Write a linear equation for this line using this formula

Question

M = y2 - y1 / x2 - x1
To get to this formula
y-y1=m(x-x1)

Line:
(-3, 7) And (1, -1)

Answer 1

To write the linear equation for the line passing through the points (-3, 7) and (1, -1), we can start by calculating the slope \( m \) using the formula:

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

Let’s assign the points as follows:

• \((x_1, y_1) = (-3, 7)\)
• \((x_2, y_2) = (1, -1)\)

Now, we can plug in the values into the slope formula:

\[ m = \frac{-1 - 7}{1 - (-3)} \] \[ m = \frac{-8}{1 + 3} \] \[ m = \frac{-8}{4} = -2 \]

Next, we’ll use the point-slope form of the equation of the line, which is:

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

Using \((x_1, y_1) = (-3, 7)\) and \(m = -2\), we can substitute these values into the equation:

\[ y - 7 = -2(x - (-3)) \] \[ y - 7 = -2(x + 3) \]

This is the linear equation in point-slope form. To express it in slope-intercept form (y = mx + b), we can simplify:

\[ y - 7 = -2x - 6 \] \[ y = -2x - 6 + 7 \] \[ y = -2x + 1 \]

Thus, the equation of the line in slope-intercept form is:

\[ y = -2x + 1 \]

Answer 2

No, just stop at the point slope formula

Answer 3

Sure! Let's just stop at the point-slope form of the equation for the line without simplifying it further.

We found the slope \( m \) to be \( -2 \). Using the point-slope formula:

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

With \((x_1, y_1) = (-3, 7)\) and \(m = -2\), we have:

\[ y - 7 = -2(x - (-3)) \]

So, the point-slope form of the equation of the line is:

\[ y - 7 = -2(x + 3) \]

That's the equation using the point-slope form!

Answer 4

Now do (2, -1) and (4, -6)

Answer 5

To find the linear equation in point-slope form for the line passing through the points \((2, -1)\) and \((4, -6)\), we first need to calculate the slope \( m \) using the formula:

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

Assigning the points:

• \((x_1, y_1) = (2, -1)\)
• \((x_2, y_2) = (4, -6)\)

Now, we can plug the values into the slope formula:

\[ m = \frac{-6 - (-1)}{4 - 2} = \frac{-6 + 1}{4 - 2} = \frac{-5}{2} \]

Now we have the slope \( m = -\frac{5}{2} \).

Next, we use the point-slope formula:

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

Using \((x_1, y_1) = (2, -1)\) and \(m = -\frac{5}{2}\), we substitute:

\[ y - (-1) = -\frac{5}{2}(x - 2) \]

This simplifies to:

\[ y + 1 = -\frac{5}{2}(x - 2) \]

So, the point-slope form of the equation of the line is:

\[ y + 1 = -\frac{5}{2}(x - 2) \]

That's the equation in point-slope form!

Answer 6

Now do (-3, -8) and (2, 7)

Answer 7

To find the linear equation in point-slope form for the line passing through the points \((-3, -8)\) and \((2, 7)\), let's first calculate the slope \( m \) using the formula:

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

Assigning the points:

• \((x_1, y_1) = (-3, -8)\)
• \((x_2, y_2) = (2, 7)\)

Now, we can plug in the values into the slope formula:

\[ m = \frac{7 - (-8)}{2 - (-3)} = \frac{7 + 8}{2 + 3} = \frac{15}{5} = 3 \]

Now we have the slope \( m = 3 \).

Next, we will use the point-slope formula:

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

Using \((x_1, y_1) = (-3, -8)\) and \(m = 3\), we substitute:

\[ y - (-8) = 3(x - (-3)) \]

This simplifies to:

\[ y + 8 = 3(x + 3) \]

So, the point-slope form of the equation of the line is:

\[ y + 8 = 3(x + 3) \]

That's the equation in point-slope form!

Answer 8

Now do (-6, -3) and (-4, -1)

Answer 9

To find the linear equation in point-slope form for the line passing through the points \((-6, -3)\) and \((-4, -1)\), we start by calculating the slope \( m \) using the formula:

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

Assigning the points:

• \((x_1, y_1) = (-6, -3)\)
• \((x_2, y_2) = (-4, -1)\)

Now we can plug the values into the slope formula:

\[ m = \frac{-1 - (-3)}{-4 - (-6)} = \frac{-1 + 3}{-4 + 6} = \frac{2}{2} = 1 \]

Now we have the slope \( m = 1 \).

