The slope is actually the y-coefficient divided by the x-coefficient.

Remember back in the first equation, the slope was 3 and the t-coefficient in x was 1/3. The t-coefficient in y was 1. 1/(1/3)=3.

In the second example, the slope was 2. The t-coefficient in y was 2 and the t-coefficient in x was 1. 2/1=2.

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