Find the equation of the line that is parallel to the line y=2/5x+1

Question

Find the equation of the line that is parallel to the line y=2/5x+1
 and passes through the point (5,5)
.(1 point)
The line that is parallel to y=2/5x+1
 and passes through the point (5,5)
 is y=2/5x+

GPT-4o mini · Answer

To find the equation of the line that is parallel to $ y = \frac{2}{5}x + 1 $ and passes through the point $ (5, 5) $, we first note that parallel lines have the same slope.

The slope of the given line $ y = \frac{2}{5}x + 1 $ is $ \frac{2}{5} $. Therefore, the slope of the line we want to find will also be $ \frac{2}{5} $.

Next, we can use the point-slope form of the equation of a line, which is given by:

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

where $ m $ is the slope, and $ (x_1, y_1) $ is the point through which the line passes.

In our case:

$ m = \frac{2}{5} $
$ (x_1, y_1) = (5, 5) $

Substituting these values into the point-slope form:

\[ y - 5 = \frac{2}{5}(x - 5) \]

Now, we can simplify this equation.

First, distribute $ \frac{2}{5} $:

\[ y - 5 = \frac{2}{5}x - \frac{2}{5} \cdot 5 \]

\[ y - 5 = \frac{2}{5}x - 2 \]

Next, add 5 to both sides:

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

Simplifying further:

\[ y = \frac{2}{5}x + 3 \]

Thus, the equation of the line that is parallel to $ y = \frac{2}{5}x + 1 $ and passes through the point $ (5, 5) $ is:

\[ y = \frac{2}{5}x + 3 \]

So, filling in the blank, you get:

The line that is parallel to $ y = \frac{2}{5}x + 1 $ and passes through the point $ (5,5) $ is $ y = \frac{2}{5}x + 3 $.

x · Answer

What value completes the equation of the line that is perpendicular to the line y=−3/4x+7
and goes through the point (6,1)
?(1 point)
The line that is perpendicular to y=−3/4x+7
and passes through the point (6,1)
is y=4/3x+

GPT-4o mini · Answer

To find the equation of a line that is perpendicular to the line given by $ y = -\frac{3}{4}x + 7 $ and passes through the point $ (6, 1) $, we first need to determine the slope of the perpendicular line.

Identify the slope of the original line: The slope of the line $ y = -\frac{3}{4}x + 7 $ is $ -\frac{3}{4} $.
Find the slope of the perpendicular line: The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope $ m $ of the perpendicular line is: \[ m = -\frac{1}{\left(-\frac{3}{4}\right)} = \frac{4}{3} \]
Use the point-slope form: Now we can use the point-slope form of the equation of a line, which is given by: \[ y - y_1 = m(x - x_1) \] where $ (x_1, y_1) $ is the point through which the line passes, and $ m $ is the slope.

Plugging in our values:
- $ m = \frac{4}{3} $
- $ (x_1, y_1) = (6, 1) $
We get: \[ y - 1 = \frac{4}{3}(x - 6) \]
Simplify the equation: Distributing $ \frac{4}{3} $: \[ y - 1 = \frac{4}{3}x - \frac{4}{3} \cdot 6 \] \[ y - 1 = \frac{4}{3}x - 8 \]

Now, add 1 to both sides: \[ y = \frac{4}{3}x - 8 + 1 \] \[ y = \frac{4}{3}x - 7 \]

So the value that completes the equation of the line that is perpendicular to $ y = -\frac{3}{4}x + 7 $ and passes through the point $ (6, 1) $ is:

The line that is perpendicular to $ y = -\frac{3}{4}x + 7 $ and passes through the point $ (6,1) $ is $ y = \frac{4}{3}x - 7 $.

Therefore, filling in the blank, we have:

The line that is perpendicular to $ y = -\frac{3}{4}x + 7 $ and passes through the point $ (6, 1) $ is $ y = \frac{4}{3}x - 7 $.

x · Answer

Which of the following lines would be parallel to y=3/4x−2 ?(1 point) Responses 4x−3y=12 4 x minus 3 y equals 12 3x−4y=8 3 x minus 4 y equals 8 3x−4y=12 3 x minus 4 y equals 12 4x−3y=6

GPT-4o mini · Answer

To determine which of the lines is parallel to $ y = \frac{3}{4}x - 2 $, we need to find the slope of the given line and then compare it with the slopes of the lines provided in the responses.

The slope of the line $ y = \frac{3}{4}x - 2 $ is $ \frac{3}{4} $. For a line to be parallel, it must have the same slope.

Now, let's find the slope of each of the given lines by rewriting them in slope-intercept form $ y = mx + b $ (where $ m $ is the slope).

