Questions: The equation of the line that goes through the point (2,7) and is perpendicular to the line 5x+2y=3 can be written in the form y=mx+b where m is: and where b is:

The equation of the line that goes through the point (2,7) and is perpendicular to the line 5x+2y=3 can be written in the form y=mx+b
where m is:
and where b is:

Transcript text: The equation of the line that goes through the point $(2,7)$ and is perpendicular to the line $5 x+2 y=3$ can be written in the form $y=m x+b$ where $m$ is: $\square$ and where $b$ is: $\square$

Solution

Solution Steps

To find the equation of the line that is perpendicular to a given line and passes through a specific point, we first need to determine the slope of the given line. The slope of the line perpendicular to it will be the negative reciprocal of this slope. Then, using the point-slope form of a line equation, we can find the y-intercept of the new line.

Step 1: Determine the Slope of the Given Line

The equation of the given line is $ 5x + 2y = 3 $. To find its slope, we can rearrange it into slope-intercept form $ y = mx + b $. This gives us a slope $ m_{\text{given}} = -\frac{5}{2} = -2.5 $.

Step 2: Find the Slope of the Perpendicular Line

The slope of the line that is perpendicular to the given line is the negative reciprocal of $ m_{\text{given}} $. Thus, we have: \[ m_{\text{perpendicular}} = -\frac{1}{m_{\text{given}}} = -\frac{1}{-\frac{5}{2}} = \frac{2}{5} = 0.4 \]

Step 3: Use the Point-Slope Form to Find the y-Intercept

Using the point $ (2, 7) $ and the slope $ m_{\text{perpendicular}} = 0.4 $, we can apply the point-slope form of the line equation: \[ y - y_1 = m(x - x_1) \] Substituting in the values, we get: \[ y - 7 = 0.4(x - 2) \] To find the y-intercept $ b $, we can rearrange this equation to the form $ y = mx + b $ and find $ b $ when $ x = 0 $: \[ y = 0.4x + 6.2 \] Thus, the y-intercept is $ b = 6.2 $.