To determine which point Harold used to write the equation [tex]\( y = 3(x - 7) \)[/tex] in point-slope form, let's start by examining the given equation in its equivalent point-slope form:
The point-slope form of a linear equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
In the given equation:
[tex]\[ y = 3(x - 7) \][/tex]
We can rewrite it to match the point-slope form:
[tex]\[ y - y_1 = 3(x - 7) \][/tex]
By comparing this with [tex]\( y - y_1 = m(x - x_1) \)[/tex], we can identify the slope [tex]\( m \)[/tex] and the coordinates of the point [tex]\( (x_1, y_1) \)[/tex]:
- The slope [tex]\( m \)[/tex] is the coefficient of [tex]\( (x - x_1) \)[/tex], which is 3.
- The term [tex]\( (x - 7) \)[/tex] indicates that [tex]\( x_1 \)[/tex] is 7.
Therefore, the equation [tex]\( y - y_1 = 3(x - 7) \)[/tex] implies that:
- [tex]\( x_1 = 7 \)[/tex]
- We need to determine [tex]\( y_1 \)[/tex] by ensuring the equation matches up.
Expanding the given equation:
[tex]\[ y = 3(x - 7) \][/tex]
[tex]\[ y = 3x - 21 \][/tex]
By comparing this with the expanded form of the point-slope equation:
[tex]\[ y - y_1 = 3x - 21 \][/tex]
We can see that:
[tex]\[ y - y_1 = 3x - 21 \][/tex]
Rewriting it, we get:
[tex]\[ y = 3x - 21 + y_1 \][/tex]
For the equation [tex]\( y = 3x - 21 \)[/tex] to match the given form, [tex]\( y_1 \)[/tex] must be zero. Thus:
[tex]\[ y_1 = 0 \][/tex]
So, Harold used the point [tex]\((7, 0)\)[/tex] to write the equation [tex]\( y = 3(x - 7) \)[/tex].
Hence, the point used by Harold is:
\[ (7, 0) \)