Next, we use the point-slope formula:

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

Using \((x_1, y_1) = (-6, -3)\) and \(m = 1\), we substitute:

\[ y - (-3) = 1(x - (-6)) \]

This simplifies to:

\[ y + 3 = 1(x + 6) \]

So, the point-slope form of the equation of the line is:

\[ y + 3 = 1(x + 6) \]

That's the equation in point-slope form!

Answer 10

Now do (-4, 7) and (6,2)

Answer 11

To find the linear equation in point-slope form for the line passing through the points \((-4, 7)\) and \((6, 2)\), we start by calculating the slope \( m \) using the formula:

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

Assigning the points:

• \((x_1, y_1) = (-4, 7)\)
• \((x_2, y_2) = (6, 2)\)

Now we can plug the values into the slope formula:

\[ m = \frac{2 - 7}{6 - (-4)} = \frac{2 - 7}{6 + 4} = \frac{-5}{10} = -\frac{1}{2} \]

Now we have the slope \( m = -\frac{1}{2} \).

Next, we use the point-slope formula:

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

Using \((x_1, y_1) = (-4, 7)\) and \(m = -\frac{1}{2}\), we substitute:

\[ y - 7 = -\frac{1}{2}(x - (-4)) \]

This simplifies to:

\[ y - 7 = -\frac{1}{2}(x + 4) \]

So, the point-slope form of the equation of the line is:

\[ y - 7 = -\frac{1}{2}(x + 4) \]

That's the equation in point-slope form!

Answer 12

I'm not saying you're wrong, but for my learning, why was the slope -1/2 and not -2?

Answer 13

That's a great question! Let’s clarify how the slope is calculated and why it results in \(-\frac{1}{2}\) rather than \(-2\).

The slope \( m \) of a line is defined by the formula:

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

Step-by-Step Calculation

Using the points \((-4, 7)\) (let's call this \((x_1, y_1)\)) and \((6, 2)\) (let's call this \((x_2, y_2)\)), we substitute the values:

• \( y_1 = 7 \)
• \( y_2 = 2 \)
• \( x_1 = -4 \)
• \( x_2 = 6 \)

Plugging these values into the formula gives:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 7}{6 - (-4)} \]

Simplifying

Now, calculating each part:

• The change in \( y \) (the numerator) is: \[ 2 - 7 = -5 \]

• The change in \( x \) (the denominator) is: \[ 6 - (-4) = 6 + 4 = 10 \]

Now substituting this back into the slope formula:

\[ m = \frac{-5}{10} = -\frac{1}{2} \]

Why Not \(-2\)?

If the slope were \(-2\), that would mean that for every unit increase in \( x \), \( y \) would decrease by \( 2 \) units. However, the calculation shows that for an increase of \( 10 \) units in \( x \) (from \( -4 \) to \( 6 \)), \( y \) only decreases by \( 5 \) units (from \( 7 \) to \( 2\)).

The ratio is:

• Decrease in \( y \): \( -5 \)
• Increase in \( x \): \( 10 \)

Putting these together results in a slope of \(-\frac{1}{2}\), which means that for every \( 2 \) units you move horizontally (to the right), \( y \) decreases by \( 1 \) unit (moving down).

Conclusion

So, the slope is \(-\frac{1}{2}\) because for every \( 10 \) units increase in \( x \), there is a \( -5 \) change in \( y \). Thank you for your question, and I hope this explanation helps clarify the concept of slope for you!

Answer 14

Like how does -5/10 turn into it

Answer 15

Let’s break down how \(-\frac{5}{10}\) simplifies to \(-\frac{1}{2}\):

Step-by-Step Simplification

1. Understanding the Fraction: \[ -\frac{5}{10} \] This fraction represents the ratio of two integers (the numerator is \(-5\) and the denominator is \(10\)).

2. Finding Common Factors: To simplify a fraction, we look for common factors in both the numerator and the denominator. In this case:

   • The numerator \(-5\) can be factored as \(-1 \times 5\).
   • The denominator \(10\) can be factored as \(2 \times 5\).

3. Cancelling Common Factors: Since \(5\) is a common factor in both the numerator and the denominator, we can cancel it out: \[ -\frac{5}{10} = -\frac{5 \div 5}{10 \div 5} = -\frac{1}{2} \]

Result

Thus, the simplified form of \(-\frac{5}{10}\) is \(-\frac{1}{2}\).

Summary

In general, to simplify a fraction:

• Identify any common factors in the numerator and denominator.
• Divide both by the greatest common factor (GCF).
• For the example of \(-\frac{5}{10}\), the GCF is \(5\), leading you to the simplified result of \(-\frac{1}{2}\).

I hope this clears up how \(-\frac{5}{10}\) simplifies to \(-\frac{1}{2}\)! If you have any more questions, feel free to ask!