BECA Classroom assignments and resources
AI-SAT-slope-problem
Query (5 December 2022): write a math word problem about slope in the style of the SAT test
Response: A line has a slope of -3. If a point on the line has coordinates (6, 2), what is the y-intercept of the line?
To solve this problem, we need to first understand what the y-intercept of a line is. The y-intercept is the point where a line crosses the y-axis. In other words, it is the point where the x-coordinate is 0.
Next, we need to understand what the slope of a line is. The slope of a line is a measure of how steep the line is. It is calculated by taking the difference in the y-coordinates of two points on the line and dividing it by the difference in the x-coordinates of those two points. In this case, the slope of the line is -3, which means that for every unit increase in the x-coordinate, the y-coordinate decreases by 3 units.
Now, we can use the point (6, 2) and the slope of -3 to find the y-intercept of the line. To do this, we first need to find another point on the line that has an x-coordinate of 0. We can do this by using the slope to find the corresponding y-coordinate of the point. Since the slope is -3, we know that for every unit increase in the x-coordinate, the y-coordinate decreases by 3 units. Therefore, if we decrease the x-coordinate by 6 units, the y-coordinate will decrease by 6 * -3 = -18 units. This means that the point (0, -18) is also on the line.
Finally, we can use the point-slope formula to find the equation of the line. The point-slope formula is:
y - y1 = m(x - x1)
where m is the slope of the line, (x1, y1) is a point on the line, and (x, y) is a variable point on the line. In this case, we know that the slope is -3 and the point is (6, 2), so we can plug these values into the formula to get:
y - 2 = -3(x - 6)
We can then simplify this equation to get:
y = -3x + 20
The y-intercept is the point where the line crosses the y-axis, which means that the x-coordinate is 0. Therefore, we can plug 0 in for x in the equation above to find the y-intercept:
y = -3(0) + 20
This simplifies to:
y = 20
Therefore, the y-intercept of the line is (0, 20).
In summary, to solve this problem, we used the point (6, 2) and the slope of -3 to find another point on the line with an x-coordinate of 0. We then used the point-slope formula to find the equation of the line, and finally we plugged 0 in for the x-coordinate to find the y-intercept. The y-intercept of the line is (0, 20).