$ 4x - 3y = 12 $: \[ -3y = -4x + 12 \quad \Rightarrow \quad y = \frac{4}{3}x - 4 \] Slope: $ \frac{4}{3} $
$ 3x - 4y = 8 $: \[ -4y = -3x + 8 \quad \Rightarrow \quad y = \frac{3}{4}x - 2 \] Slope: $ \frac{3}{4} $
$ 3x - 4y = 12 $: \[ -4y = -3x + 12 \quad \Rightarrow \quad y = \frac{3}{4}x - 3 \] Slope: $ \frac{3}{4} $
$ 4x - 3y = 6 $: \[ -3y = -4x + 6 \quad \Rightarrow \quad y = \frac{4}{3}x - 2 \] Slope: $ \frac{4}{3} $

Now that we have calculated the slopes:

The slope of $ 4x - 3y = 12 $ is $ \frac{4}{3} $ (not parallel).
The slope of $ 3x - 4y = 8 $ is $ \frac{3}{4} $ (parallel).
The slope of $ 3x - 4y = 12 $ is $ \frac{3}{4} $ (parallel).
The slope of $ 4x - 3y = 6 $ is $ \frac{4}{3} $ (not parallel).

Conclusion: The lines $ 3x - 4y = 8 $ and $ 3x - 4y = 12 $ are parallel to $ y = \frac{3}{4}x - 2 $.

x · Answer

just pick 1

GPT-4o mini · Answer

The line that is parallel to $ y = \frac{3}{4}x - 2 $ is: **$ 3x - 4y = 8 $**.

x · Answer

Which line is parallel to the line y=5/6x and goes through the point (12,6)?(1 point)
Responses

6y=5x
6 y equals 5 x

y=5/6x−4
y equals Start Fraction 5 over 6 End Fraction x minus 4

y=5/6x+6
y equals Start Fraction 5 over 6 End Fraction x plus 6

y=−6/5x+6

GPT-4o mini · Answer

To find a line that is parallel to $ y = \frac{5}{6}x $ and passes through the point $ (12, 6) $, we first note that parallel lines have the same slope. The slope of the line $ y = \frac{5}{6}x $ is $ \frac{5}{6} $. Now we can use the point-slope form of the equation of a line, which is given by: $$ y - y_1 = m(x - x_1) $$ Where: - $ m = \frac{5}{6} $ (the slope) - $ (x_1, y_1) = (12, 6) $ Substituting the known values into the point-slope form: $$ y - 6 = \frac{5}{6}(x - 12) $$ Let's simplify this: 1. Distribute $ \frac{5}{6} $: $$ y - 6 = \frac{5}{6}x - \frac{5}{6} \cdot 12 $$ $$ y - 6 = \frac{5}{6}x - 10 $$ 2. Add 6 to both sides: $$ y = \frac{5}{6}x - 10 + 6 $$ $$ y = \frac{5}{6}x - 4 $$ Now, we need to check which of the provided options is correct. The options given are: 1. $ 6y = 5x $ 2. $ y = \frac{5}{6}x - 4 $ 3. $ y = \frac{5}{6}x + 6 $ 4. $ y = -\frac{6}{5}x + 6 $ The line that matches our result $ y = \frac{5}{6}x - 4 $ is: **$ y = \frac{5}{6}x - 4 $**.

x · Answer

Which of the following options would be the equation of a line perpendicular to the line y=−3x+5?(1 point)
Responses

y=1/3x+9
y equals Start Fraction 1 over 3 End Fraction x plus 9

y=−1/3x+9
y equals negative Start Fraction 1 over 3 End Fraction x plus 9

y=3x+9
y equals 3 x plus 9

y=−3x+9

GPT-4o mini · Answer

To find the equation of a line that is **perpendicular** to the line given by $ y = -3x + 5 $, we first need to determine the slope of the original line. 1. **Identify the slope of the original line**: The slope ($ m $) of the line $ y = -3x + 5 $ is **-3**. 2. **Find the slope of the perpendicular line**: The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of $-3$ is: $$ m = \frac{1}{3} $$ Now, let's look at the provided options and identify the one with the slope of $\frac{1}{3}$: 1. **$ y = \frac{1}{3}x + 9 $**: Slope is $\frac{1}{3}$ (this could be our answer). 2. **$ y = -\frac{1}{3}x + 9 $**: Slope is $-\frac{1}{3}$. 3. **$ y = 3x + 9 $**: Slope is $3$. 4. **$ y = -3x + 9 $**: Slope is $-3$. The only line with a slope of $\frac{1}{3}$ is: **$ y = \frac{1}{3}x + 9 $**. Thus, the answer is $ y = \frac{1}{3}x + 9 $, which is the equation of the line perpendicular to $ y = -3x + 5 $.