Answer to Question #134433 in Algebra for Alexandria Allen

Question

write the equation that has the listed conditions

Contains a y-intercept of 6 and has a slope of 10

And

Contains a y-intercept of -11 and is perpendicular to 5

Answer 1

$(1) \hspace{0.1cm}y = mx + c \hspace{0.1cm} \textsf{is the equation of a}\\\textsf{straight line in the gradient(slope) - intercept form}\\ \textsf{y-intercept of}\hspace{0.1cm}6\hspace{0.1cm}\textsf{means at}\hspace{0.1cm} x = 0, y = 6. \\ y = mx + c, m = 6 \hspace{0.1cm}(is \hspace{0.1cm}given)\\ \textsf{at}\hspace{0.1cm} x = 0, y = 6 \\ \Rightarrow y = 6 = m(0) + c, c = 6.\\ \therefore y = 10x + 6\hspace{0.1cm} \textsf{is the required equation}\\ (2) \hspace{0.1cm}\textsf{Similarly,}\hspace{0.1cm} c = -11\\ \textsf{But, since the line is perpendicular}\\\textsf{to a line whose gradient is 5, therefore,}\\\textsf{we must apply the condition of perpendicularity.}\\\textsf{That is, the product of the two}\\\textsf{gradients is equal to}\hspace{0.1cm} -1.\\ \therefore m = -\frac{1}{5}\\\Rightarrow y = -\frac{1}{5}x - 11$