Final answer:
The equation y = 5x - [box] represents direct variation only if the value in the box is 0, making the equation y = 5x with no y-intercept, which is the requirement for a direct variation.
Step-by-step explanation:
The question refers to whether the equation y = 5x - [box] can represent a direct variation. Direct variation occurs when a linear equation has the form y = mx, where m is the non-zero slope of the line and y is the dependent variable while x is the independent variable. For a linear equation to represent a direct variation, there must be no y-intercept, which means the value that would go into the 'box' needs to be 0. In this case, the equation would become y = 5x.
In general, a linear equation can be expressed as y = a + bx, where a is the y-intercept and b is the slope. If the equation is to represent a direct variation, the value of a must be zero. Therefore, if Lydia writes 0 in the box, the equation becomes a direct variation as there would be no y-intercept, matching the required form of y = mx where the y-intercept is zero.
Understanding direct variation is crucial in many fields such as physics and economics, where one variable changes in direct proportion to another. For example, in physics, when discussing distance and displacement, the slope of a position versus time graph would indicate the speed, and the y-intercept would typically be the initial position. But for direct variation, such as speed relating directly to time without an initial value, the y-intercept must be zero.