For example, you might be given the equation yx=32{\displaystyle {\frac {y}{x}}={\frac {3}{2}}}.
For example, if you multiply both sides of the equation yx=32{\displaystyle {\frac {y}{x}}={\frac {3}{2}}} by x{\displaystyle x}, the equation becomes y=32x{\displaystyle y={\frac {3}{2}}x}, which is in the form of y=kx{\displaystyle y=kx}, with 32{\displaystyle {\frac {3}{2}}} being the constant.
For example, you might be given the set of points xy214263{\displaystyle {\begin{matrix}x&y\\hline \2&1\4&2\6&3\end{matrix}}} The x-coordinate of the first point is 2, and the x-coordinate of the second point is 4.
For example, if the first x-coordinate is 2, and the second x-coordinate is 4, you need to determine what you multiply 2 by to get 4:2k=4{\displaystyle 2k=4}2k2=42{\displaystyle {\frac {2k}{2}}={\frac {4}{2}}}k=2{\displaystyle k=2}So, the x{\displaystyle x} variable grows by the constant 2.
For example, if the first y-coordinate is 1, and the second y-coordinate is 2, you need to determine what you multiply 1 by to get 2:1k=2{\displaystyle 1k=2}1k1=21{\displaystyle {\frac {1k}{1}}={\frac {2}{1}}}k=2{\displaystyle k=2}So, the variable y{\displaystyle y} grows by the constant 2.
For example, since the x-coordinates changed by a factor of 2 while the y-coordinates also changed by a factor of 2, the two variables are directly proportional.
The y-axis is the vertical axis.
For example, if the first point is (1,3){\displaystyle (1,3)}, and the second point is (2,6){\displaystyle (2,6)}, the x-coordinate changed by a factor of 2, since 1(2)=2{\displaystyle 1(2)=2}. The y-coordinate also changed by a factor of 2, since 3(2)=6{\displaystyle 3(2)=6}. Thus, you can confirm that the line represents two variables that are directly proportional.
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 {\frac {xy}{x}}={\frac {6}{x}}}y=61x{\displaystyle y=6{\frac {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. [7] X Research source
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 {\frac {1k}{1}}={\frac {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 {\frac {3k}{3}}={\frac {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.
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.
