Question

Which equation represents the line passing through the point (-5, 12) and perpendicular to the line y=2/3x +14

Answer 1

Step 1:

In this question, we are given the following:

Which equation represents the line passing through the point (-5, 12) and perpendicular to the line y=2/3x +14 ?

Step 2:

The details of the solution are as follows:

`\begin{gathered} Given\text{ that the equation is given as:} \\ \text{y = }(2x)/(3)\text{ + 14} \\ compari\text{ng this with:} \\ \text{y = mx + c , we have that:} \\ \text{m}_1\text{ =}(2)/(3) \end{gathered}`
`\begin{gathered} Now,\text{ for perpendicular lines, we have that:} \\ m_1m_2\text{ = -1} \\ Then,\text{ we have that:} \\ m_2\text{ = }(-1)/(m_1)\text{ , where m}_1\text{ = }(2)/(3) \\ Then, \\ m_2\text{ = -1 }/\text{ }(2)/(3) \\ m_2=\text{ -1 x }(3)/(2) \\ m_2\text{ = }(-3)/(2) \end{gathered}`

Step 3:

`\begin{gathered} Given\text{ that:} \\ m_2=(-3)/(2) \\ (\text{ x}_(1,)y_1)\text{ = \lparen -5, 12 \rparen} \end{gathered}`
`\begin{gathered} Using\text{ the new equation:} \\ y\text{ -y}_1\text{ = m \lparen x - x}_1) \\ y\text{ - \lparen12\rparen = }(-3)/(2)\text{ \lparen x --5\rparen} \\ \text{y -12 = }(-3)/(2)(\text{ x+ 5 \rparen} \\ Multiply\text{ through by 2, we have that:} \\ 2y\text{ - 24 = - 3 \lparen x + 5\rparen} \\ 2y\text{ - 24 = -3x - 15} \\ Re-arranging,\text{ we have that:} \\ 3x+\text{ 2y - 24 + 15 = 0} \\ 3x+\text{ 2y -9 = 0} \end{gathered}`

CONCLUSION:

The final answer is:

`3x\text{ + 2y -9 = 0}`