- For 3 candy bars: $3.75 ($3.75 / 3 = $1.25 per bar)
- For 4 candy bars: $5.00 ($5.00 / 4 = $1.25 per bar)
- For 5 candy bars: $6.25 ($6.25 / 5 = $1.25 per bar)
The cost per candy bar for Candy Bar A is consistently \$1.25.
Now, let's examine Candy Bar B, represented by the equation \( y = x \). This means each candy bar costs \$1.00, so you have:
- Cost for Candy Bar B:
- For 1 candy bar: $1.00
- For 2 candy bars: $2.00 ($2.00 / 2 = $1.00 per bar)
- For 3 candy bars: $3.00 ($3.00 / 3 = $1.00 per bar)
- For 4 candy bars: $4.00 ($4.00 / 4 = $1.00 per bar)
- For 5 candy bars: $5.00 ($5.00 / 5 = $1.00 per bar)
The cost per candy bar for Candy Bar B is consistently \$1.00.
To summarize:
- Cost per bar for Candy Bar A: \$1.25
- Cost per bar for Candy Bar B: \$1.00
Since \$1.00 (Candy Bar B) is cheaper than \$1.25 (Candy Bar A), the answer is:
**2 for Candy Bar B.**
- Cost for Candy Bar A:
- For 1 candy bar: $1.25
- For 2 candy bars: $2.50 ($2.50 / 2 = $1.25 per bar)
- For 3 candy bars: $3.75 ($3.75 / 3 = $1.25 per bar)
- For 4 candy bars: $5.00 ($5.00 / 4 = $1.25 per bar)
- For 5 candy bars: $6.25 ($6.25 / 5 = $1.25 per bar)
The cost per candy bar for Candy Bar A is consistently \$1.25.
Now, let's examine Candy Bar B, represented by the equation \( y = x \). This means each candy bar costs \$1.00, so you have:
- Cost for Candy Bar B:
- For 1 candy bar: $1.00
- For 2 candy bars: $2.00 ($2.00 / 2 = $1.00 per bar)
- For 3 candy bars: $3.00 ($3.00 / 3 = $1.00 per bar)
- For 4 candy bars: $4.00 ($4.00 / 4 = $1.00 per bar)
- For 5 candy bars: $5.00 ($5.00 / 5 = $1.00 per bar)
The cost per candy bar for Candy Bar B is consistently \$1.00.
To summarize:
- Cost per bar for Candy Bar A: \$1.25
- Cost per bar for Candy Bar B: \$1.00
Since \$1.00 (Candy Bar B) is cheaper than \$1.25 (Candy Bar A), the answer is:
**2 for Candy Bar B.**
Answer
Graph the linear equation y = 2x - 3.
a. (1, 4) & (0, 1)
b. (1, 1) & (0,3)
c. (1, 5) & (0, 3)
d. (0, -1) & (1, -3)
a. (1, 4) & (0, 1)
b. (1, 1) & (0,3)
c. (1, 5) & (0, 3)
d. (0, -1) & (1, -3)
Answered by
GPT-4o mini
To graph the linear equation \( y = 2x - 3 \), we can find points that satisfy this equation by substituting values for \( x \) and calculating the corresponding \( y \) values.
1. When \( x = 0 \):
- \( y = 2(0) - 3 = -3 \)
- Point: \( (0, -3) \)
2. When \( x = 1 \):
- \( y = 2(1) - 3 = 2 - 3 = -1 \)
- Point: \( (1, -1) \)
3. When \( x = 2 \):
- \( y = 2(2) - 3 = 4 - 3 = 1 \)
- Point: \( (2, 1) \)
Now let's evaluate the options based on the points we found:
- **Option a**: (1, 4) & (0, 1)
- (1, 4) does not satisfy the equation, since \( 2(1) - 3 = -1 \).
- **Option b**: (1, 1) & (0, 3)
- (1, 1) does not satisfy the equation, since \( 2(1) - 3 = -1 \) and (0, 3) does not satisfy \( y = -3 \).
- **Option c**: (1, 5) & (0, 3)
- Both points do not satisfy, since \( 2(1) - 3 = -1 \) and (0, 3) does not satisfy \( y = -3 \).
- **Option d**: (0, -1) & (1, -3)
- (0, -1) does not satisfy the equation, since \( y = -3 \) for \( x = 0\).
- (1, -3) does not satisfy \( y = -1 \).
Since none of the options provided contain points that satisfy the equation \( y = 2x - 3 \), it appears that none of the given options are correct based on our calculations. The correct points based on our analysis are:
- \( (0, -3) \)
- \( (1, -1) \)
If you'd like to confirm any specific points or a correction in any of the options, or if you have more data to provide, please let me know!
1. When \( x = 0 \):
- \( y = 2(0) - 3 = -3 \)
- Point: \( (0, -3) \)
2. When \( x = 1 \):
- \( y = 2(1) - 3 = 2 - 3 = -1 \)
- Point: \( (1, -1) \)
3. When \( x = 2 \):
- \( y = 2(2) - 3 = 4 - 3 = 1 \)
- Point: \( (2, 1) \)
Now let's evaluate the options based on the points we found:
- **Option a**: (1, 4) & (0, 1)
- (1, 4) does not satisfy the equation, since \( 2(1) - 3 = -1 \).
- **Option b**: (1, 1) & (0, 3)
- (1, 1) does not satisfy the equation, since \( 2(1) - 3 = -1 \) and (0, 3) does not satisfy \( y = -3 \).
