Sure, let's solve the problem step by step.
We have a point [tex]\( A \)[/tex] with coordinates [tex]\( (9,12) \)[/tex] and the equation of a line given by [tex]\( y = x - 3 \)[/tex].
Step 1: Understand the Equation of the Line
The equation of the line is [tex]\( y = x - 3 \)[/tex]. This is a linear equation in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m = 1 \)[/tex] (the slope) and [tex]\( b = -3 \)[/tex] (the y-intercept).
Step 2: Locate the Point on the Line Corresponding to the x-coordinate of A
The x-coordinate of point [tex]\( A \)[/tex] is [tex]\( 9 \)[/tex]. To find the corresponding y-coordinate on the line, we substitute [tex]\( x = 9 \)[/tex] into the line’s equation:
[tex]\[ y = 9 - 3 \][/tex]
[tex]\[ y = 6 \][/tex]
So, the point on the line with the same x-coordinate as [tex]\( A \)[/tex] is [tex]\( (9, 6) \)[/tex].
Step 3: Calculate the Difference in the y-coordinates
Now, we need to find the difference between the y-coordinate of point [tex]\( A \)[/tex] and the y-coordinate of the corresponding point on the line.
The y-coordinate of point [tex]\( A \)[/tex] is [tex]\( 12 \)[/tex].
The y-coordinate of the point on the line is [tex]\( 6 \)[/tex].
Thus, the difference in the y-coordinates is:
[tex]\[ 12 - 6 = 6 \][/tex]
So the point [tex]\( A \)[/tex] is 6 units above the corresponding point on the line.
Final Answer
Given the information provided and the steps outlined:
- The coordinates of point [tex]\( A \)[/tex] are [tex]\( (9, 12) \)[/tex].
- The coordinates of the corresponding point on the line are [tex]\( (9, 6) \)[/tex].
- The difference in the y-coordinates is [tex]\( 6 \)[/tex].
Thus, the detailed step-by-step solution confirms that:
[tex]\[ A = (9, 12) \][/tex]
[tex]\[ \text{Point on the line} = (9, 6) \][/tex]
[tex]\[ \text{Difference in y-coordinates} = 6 \][/tex]
