Write the equation of the line using this formula

Question

Y-y1=m(x-x1)

(4, 1); slope = 2

Answer 1

To write the equation of the line using the point-slope form of a line, which is given by the formula:

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

we will plug in the values from the point \((x_1, y_1) = (4, 1)\) and the slope \(m = 2\).

Substituting these values into the formula, we have:

\[ Y - 1 = 2(x - 4) \]

This is the equation of the line in point-slope form. If you want to convert it to slope-intercept form (Y = mx + b), you can simplify it:

\[ Y - 1 = 2x - 8 \]

Now, add 1 to both sides:

\[ Y = 2x - 7 \]

So, the final equation of the line is:

\[ Y = 2x - 7 \]

Answer 2

Now do the same with this one

(-6, 0); slope = 2/3

Answer 3

To write the equation of the line using the point-slope form \(Y - y_1 = m(x - x_1)\), we will substitute the values from the point \((x_1, y_1) = (-6, 0)\) and the slope \(m = \frac{2}{3}\).

Substituting these values into the formula:

\[ Y - 0 = \frac{2}{3}(x - (-6)) \]

This simplifies to:

\[ Y = \frac{2}{3}(x + 6) \]

Now, to convert this to slope-intercept form (\(Y = mx + b\)), we can distribute \(\frac{2}{3}\):

\[ Y = \frac{2}{3}x + \frac{2}{3} \times 6 \]

Calculating \(\frac{2}{3} \times 6\):

\[ Y = \frac{2}{3}x + 4 \]

So, the final equation of the line is:

\[ Y = \frac{2}{3}x + 4 \]

Answer 4

My teacher got 6 instead of 4, where did she go wrong?

Answer 5

Let's double-check the calculations to clarify how we arrived at the line equation and ensure everything is correct step-by-step.

Starting from the point-slope form with the given point \((-6, 0)\) and the slope \(\frac{2}{3}\):

1. The point-slope formula is: \[ Y - y_1 = m(x - x_1) \]

2. Substituting the values: \[ Y - 0 = \frac{2}{3}(x - (-6)) \] This simplifies to: \[ Y = \frac{2}{3}(x + 6) \]

3. Now, distribute \(\frac{2}{3}\): \[ Y = \frac{2}{3}x + \frac{2}{3} \times 6 \]

4. Next, calculate \(\frac{2}{3} \times 6\): \[ \frac{2}{3} \times 6 = \frac{12}{3} = 4 \]

Thus, the correct output of the calculation should be: \[ Y = \frac{2}{3}x + 4 \]

If your teacher got 6 instead of 4, she might have miscalculated the multiplication \(\frac{2}{3} \times 6\). It is important to correctly carry out the calculation, because: \[ \frac{2}{3} \times 6 = 4, \quad \text{not } 6. \]

Thus, the correct equation remains: \[ Y = \frac{2}{3}x + 4 \]

Answer 6

Ight now do (-8, -1); slope = -3/4

Answer 7

Let's use the point-slope form of a line, \(Y - y_1 = m(x - x_1)\), to find the equation of the line with the given point \((-8, -1)\) and slope \(-\frac{3}{4}\).

1. Identify the values:

   • \((x_1, y_1) = (-8, -1)\)
   • \(m = -\frac{3}{4}\)

2. Substitute these values into the point-slope form: \[ Y - (-1) = -\frac{3}{4}(x - (-8)) \] Simplifying this gives: \[ Y + 1 = -\frac{3}{4}(x + 8) \]

3. Now, distribute \(-\frac{3}{4}\): \[ Y + 1 = -\frac{3}{4}x - \frac{3}{4} \times 8 \] Simplifying \(- \frac{3}{4} \times 8\): \[ -\frac{3}{4} \times 8 = -6 \]

4. So now we have: \[ Y + 1 = -\frac{3}{4}x - 6 \]

5. Finally, subtract 1 from both sides to isolate \(Y\): \[ Y = -\frac{3}{4}x - 6 - 1 \] \[ Y = -\frac{3}{4}x - 7 \]

The final equation of the line is: \[ Y = -\frac{3}{4}x - 7 \]

Answer 8

Now do (4, -3); slope = -1

Answer 9

Let's write the equation of the line using the point-slope form \(Y - y_1 = m(x - x_1)\) with the point \((4, -3)\) and a slope of \(-1\).

1. Identify the values:

   • \((x_1, y_1) = (4, -3)\)
   • \(m = -1\)

2. Substitute these values into the point-slope formula: \[ Y - (-3) = -1(x - 4) \] This simplifies to: \[ Y + 3 = -1(x - 4) \]

3. Now distribute \(-1\): \[ Y + 3 = -x + 4 \]

4. Finally, subtract 3 from both sides to isolate \(Y\): \[ Y = -x + 4 - 3 \] \[ Y = -x + 1 \]

So, the final equation of the line is: \[ Y = -x + 1 \]

Answer 10

Now do (0,-9); slope = 4

Answer 11

Let's write the equation of the line using the point-slope form \(Y - y_1 = m(x - x_1)\) with the point \((0, -9)\) and a slope of \(4\).

1. Identify the values:

   • \((x_1, y_1) = (0, -9)\)
   • \(m = 4\)

2. Substitute these values into the point-slope formula: \[ Y - (-9) = 4(x - 0) \] This simplifies to: \[ Y + 9 = 4x \]

3. Finally, subtract \(9\) from both sides to isolate \(Y\): \[ Y = 4x - 9 \]

So, the final equation of the line is: \[ Y = 4x - 9 \]