Write y−9=−6(x−3)

Question

Write  y−9=−6(x−3)
  in slope-intercept form.

GPT-4o mini · Answer

To convert the equation $ y - 9 = -6(x - 3) $ into slope-intercept form (which is $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept), we can follow these steps: 1. Start with the given equation: $$ y - 9 = -6(x - 3) $$ 2. Distribute the $-6$ on the right side: $$ y - 9 = -6x + 18 $$ 3. Add $ 9 $ to both sides to isolate $ y $: $$ y = -6x + 18 + 9 $$ 4. Combine the constant terms: $$ y = -6x + 27 $$ So, the equation in slope-intercept form is: $$ y = -6x + 27 $$

bigepic · Answer

where did 18 come from

GPT-4o mini · Answer

Let's break down the distribution step in the equation $ y - 9 = -6(x - 3) $ to clarify how we obtained the $ 18 $.

When we distribute $-6$ to both terms inside the parentheses $(x - 3)$, here's what happens:

Distributing $-6$ to $x$: \[ -6 \cdot x = -6x \]
Distributing $-6$ to $-3$: \[ -6 \cdot (-3) = +18 \]

Combining these, we get: \[ -6(x - 3) = -6x + 18 \]

That's how $18$ comes from distributing $-6$ to $-3$.

The equation simplifies as follows:

Starting with: \[ y - 9 = -6(x - 3) \]
Distributing gives: \[ y - 9 = -6x + 18 \]
Then adding $9$ to both sides results in: \[ y = -6x + 27 \]

I hope that clarifies it! If you have any more questions, feel free to ask!

bigepic · Answer

no when -6 was distributed there was an 18, where did it come from. explain

GPT-4o mini · Answer

I apologize for any confusion earlier. Let's clarify the distribution step in detail to eliminate any doubts about where the $18$ comes from.

Starting from the original equation:

\[ y - 9 = -6(x - 3) \]

When we distribute $-6$ across the terms inside the parentheses $(x - 3)$, we need to apply multiplication to both terms:

The first term is $x$: \[ -6 \cdot x = -6x \]
The second term is $-3$: \[ -6 \cdot (-3) = 18 \]

Now, let's write that out step by step:

We take the $-6$ and apply it to both $x$ and $-3$: \[ -6 \cdot (x - 3) = -6 \cdot x + -6 \cdot (-3) \] This simplifies to: \[ -6x + 18 \]

So, when we say $ -6 $ distributes to $-3$, keep in mind that multiplying two negative numbers gives a positive result, thus:

$-6$ times $-3$ equals $+18$.

Putting it all together, after distributing, we have:

\[ y - 9 = -6x + 18 \]

If we add $9$ to both sides to solve for $y$:

\[ y = -6x + 18 + 9 \]

\[ y = -6x + 27 \]

So the final equation in slope-intercept form is:

\[ y = -6x + 27 \]

Thank you for pointing that out! If you have further questions, feel free to ask.

bigepic · Answer

Write −1/2y+5=−10x in slope-intercept form.

GPT-4o mini · Answer

To write the equation $ -\frac{1}{2}y + 5 = -10x $ in slope-intercept form (which is $ y = mx + b $), follow these steps: 1. Start with the given equation: $$ -\frac{1}{2}y + 5 = -10x $$ 2. First, isolate the term with $ y $ by subtracting $ 5 $ from both sides: $$ -\frac{1}{2}y = -10x - 5 $$ 3. To eliminate the fraction, multiply every term by $-2$ (which is the reciprocal of $-\frac{1}{2}$): $$ y = (-2)(-10x) + (-2)(-5) $$ Simplifying this gives: $$ y = 20x + 10 $$ Now the equation is in slope-intercept form, where: - The slope ($ m $) is $ 20 $, and - The y-intercept ($ b $) is $ 10 $. Thus, the final equation in slope-intercept form is: $$ y = 20x + 10 $$

bigepic · Answer

Transform the linear equation −2y=14−6x to slope-intercept form.

GPT-4o mini · Answer

To transform the equation $ -2y = 14 - 6x $ into slope-intercept form (which is $ y = mx + b $), we can follow these steps: 1. Start with the original equation: $$ -2y = 14 - 6x $$ 2. First, we want to isolate $ y $. To do this, divide every term by $-2$: $$ y = \frac{14}{-2} - \frac{6x}{-2} $$ 3. Simplifying each term gives: $$ y = -7 + 3x $$ 4. Rearranging it into the standard slope-intercept form ($ y = mx + b $): $$ y = 3x - 7 $$ Now the equation is in slope-intercept form, where: - The slope ($ m $) is $ 3 $, and - The y-intercept ($ b $) is $-7$. Thus, the final equation in slope-intercept form is: $$ y = 3x - 7 $$

bigepic · Answer

Write 0.4x+0.2y=0.6 in slope-intercept form.(1 point)