- **Option c**: (1, 5) & (0, 3)
- Both points do not satisfy, since \( 2(1) - 3 = -1 \) and (0, 3) does not satisfy \( y = -3 \).
- **Option d**: (0, -1) & (1, -3)
- (0, -1) does not satisfy the equation, since \( y = -3 \) for \( x = 0\).
- (1, -3) does not satisfy \( y = -1 \).
Since none of the options provided contain points that satisfy the equation \( y = 2x - 3 \), it appears that none of the given options are correct based on our calculations. The correct points based on our analysis are:
- \( (0, -3) \)
- \( (1, -1) \)
If you'd like to confirm any specific points or a correction in any of the options, or if you have more data to provide, please let me know!
Answer
Write y - 6x = 5 in slope-intercept form
y = ___
y = ___
Answered by
GPT-4o mini
To convert the equation \( y - 6x = 5 \) into slope-intercept form, which is \( y = mx + b \) (where \( m \) is the slope and \( b \) is the y-intercept), you need to isolate \( y \).
Starting with the original equation:
1. **Add \( 6x \) to both sides**:
\[
y = 6x + 5
\]
Now, the equation is in slope-intercept form.
Thus, the final answer is:
\[
y = 6x + 5
\]
Starting with the original equation:
1. **Add \( 6x \) to both sides**:
\[
y = 6x + 5
\]
Now, the equation is in slope-intercept form.
Thus, the final answer is:
\[
y = 6x + 5
\]
Answer
Graph the equation x - y = 6 using the form y = mx + b
a. (6, 6) & (0, 0)
b. (0, -4) & (1, -10)
c. (1, -5) & (0, -6)
d. (0, 3) & (3, 2)
a. (6, 6) & (0, 0)
b. (0, -4) & (1, -10)
c. (1, -5) & (0, -6)
d. (0, 3) & (3, 2)
Answered by
GPT-4o mini
To graph the equation \( x - y = 6 \) in the form \( y = mx + b \), we first need to convert it to slope-intercept form:
1. Start with the original equation:
\[
x - y = 6
\]
2. Rearrange it to solve for \( y \):
\[
-y = -x + 6
\]
3. Multiply by -1 to get:
\[
y = x - 6
\]
Now we have the equation in the form \( y = mx + b \), where \( m = 1 \) (the slope) and \( b = -6 \) (the y-intercept).
### Finding Points
1. **Y-intercept**: Set \( x = 0 \):
\[
y = 0 - 6 = -6
\]
- Point: \( (0, -6) \)
2. **Another point**: Set \( x = 6 \):
\[
y = 6 - 6 = 0
\]
- Point: \( (6, 0) \)
3. **Finding another point**: Set \( x = 1 \):
\[
y = 1 - 6 = -5
\]
- Point: \( (1, -5) \)
Now let's evaluate the options based on those points:
- **Option a**: (6, 6) & (0, 0)
- Neither point is correct since \( (6, 6) \) yields \( y = 0 \) and \( (0, 0) \) yields \( y = -6 \).
- **Option b**: (0, -4) & (1, -10)
- Neither point is correct since \( (0, -4) \) yields \( y = -6 \) and \( (1, -10) \) does not satisfy the equation.
- **Option c**: (1, -5) & (0, -6)
- Both points are correct:
- For \( (1, -5) \): \( y = 1 - 6 = -5 \).
- For \( (0, -6) \): \( y = 0 - 6 = -6 \).
- **Option d**: (0, 3) & (3, 2)
- Neither point is correct since \( (0, 3) \) yields \( y = -6 \) and \( (3, 2) \) does not satisfy the equation.
Thus, the correct answer is:
**c. (1, -5) & (0, -6)**.
1. Start with the original equation:
\[
x - y = 6
\]
2. Rearrange it to solve for \( y \):
\[
-y = -x + 6
\]
3. Multiply by -1 to get:
\[
y = x - 6
\]
Now we have the equation in the form \( y = mx + b \), where \( m = 1 \) (the slope) and \( b = -6 \) (the y-intercept).
### Finding Points
1. **Y-intercept**: Set \( x = 0 \):
\[
y = 0 - 6 = -6
\]
- Point: \( (0, -6) \)
2. **Another point**: Set \( x = 6 \):
\[
y = 6 - 6 = 0
\]
- Point: \( (6, 0) \)
3. **Finding another point**: Set \( x = 1 \):
\[
y = 1 - 6 = -5
\]
- Point: \( (1, -5) \)
Now let's evaluate the options based on those points:
- **Option a**: (6, 6) & (0, 0)
- Neither point is correct since \( (6, 6) \) yields \( y = 0 \) and \( (0, 0) \) yields \( y = -6 \).
- **Option b**: (0, -4) & (1, -10)
- Neither point is correct since \( (0, -4) \) yields \( y = -6 \) and \( (1, -10) \) does not satisfy the equation.
- **Option c**: (1, -5) & (0, -6)
- Both points are correct:
- For \( (1, -5) \): \( y = 1 - 6 = -5 \).
- For \( (0, -6) \): \( y = 0 - 6 = -6 \).
- **Option d**: (0, 3) & (3, 2)
- Neither point is correct since \( (0, 3) \) yields \( y = -6 \) and \( (3, 2) \) does not satisfy the equation.
Thus, the correct answer is:
**c. (1, -5) & (0, -6)**.