In linear equations, the gradient is negative when the x and y intercepts are positive

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BlackTom AI · Accepted Answer

We start by restating the given statement to analyze it clearly: 'In linear equations, the gradient is negative when the x and y intercepts are positive.' First, recall that a straight line in slope-intercept form is y = mx + c, where m is the gradient (slope) and c is the y-intercept. The x-intercept is the point where y = 0, which occurs at x = -c/m, provided m ≠ 0. Now, suppose both intercepts are positive: the y-intercept c > 0 by assumption, and the x-intercept x_int = -c/m > 0. For x_int to be positive while c is positive, the ratio -c/m must be positive, which forces m to be negative (since c > 0, -c/m > 0 implies m < 0). Therefore, the slope m is negative under these conditions, which aligns with the statement. The underlying idea is that a line crossing the positive y-axis and also crossing the positive x-axis must descend as x increases, hence a negative gradient. If we test other scenarios to contrast: - If the y-intercept were positive and the x-intercept negative, then -c/m > 0 would not hold; instead, the slope could be positive, reflecting a line that rises from left to right and crosses the x-axis on the negative side. - If either intercept were zero or negative, the sign of the slope could differ, confirming that the positivity of both intercepts is what enforces a negative slope. In summary, with c > 0 and x_int > 0, the equation y = mx + c forces m < 0, making the gradient negative. This outcome is a direct consequence of how intercepts relate to the slope via the x-intercept formula x_int = -c/m.

FMT002 Week 1 Quiz |LA004

In linear equations, the gradient is negative when the x and y intercepts are positive

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