Questions: Problem Find the equation of the line passing through the point (6,3) that is perpendicular to the line 4 x-5 y=-10. Enter your answers below. Use a forward slash (i.e. "/") for fractions (e.g. 1 / 2 for 1/2 ). Solution Step 1: Find the slope of the line 4 x-5 y=-10. Use a forward slash (i.e. "/") for all fractions (e.g. 1 / 2 for 1/2 ). m= What would the perpendicular slope be? m= Step 2: Use the slope to find the y-intercept of the perpendicular line. b= Step 3: Write the equation of the line that passes through the point (6,3) that is perpendicular to the line 4 x-5 y=-10 y= x+

Problem Find the equation of the line passing through the point (6,3) that is perpendicular to the line 4 x-5 y=-10. Enter your answers below. Use a forward slash (i.e. "/") for fractions (e.g. 1 / 2 for 1/2 ). Solution Step 1: Find the slope of the line 4 x-5 y=-10. Use a forward slash (i.e. "/") for all fractions (e.g. 1 / 2 for 1/2 ). m= What would the perpendicular slope be? m= Step 2: Use the slope to find the y-intercept of the perpendicular line. b= Step 3: Write the equation of the line that passes through the point (6,3) that is perpendicular to the line 4 x-5 y=-10 y= x+
Transcript text: Problem Find the equation of the line passing through the point $(6,3)$ that is perpendicular to the line $4 x-5 y=-10$. Enter your answers below. Use a forward slash (i.e. "/") for fractions (e.g. $1 / 2$ for $\frac{1}{2}$ ). Solution Step 1: Find the slope of the line $4 x-5 y=-10$. Use a forward slash (i.e. "/") for all fractions (e.g. $1 / 2$ for $\frac{1}{2}$ ). \[ m= \] What would the perpendicular slope be? \[ m= \] Step 2: Use the slope to find the $y$-intercept of the perpendicular line. \[ b= \] Step 3: Write the equation of the line that passes through the point $(6,3)$ that is perpendicular to the line $4 x-5 y=-10$ \[ y=\square x+\square \]
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Solution

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Solution Steps

Solution Approach
  1. First, convert the given line equation 4x−5y=−104x - 5y = -10 into the slope-intercept form y=mx+by = mx + b to find its slope.
  2. The slope of the line perpendicular to the given line is the negative reciprocal of the original slope.
  3. Use the point-slope form of the line equation with the perpendicular slope and the given point (6,3)(6, 3) to find the equation of the new line.
Step 1: Find the Slope of the Given Line

The equation of the line is given by 4x−5y=−104x - 5y = -10. To find the slope, we can rearrange this equation into the slope-intercept form y=mx+by = mx + b. Solving for yy, we get: y=45x+2 y = \frac{4}{5}x + 2 Thus, the slope of the original line is m=45m = \frac{4}{5}.

Step 2: Find the Perpendicular Slope

The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, the perpendicular slope is: mperpendicular=−54 m_{\text{perpendicular}} = -\frac{5}{4}

Step 3: Find the Equation of the Perpendicular Line

Using the point-slope form of the line equation, which is y−y1=m(x−x1)y - y_1 = m(x - x_1), where (x1,y1)=(6,3)(x_1, y_1) = (6, 3) and m=−54m = -\frac{5}{4}, we can write: y−3=−54(x−6) y - 3 = -\frac{5}{4}(x - 6) Expanding this, we find: y−3=−54x+304 y - 3 = -\frac{5}{4}x + \frac{30}{4} Simplifying gives: y=−54x+212 y = -\frac{5}{4}x + \frac{21}{2}

Final Answer

The equation of the line that passes through the point (6,3)(6, 3) and is perpendicular to the line 4x−5y=−104x - 5y = -10 is: y=−54x+212 \boxed{y = -\frac{5}{4}x + \frac{21}{2}}

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