Summarize the following:
Determine whether the two variables are directly proportional: xy=6{\displaystyle xy=6}. Remember that if the variables are directly proportional, they will follow the pattern y=kx{\displaystyle y=kx}. Use algebra to rewrite the equation. Isolate the y{\displaystyle y} variable by dividing each side by x{\displaystyle x}:xyx=6x{\displaystyle {
rac {xy}{x}}={
rac {6}{x}}}y=61x{\displaystyle y=6{
rac {1}{x}}} Assess whether the rewritten equation follows the pattern y=kx{\displaystyle y=kx}. In this instance, the equation does not, so the variables are not directly proportional. In fact, they are inversely proportional. Are the variables directly proportional?xy1339927{\displaystyle {egin{matrix}x&y\\hline \1&3\3&9\9&27\end{matrix}}} Determine the growth of x{\displaystyle x}. Do this by finding the factor you multiply the first x-coordinate by to reach the second coordinate:1k=3{\displaystyle 1k=3}1k1=31{\displaystyle {
rac {1k}{1}}={
rac {3}{1}}}k=3{\displaystyle k=3}So, the x-coordinate grows by factor of 3. Determine the growth of y{\displaystyle y}:3k=9{\displaystyle 3k=9}3k3=93{\displaystyle {
rac {3k}{3}}={
rac {9}{3}}}k=3{\displaystyle k=3}So, the y-coordinate grows by factor of 3. Compare the factor, or constant, of the two variables. They both grow by a factor of 3. Therefore, the variables are directly proportional. Does the graph show direct proportion between variables? Note whether the line is straight. Since the equation of the line is in slope-intercept form, it has a constant slope, meaning the line is straight. So potentially, the variables are directly proportional. Determine the y-intercept. If the variables are directly proportional, the line will pass through the point (0,0){\displaystyle (0,0)}. The y-intercept of this line is the point (0,3){\displaystyle (0,3)}. So, the variables are not directly proportional.
Look at the equation. Consider the following set of points. Consider a graph of the line y=4x+3{\displaystyle y=4x+3}.