GPT-4o mini · Answer

To write the equation $ 0.4x + 0.2y = 0.6 $ in slope-intercept form (which is $ y = mx + b $), follow these steps: 1. Start with the original equation: $$ 0.4x + 0.2y = 0.6 $$ 2. To isolate $ y $, first subtract $ 0.4x $ from both sides: $$ 0.2y = 0.6 - 0.4x $$ 3. Next, divide every term by $ 0.2 $ to solve for $ y $: $$ y = \frac{0.6}{0.2} - \frac{0.4x}{0.2} $$ 4. Simplify each term: $$ y = 3 - 2x $$ 5. Rearranging it to put $ x $ first gives: $$ y = -2x + 3 $$ Now we have the equation in slope-intercept form. Thus, the final equation is: $$ y = -2x + 3 $$ Here, the slope ($ m $) is $-2$ and the y-intercept ($ b $) is $3$.

bigepic · Answer

Write the linear equation 3y=6x−18
in slope-intercept form.(1 point)
What is the slope and the y
-intercept of the equation 9+27/2x=−3/2y
alt="An illustration shows a coordinate plane with four quadrants. The x and y axes range from negative 10 to 10 in one unit increments. A downward slanting line is graphed on the coordinate plane. The downward slanting line passes through closed points plotted at left parenthesis 0 comma 10 right parenthesis and left parenthesis 1 comma 7 right parenthesis. "
Annie has created a graph of 3x+y=10
on the coordinate plane. Did Annie create her graph correctly?

Option 1: No; Annie's y
-intercept is correct but her slope is incorrect.

Option 2: No; Annie's slope is correct but her y
-intercept is incorrect.

Option 3: Yes; Annie's graph is correct.

Enter "1" for Option 1, "2" for Option 2, or "3" for Option 3.

(1 point)

GPT-4o mini · Answer

Let's tackle each part of your request step by step.

Part 1: Convert $3y = 6x - 18$ to slope-intercept form

Start with the equation: \[ 3y = 6x - 18 \]
Divide every term by $3$ to solve for $y$: \[ y = \frac{6}{3}x - \frac{18}{3} \]
Simplifying gives: \[ y = 2x - 6 \]

Conclusion: The equation in slope-intercept form is $y = 2x - 6$.

Slope (m): $2$
y-intercept (b): $-6$

Part 2: Find the slope and y-intercept of $9 + \frac{27}{2}x = -\frac{3}{2}y$

Start with the equation: \[ 9 + \frac{27}{2}x = -\frac{3}{2}y \]
To isolate $y$, first rearrange the equation: \[ -\frac{3}{2}y = \frac{27}{2}x + 9 \]
Multiply every term by $-\frac{2}{3}$ (the reciprocal of $-\frac{3}{2}$): \[ y = -\frac{2}{3} \left(\frac{27}{2}x + 9\right) \]
Distributing gives: \[ y = -\frac{2}{3} \cdot \frac{27}{2}x - \frac{2}{3} \cdot 9 \] Simplifying each term:
- For the first term: $-\frac{2}{3} \cdot \frac{27}{2} = -9$
- For the second term: $-\frac{2}{3} \cdot 9 = -6$

So the slope-intercept form is: \[ y = -9x - 6 \]

Slope (m): $-9$
y-intercept (b): $-6$

Part 3: Evaluate Annie's graph of $3x + y = 10$

Start with the equation: \[ 3x + y = 10 \]
Rearranging to slope-intercept form: \[ y = -3x + 10 \]

From this form:

Slope (m): $-3$
y-intercept (b): $10$

Based on the problem statements:

If Annie's graph shows a slope of $-3$ and a y-intercept of $10$, then her graph is correct.
If not, it would depend on what she plotted.

So, based on your question, if Annie's graph accurately reflects these values:

Conclusion: If Annie represented both correctly:

Enter "3" for Option 3 (Yes; Annie's graph is correct).

If the graph does not align with these values, choose accordingly between options 1 or 2.

Final Answers:

For the first part: $y = 2x - 6$ (Slope: $2$, y-intercept: $-6$).
For the second part: Slope is $-9$, y-intercept is $-6$.
For Annie's graph: Enter "3" for Option 3 (assuming everything is correct).

Write y−9=−6(x−3)

Part 1: Convert \(3y = 6x - 18\) to slope-intercept form

Part 2: Find the slope and y-intercept of \(9 + \frac{27}{2}x = -\frac{3}{2}y\)

Part 3: Evaluate Annie's graph of \(3x + y = 10\)

Final Answers: