Before you get to that, what does it mean for the points to be symmetric about y = mx + b?RTCNTC said:Determine values for m and b so that the points (8, 2) and (4, 8) are symmetric about the line y = mx + b.
Do I plug the coordinates of each point into the formula
y = mx + b individually to find values for m and b?
Absolutely NOT! That would give you the equation of the line that contains (8, 2) and (4, 6) but these two points do NOT lie on the line you seek! They are symmetric about that line. In particular, the line you seek must be the perpendicular bisector of the line segment between (8, 2) and (4, 8).RTCNTC said:Determine values for m and b so that the points (8, 2) and (4, 8) are symmetric about the line y = mx + b.
Do I plug the coordinates of each point into the formula
y = mx + b individually to find values for m and b?
HallsofIvy said:Absolutely NOT! That would give you the equation of the line that contains (8, 2) and (4, 6) but these two points do NOT lie on the line you seek! They are symmetric about that line. In particular, the line you seek must be the perpendicular bisector of the line segment between (8, 2) and (4, 8).
What are the coordinates of the point midway between (8, 2) and (4, 6)? What is the slope of the line through (8, 2) and (4, 6)? What is the slope of a line perpendicular to that line? Finally, what is the equation of the line through that midpoint perpendicular to that line?
The variables m and b represent the slope and y-intercept, respectively, in a linear equation. The slope (m) is the measure of how steep the line is, while the y-intercept (b) is the point where the line intersects the y-axis.
To find the values for m and b, you can use two points on the line. First, calculate the slope by finding the change in y over the change in x between the two points. Then, plug in the values of one of the points and the calculated slope into the equation y = mx + b and solve for b. This will give you the value of b, and you can then plug it back into the equation to find the value of m.
Yes, both m and b can have negative values. The slope (m) can be positive or negative depending on the direction of the line, and the y-intercept (b) can also be negative if the line crosses the y-axis below the origin.
The value of m determines the steepness of the line, with a larger absolute value of m resulting in a steeper line. The value of b determines the y-intercept, or where the line crosses the y-axis. Changing the values of m and b will result in a different line with a different slope and y-intercept.
Yes, the values of m and b can change in a linear equation. If the equation is in slope-intercept form (y = mx + b), then m and b can be adjusted to create different lines with different slopes and y-intercepts. However, if the equation is in standard form (Ax + By = C), then m and b cannot be changed as they are determined by the coefficients A and B.