The line represented by 2y = 6x + 8 is dilated by a scale factor of k centered at the origin, such that the image of the line has an equation of y = 3x+ 12. What is the scale factor?
2y=6x+8 is the same as
y = 3x+4
The new line is twice as far from the origin, so the dilation must have been a factor of 2.
To find the scale factor by which the line represented by 2y = 6x + 8 is dilated, we need to compare the coefficients of x and y in the original equation to the coefficients in the image equation.
The original equation is 2y = 6x + 8. To make it easier to compare, let's rearrange it to the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
Dividing both sides of the equation by 2, we get:
y = 3x + 4
Now, let's compare this to the equation of the image, which is y = 3x + 12.
Looking at the coefficients of x, we can see that they are both equal to 3. This means that the line has not been stretched or compressed horizontally. So, the scale factor for the x-coordinate is 1.
Next, looking at the y-intercepts, we can see that the original equation had a y-intercept of 4, while the image equation has a y-intercept of 12. The y-coordinate has been stretched vertically by a factor of (12/4) = 3.
Therefore, the scale factor, k, is equal to 3.
To find the scale factor, we compare the coefficients of x and y in the original equation 2y = 6x + 8 with the coefficients in the image equation y = 3x + 12.
1. Let's start by comparing the coefficient of x in both equations:
In the original equation, the coefficient of x is 6.
In the image equation, the coefficient of x is 3.
Therefore, the x-coefficient has been divided by a factor of k.
2. Next, let's compare the coefficients of y in both equations:
In the original equation, the coefficient of y is 2.
In the image equation, the coefficient of y is 1 (since it is y = ...).
Therefore, the y-coefficient has been divided by a factor of k.
3. Since the x-coefficient and y-coefficient are both divided by the same factor of k, we can set up the equation:
3 (x-coefficient in image equation) = 6 (x-coefficient in original equation) / k
4. Solving for k:
3 = 6 / k
Multiply both sides by k:
3k = 6
Divide both sides by 3:
k = 6 / 3
Simplify:
k = 2
Therefore, the scale factor is 2.