Question

Use the equation \(2 x+7 y=14\). Find a relationship between the denominators on the left side of the equation and the \(x\) -intercept and the \(y\) -intercept.

Short answer

The denominators of the variables (i.e., coefficients 2 and 7) on the left side of the equation are the reciprocals of the x and y intercepts of the equation respectively.

Step-by-step solution

Calculate the x-intercept

To calculate the x-intercept, set y to 0 in the equation and solve for x.Doing this gives 2x + 7(0) = 14. Simplify to 2x = 14. Then divide each side by 2 to solve for x, thus x = 7.

Calculate the y-intercept

Now calculate the y-intercept: To do this, set x to 0 in the equation and solve for y. Thus, 2(0) + 7y = 14. Simplify to 7y = 14. Then divide each side by 7 to solve for y, so y = 2.

Identify the relationship

Now that the x-intercept (7) and y-intercept (2) have been determined, it's clear that they are reciprocal to the denominators on the left side of the equation (2 for x and 7 for y). Thus, the relationship is the reciprocal relation between coefficients of variables (x and y) and their respective intercepts. While '2' turns out to be the y-intercept, its reciprocal '1/2' is the coefficient of x in the given equation. Similarly for x-intercept '7', its reciprocal '1/7' is the coefficient of y in the equation.