To find a vector parallel to the line 2x + y = 10, we can rewrite the equation in slope-intercept form, y = -2x + 10.
The slope of the line is -2, indicating that for every increase of 1 in the x-coordinate, the y-coordinate decreases by 2. Therefore, the vector (1, -2) has the same slope and is parallel to the line 2x + y = 10.
In the above, can you explain how the y-coordinate is decreased by 2 for every increase of 1 in the x-coordinate? How do we know is every increase of 1 in the x-coordinate?
To understand how the y-coordinate is decreased by 2 for every increase of 1 in the x-coordinate, we can look at the coefficient of x in the slope-intercept form of the equation, y = -2x + 10.
In slope-intercept form, the coefficient of x represents the slope of the line. In this case, the coefficient of x is -2. The slope is a measure of how steep a line is and tells us how much y changes for a given change in x.
A slope of -2 means that for every increase of 1 in the x-coordinate, the y-coordinate decreases by 2. This is because the negative sign indicates a downward slope, and the absolute value of the slope, 2, tells us the extent of the change.
So, if we start at a certain point (x, y) on the line, and then increase x by 1 unit, the new point (x + 1, y - 2) will lie on the line. We can see that the y-coordinate is decreased by 2 units for every increase of 1 in the x-coordinate, as determined by the slope of the line (-2).
As for how we know that every increase of 1 in the x-coordinate results in a decrease of 2 in the y-coordinate, it is based on the definition of the slope. The slope is calculated by dividing the change in y by the change in x between two points on the line. In this case, since the slope is -2, it means that for every increase of 1 in the x-coordinate, the y-coordinate decreases by 2. This is a characteristic property of lines with a slope of -2.