How to Find the Equation of a Line Parallel to $y=\frac{2}{3}x-9$ Passing Through (12,15)

Question

Line K passes through the point (12,15) and is parallel to the line $y = \frac{2}{3}x - 9$. Work out the equation of line K. Give your answer in the form $y = mx + c$.

Accepted Answer

$y = \frac{2}{3}x + 7$

Answer

Step 1: Identify the slope of the parallel line. For a line in the form $y=mx+c$, $m$ is the slope. The given line is $y = \frac{2}{3}x - 9$, so its slope is $\frac{2}{3}$. Parallel lines have equal slopes, so line K also has a slope $m = \frac{2}{3}$. Step 2: Substitute the known slope and the coordinates of the point (12,15) into the slope-intercept form $y=mx+c$. Replace $m$ with $\frac{2}{3}$, $x$ with 12, and $y$ with 15: $15 = \frac{2}{3}(12) + c$. Step 3: Calculate the value of $c$. First compute $\frac{2}{3}(12) = 8$, so the equation becomes $15 = 8 + c$. Subtract 8 from both sides: $c = 15 - 8 = 7$. Step 4: Write the final equation of line K by substituting $m = \frac{2}{3}$ and $c = 7$ into $y=mx+c$, resulting in $y = \frac{2}{3}x + 7$.