Question

What is a point on the line represented by the equation y−1=
2
3
​
(x−6.5)?

(1, 6.5)

(−1, −6.5)

(−6.5, −1)

(6.5, 1)

Answer 1

To find a point on the line represented by the equation \( y - 1 = \frac{2}{3}(x - 6.5) \), we can substitute the values from the given options into the equation and check if they satisfy it.

Starting with the equation:

\[
y - 1 = \frac{2}{3}(x - 6.5)
\]

We rearrange it to find \( y \):

\[
y = \frac{2}{3}(x - 6.5) + 1
\]

Now, let's evaluate each of the given points:

1. **Point (1, 6.5)**:
\[
y = 6.5, \quad x = 1
\]
\[
6.5 = \frac{2}{3}(1 - 6.5) + 1
\]
\[
6.5 = \frac{2}{3}(-5.5) + 1
\]
\[
6.5 = -\frac{11}{3} + 1 = -\frac{11}{3} + \frac{3}{3} = -\frac{8}{3}
\] (Not satisfied)

2. **Point (-1, -6.5)**:
\[
y = -6.5, \quad x = -1
\]
\[
-6.5 = \frac{2}{3}(-1 - 6.5) + 1
\]
\[
-6.5 = \frac{2}{3}(-7.5) + 1
\]
\[
-6.5 = -\frac{15}{3} + 1 = -5 + 1 = -4
\] (Not satisfied)

3. **Point (-6.5, -1)**:
\[
y = -1, \quad x = -6.5
\]
\[
-1 = \frac{2}{3}(-6.5 - 6.5) + 1
\]
\[
-1 = \frac{2}{3}(-13) + 1
\]
\[
-1 = -\frac{26}{3} + 1 = -\frac{26}{3} + \frac{3}{3} = -\frac{23}{3}
\] (Not satisfied)

4. **Point (6.5, 1)**:
\[
y = 1, \quad x = 6.5
\]
\[
1 = \frac{2}{3}(6.5 - 6.5) + 1
\]
\[
1 = \frac{2}{3}(0) + 1
\]
\[
1 = 0 + 1
\]

Since the last equation holds true, the point **(6.5, 1)** lies on the line represented by the given equation.

Thus, the answer is \((6.5, 1)\).